Molecular Thermodynamcs of Fluid Phase Equilibria: Solutions Manual

Solutions Manual to Accompany

Molecular Thermodynamics of FluidPhase Equilibria Third Edition

John M. Prausnitz Rüdiger N. Lichtenthaler Edmundo Gomes de Azevedo

Prentice Hall PTR Upper Saddle River, New Jersey 07458 http://www.phptr.com

” 2000 Prentice Hall PTR Prentice Hall, Inc. Upper Saddle River, NJ 07458

Electronic Typesetting: Edmundo Gomes de Azevedo

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Printed in the United States of America 10 9 8 7 6 5 4 3 2

ISBN 0-13-018388-1

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C O N T E N T S

Preface

i

Solutions to Problems Chapter 2

1

Solutions to Problems Chapter 3

17

Solutions to Problems Chapter 4

29

Solutions to Problems Chapter 5

49

Solutions to Problems Chapter 6

81

Solutions to Problems Chapter 7

107

Solutions to Problems Chapter 8

121

Solutions to Problems Chapter 9

133

Solutions to Problems Chapter 10

153

Solutions to Problems Chapter 11

165

Solutions to Problems Chapter 12

177

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

b

From problem statement, we want to find wP / wT Using the product-rule,

FG wP IJ H wT K

 v

g

v

.

FG wv IJ FG wP IJ H wT K H w v K P

By definition,

FG IJ H K

1 wv v wT

DP and

NT



T

P

FG IJ H K

1 wv v wP

2

T

Then,

FG wP IJ H wT K

v

DP NT

. u 10 5 18 5.32 u 10 6

33.8 bar D C -1

Integrating the above equation and assuming DPand NT constant over the temperature range, we obtain 'P

DP 'T NT

'P

33.8 bar

For 'T = 1qC, we get

1

2

2.

Solutions Manual

Given the equation of state, P

FG V  bIJ Hn K

RT

we find:

FG wS IJ H wV K FG wS IJ H wP K FG wU IJ H wV K FG wU IJ H wP K FG wH IJ H wP K

T

T

T

For an isothermal change,

z FGH V2

'S

V1

V

T

T

FG wP IJ nR H wT K V  nb F wV I  nR G J H wT K P F wP I TG J  P 0 H wT K FG wU IJ FG wV IJ 0 H wV K H wP K F wV IJ  V nb T G H wT K P

V

T

wP wT

IJ K

dV V

T

V  nb nR ln 2 V1  nb

P  nR ln 1 P2

z

LMFG  wU IJ FG wV IJ OP dP 0 MNH wV K H wP K PQ LMT FG wV IJ  V OP dP nbbP  P g MN H wT K PQ FP I 'H  T'S nbb P  P g  nRT lnG J HP K FP I 'A 'U  T'S  nRT lnG J HP K

'U

'H

z

P1

T

T

P2

P1

'G

P2

2

1

P

2

1

1

2

1 2

3

3.

Solutions Manual

This entropy calculation corresponds to a series of steps as follows: s3 (vapor, T = 298.15 K P = 1 bar)

s1 (saturated liq. T = 298.15 K P = 0.03168 bar)

's1o 2

' vap s

s2 (saturated vapor, T = 298.15 K P = 0.03168 bar)

s3

's1o 2  's2o 3  s1

' vap h

(2436) u (18.015) 298.15

T 's2o 3

Because v

z LMMN FGH P3



P2

IJ K

147.19 J K 1 mol 1

OP PQ

wv dP wT P

RT (ideal gas), P 's2 o 3

§P ·  R ln ¨ 3 ¸ © P2 ¹ § 1.0 · (8.31451) u ln ¨ ¸ © 0.03168 ¹ 28.70 J K 1 mol 1

s3

s0 (H2 O, vapor) 147.19  28.70  69.96 188.45 J K1 mol 1

4

4.

Solutions Manual

Because D

RT  v, P

P

RT Dv

RT 2  3 / v2  v

or P

§ wP · ¨ wv ¸ © ¹T

RTv 2 2v 2  3  v 3



RT v (v 3  6) (2v 2  3  v 3 )2

2.3 L mol 1 , T = 373.15 K, R = 0.0831451 bar L K-1 mol-1, and molar mass is 100 g

As v mol-1,

FG wP IJ H wv K w2

 g c kv 2

FG H

 1

FG wP IJ H wv K

kg m N

s2

3.3245 bar L-1 mol

3.3245 u 10 8 Pa m3 mol

T

T

IJ u (1.4) u FG 1 K H 100 u 10

IJ F K GH

mol m3 u 2.3 u 10 3 3 kg mol

24,621 m2 s 2

w 157 m s 1

5.

Assume a three-step process:

I u FG 3.3245 u 10 JK H 2

8

N mol m2

m3

IJ K

5

Solutions Manual

Isothermal expansion to v = f (ideal gas state)

(1)

(2)

v= V

(2)

Isochoric (v is constant) cooling to T2

(3)

(1)

(T2 ,v2 )

Isothermal compression to v2

(3)

(T1 ,v1 ) T

For an isentropic process, 's

's1  's2  's3

0

Because s = s(v, T),

ds

FG ws IJ H wv K

dv  T

FG ws IJ H wT K

dT v

or

FG wP IJ H wT K

ds

dv  v

cv dT T

by using the relations

FG wS IJ FG wP IJ H wv K H wT K T

FG ws IJ H wT K

FG IJ H K

1 wu T wT

v

(Maxwell relation) v

v

cv T

then, 's

³

v f § wP ·

v1

v 2 § wP · T2 c 0 v dT   d v ¨ wT ¸ ¨ ¸ dv T1 T v f © wT ¹v © ¹v

³

³

Using van der Waals’ equation of state,

P

RT a  v  b v2

FG wP IJ H wT K Thus,

v

R vb

6

Solutions Manual

T2 c 0 §v b· v dT R ln ¨ 2  ¸ v b T  T © 1 ¹ 1

³

's

To simplify, assume

cv0

c 0p  R RT2 P2

v2

Then, RT2 b P2 ln v1  b

§ c0p  R · § T · ¨ ¸ ln ¨ 1 ¸ ¨ R ¸ © T2 ¹ © ¹

ª (82.0578) u (T2 )  45 º ln « » 600  45 ¬ ¼

T2

(3.029) u ln

203 K

6.

P

Because

b2 v2

RT a  v  b v2

623.15 T2

RT v

LM O a P 1 MM b  RTv PP MN1  v PQ

 1,

FG1  b IJ H vK

1

1

b b2  " v v2

Thus,

P

LM FG MN H

IJ K

OP PQ

RT a 1 b2  " 1 b  RT v v 2 v

or

Pv RT

FG H

1 b 

IJ K

a 1 b2  " RT v v 2

7

Solutions Manual

Because

Pv RT

z

1

B C  " v v2

the second virial coefficient for van der Waals equation is given by b

B

7.

a RT

Starting with

FG wu IJ H wP K

du Tds  Pdv T T

FG ws IJ H wP K

P T

FG wv IJ H wT K

T

FG wv IJ H wP K

P P

T

FG wv IJ H wP K

As v

RT B P

RT a b P T2

FG wv IJ H wT K FG wv IJ H wT K

R 2a  P T3

P

Then,

FG wu IJ H wP K 'u

 T

 T S

RT P2

2a T2

§ 2a ·

³0  ¨© W2 ¸¹ dP 'u



2a S W2

T

8

8.

Solutions Manual

The equation

FG P  n IJ bv  mg H v2T 1/2 K

RT

can be rewritten as

(PT 1/ 2 )v 3  (PmT 1/ 2 )v 2  nv  nm

FG H

v3  m 

FG H

IJ K

RT 3/2v 2

IJ K

RT 2 n nm v  v P PT 1/ 2 PT 1/ 2

0

(1)

At the critical point, there are three equal roots for v = vc, or, equivalently,

FG wP IJ H wv K

bv  v g

3

c

F w PI GH wv JK 2

0

2

T Tc

T Tc

v 3  3v c v 2  3v c 2v  v c 3

0

(2)

Comparing Eqs. (1) and (2) at the critical point, m

RTc Pc

3v c

(3)

n

3v 2c

(4)

v c3

(5)

PcTc1/ 2 nm PcTc1/ 2

From Eqs. (3), (4), and (5) we obtain m

vc

n

3RTc 8Pc

3v 2c PcTc1/ 2

The equation of state may be rewritten:

vc 3 or

(6)

m

RTc 8Pc

27 R 2Tc5/ 2 64 Pc

9

Solutions Manual

P

RT v

z

Pv RT

F I GG 1  n JJ GH 1  mv RT 3/2v JK

or 1 m 1 v



n RT 3/2 v

From critical data, 0.0428 L mol 1

m

63.78 bar (L mol 1 )2 K1/2

n

At 100qC and at v = (6.948)u(44)/1000 = 0.3057 L mol-1, z = 0.815

This value of z gives P = 82.7 bar. Tables of Din for carbon dioxide at 100qC and v = 6.948 cm3 g-1, give P = 81.1 bar or z = 0.799.

9.

We want to find the molar internal energy u(T , v) based on a reference state chosen so that u(T0 , v o f)

0

Then, u(T , v)

u(T , v)  u(T0 , v o f) u(T , v)  u(T , v o f)  u(T , v o f)  u(T0 , v o f )

lim

z

v

vof v f

FG wu IJ H wv K

z FGH T

IJ K

wu dT T v v w vof T0 f

dv  lim T

Schematically we have: v =v

1

2

3

Ideal gas Ref. state Ideal gas (T0 , v

)

Intermediate state Ideal gas (T, v )

State of interest Real gas (T, v)

(1)

10

Solutions Manual

In Eq. (1) we are taking 1 mol of gas from the reference state 1 to the state of interest 3 through an intermediate state 2, characterized by temperature T and volume v o f, in a two-step process consisting of an isochoric step and an isothermal step. In the step 1 o 2 the gas is infinitely rarified, and hence exhibits ideal gas behavior. Then, the second integral in Eq. (1) gives: T

§ wu · dT ¨ ¸ v of T0 © wv ¹ v vf

³

lim

because for an ideal gas c 0p  cv0

³

T

T0

cv0 dT

³

T

T0

(c0p  R) dT

(c0p  R)(T  T0 )

(2)

R and because, by the problem statement, the heat capacity at

constant pressure of the gas is temperature independent. We have now to calculate the first integral in Eq. (1). To make this calculation, we first transform the derivative involved in the integral to one expressed in terms of volumetric properties. By the fundamental equation for internal energy (see Table 2-1 of the text),

FG wu IJ T FG wP IJ  P H wv K T H wv K T

(3)

Making the derivative using the equation of state give we obtain

FG wu IJ H wv K T

RT RT a   v  b v  b v(v  b)

a v(v  b)

(4)

Then, v

§ wu · ¨ ¸ dv v f © wv ¹

³

a v ª 1 1º dv  b v f ¬« v  b v ¼»

³

a ª § vb · v º « ln ¨ » ¸  ln b ¬ © vf  b ¹ vf ¼

(5) a § v  b vf · ln ¨ ¸ b © v vf  b ¹

and

z FGH v

lim

vof v f

wu wv

IJ dv KT

FG H

vb a ln v b

IJ K

(6)

Combining Eqs. (1), (2) and (6) we obtain the desired expression for the molar internal energy, u(T , v)

FG H

a vb (c 0p  R)(T  T0 )  ln b v

IJ K

11

Solutions Manual

10. ln J w

A(1  xw )2

J w o 1 as xw o 1

such that

Using Gibbs-Duhem equation, xw d ln J w  xs d ln J s

or, because dx w

 dx s ( xw  xs

0

1) , xw

d ln J w dx w

d ln J w dx w

xs

d ln J s dx s

2 A(1  x w )(1)

2 A(1  xw )

Then, 2 Axs (1  xs ) dxs xs

d ln J s

ln J s

³0

d ln J s

2 A

³0

(1  xs ) dxs

§ x2 · 2 A ¨ x s  s ¸ ¨ 2 ¸¹ ©

ln J s

ln J s

11.

xs

2 A(1  xs ) dxs

A( x w2  1)

Henry’s law for component 1, at constant temperature, is f1

(for 0  x1  a)

k1 x1

where k1 is Henry’s constant. For a liquid phase in equilibrium with its vapor, fi L gas law, fiV

yi P.

Henry’s law can then be written: y1P

k1 x1

Taking logarithms this becomes ln( y1P)

ln k1  ln x1

Differentiation at constant temperature gives

fiV . If the vapor phase obeys ideal-

12

Solutions Manual

d ln( y1P) dx1

d ln x1 dx1

1 x1

Using the Gibbs-Duhem equation x1

d ln P1 d ln P2  x2 dx1 dx1

0

gives 1  x2

or, because dx 2

d ln( y2 P) dx1

0

 dx1 , x2

d ln( y2 P) dx2

1

or, d ln(y2 P)

d ln x2

Integration gives ln( y2 P)

ln x2  ln C

where lnC is the constant of integration. For x2

1 , y2

1 , and P

P2s . This gives C

P2s and we may write

ln x2  ln P2s

ln( y2 P)

ln( x2 P2s )

or y2 P

P2

x 2 P2s

[for (1  a)  x 2  1 ]

which is Raoult’s law for component 2.

12.

Starting from dgi

RTd ln fi ,

'gi

P* o P

f (at P) RT ln i fi (at P* )

(P* is a low pressure where gas i is ideal)

From the Steam Tables we obtain 'h and 's at T and P to calculate 'g from

'g

'h  T 's

13

Solutions Manual

Choose P * = 1 bar.

'h 's

196 J g 1

h70 bar  h1 bar

s70 bar  s1 bar

2.215 J g-1 D C 1

Then, at 320qC, 'g

1117.8 J g 1

20137 J mol 1

Thus,

b

ln fi 70 bar, 320qC

g

b

20137 8.31451 u 59315 .

g b

g

4.08

or f = 59.1 bar

13.

The virial equation for a van der Waals gas can be written (as shown in Problem 6) RT a b P RT

v

(1)

At the Boyle temperature, B

b

a RT

b

a RT

0

or

The Boyle temperature then, is given by

TB

a bR

The Joule-Thomson coefficient is

P or

FG wT IJ H wP K H

(2)

14

Solutions Manual

P

FG wT IJ FG wT IJ  H wH K H wP K FG wP IJ H wH K

P

FG wH IJ H wP K 

T

cp

H

T

Because

FG wH IJ H wP K T and because cp is never zero, when P

v  T

0,

FG wH IJ H wP K T

FG wv IJ H wT K P 0.

Substitution in Eq. (1) gives

§ wH · ¨ wP ¸ © ¹T



RT a § RT a · b ¨  P RT © P RT ¸¹

b 

2a RT

The inversion temperature is TJT

2a Rb

TJT

2TB

f1G

f1L

Comparison with Eq. (2) gives

14.

At equilibrium,

where subscript 1 stands for the solute. At constant pressure, a change in temperature may be represented by

F d ln f I GH dT JK G 1

dT P

F d ln f I GH dT JK L 1

dT

(1)

P

Since the solvent is nonvolatile, f1G (at constant pressure) depends only on T (gas composition does not change.) However, f 1L (at constant pressure) depends on T and x1 (or ln x1 ):

15

Solutions Manual

F d ln f1 I GH dT JK L

F w ln f1 I GH wT JK L

dT P

F w ln f1 I GH w ln x1 JK L

dT  P, x

d ln x1

(2)

T ,P

Further,

F d ln f GH dT F d ln f GH wT

I JK I JK

G 1

L 1

h10  h1G RT 2

P

h10  h1L RT 2

P, x

(3)

(4)

where: h10 = ideal-gas enthalpy of 1;

h1G = real-gas enthalpy of 1; h1L = partial molar enthalpy of 1 in the liquid phase.

Assuming Henry’s law, f1 x1

constant

or

F w ln f1 I GH w ln x1 JK L

1

(5)

T ,P

Substituting Eqs. (2), (3), (4), and (5) into Eq. (1), we obtain d ln x1 d (1 / T )



'h1 R

From physical reasoning we expect h1G ! h1L . Therefore x1 falls with rising temperature. This is true for most cases but not always.

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

3

The Gibbs energy of a mixture can be related to the partial molar Gibbs energies by

¦ yi e gi  gio j m

g  go

(1)

i 1

Since, at constant temperature, dg

RTd ln f , we may integrate to obtain o

g  go

RT ln f mixt  RT ln f mixt

or g  go

RT ln f mixt  RT ln P

(2)

where subscript mixt stands for mixture. For a component in a solution, d gi RTd ln fi . Integration gives gi  gio

gi  gio

RT ln fio

(3)

RT ln( yi P)

Substituting Eqs. (2) and (3) into Eq. (1) gives m

ln f mixt  ln P

m

¦ yi ln fi  ¦ yi ln( yi P) i 1

ln f mixt  ln P

i 1

m

FfI

m

i 1

i

i 1

¦ yi lnGH yi JK  ¦ yi ln P

(4)

Because

17

18

Solutions Manual

m

¦ yi ln P

ln P

i 1

Fm yI iJ GH ¦ i 1 K

ln P

Eq. (4) becomes ln f mixt

Assuming the Lewis fule, fi

m

FfI

i 1

i

¦ yi lnGH yi JK

yi f pure i , Eq. (5) becomes m

¦ yi ln fpure i

ln f mixt

i 1

or m

yi – fpure i

f mixt

i=1

2.

As shown in Problem 1, ln f mixt

m

FfI

i 1

i

¦ yi lnGH yi JK

This result is rigorous. It does not assume the Lewis fugacity rule. Using fugacity coefficients, fi

M i yi P

and ln f mixt fmixt

y A ln M A  yB ln M B  ln P y

y

M AA M BB P (0.65)0.25 u (0.90)0.75 u (50)

fmixt

41.5 bar

(5)

19

Solutions Manual

3.

Pure-component saturation pressures show that water is relatively nonvolatile at 25qC. Under these conditions the mole fraction of ethane in the vapor phase (yE) is close to unity. Henry’s law applies: fE

H (T ) x E

The equilibrium condition is f EV

f EL

or yE M E P

At 1 bar, M E | 1 and H (T )

H (T ) x E

P / xE :

H (T )

1

3.03 u 10 4 bar

0.33 u 10 4

At 35 bar we must calculate ME:

z

P

z 1 dP 0 P

ln M E

Using z

1  7.63 u 10 3 P  7.22 u 10 5 P 2

we obtain ME

0.733

Because Henry’s constant H is not a strong function of pressure, xE

xE

4.

x Ethane

MEP H

fE H

(0.733) u (35) 3.03 u 10 4

8.47 u 10 4

The change in chemical potential can be written, 'P1

P1  P10

§ RT ln ¨ ¨ ©

f1 · ¸ f10 ¸¹

( f10

1 bar )

(1)

20

Solutions Manual

1

The chemical potential may be defined as : § wG0 § wG ·  ¨ ¨ ¸ ¨ w n1 © w n1 ¹T ,P,n2 ©

P1  P10

· ¸ ¸ ¹T ,n2

(2) § w A0 § wA ·  ¨ ¨ ¸ ¨ w n1 © w n1 ¹T ,V ,n2 ©

·  RT ¸ ¸ ¹T ,n2

Combining Eqs. (1) and (2): 1 § w'A ·  1 ¨ ¸ RT © w n1 ¹T ,V ,n 2

ln f1

Using total volume, V 'A RT

nT v , nT nT ln

n1  n2 ,

FG V IJ  n1 lnFG V IJ  n2 lnFG V IJ H V  n b K H n1RT K H n2 RT K T

Taking the partial derivative and substituting gives ln

f1 y1 RT vb

or

b1 vb

FG H

y1RT b1 exp vb vb

f1

IJ K

The same expression for the fugacity can be obtained with an alternative (but equivalent) derivation:

1

A0

U 0  TS 0 ;

P10

§ wG0 ¨ ¨ w n1 ©

G0

U 0  PV 0  TS 0 ;

§ wU0 ¨ ¨ w n1 ©

· ¸ ¸ ¹T ,n j z1

PV 0

nT RT

· § w S0  RT  T ¨ ¸ ¸ ¨ w n1 ¹T ,n j z1 ©

· ¸ ¸ ¹T ,n j z1

and § w A0 ¨ ¨ w n1 ©

· ¸ ¸ ¹T ,n j z1

§ wU0 ¨ ¨ w n1 ©

· § w S0  RT  T ¨ ¸ ¸ ¨ w n1 ¹T ,n j z1 ©

then P10

§ w A0 ¨ ¨ w n1 ©

·  RT ¸ ¸ ¹T ,n j z1

· ¸ ¸ ¹T ,n j z1

21

Solutions Manual

§ f · RT ln ¨ 1 ¸ ¨ f0 ¸ © 1 ¹

P1  P10

By definition, P1

FG w A IJ H wn K

( f10

1 bar)

. The Helmholtz energy change 'a can be written as

1 T ,V ,n j z1

nT 'a

A

¦ ni ai0 i

A

¦ ni (P10  RT ) i

Then, § wnT 'a · ¨ ¸ © wn1 ¹n j z1,T ,V

§ wA ·  P10  RT ¨ ¸ n w © 1 ¹n j z1,T ,V P1  P10  RT

and P1  P10 RT

ª w(nT 'a / RT ) º  1 « » wn1 ¬ ¼ n j z1,T ,V

Using the equation for 'a , ª w(nT 'a / RT ) º « » wn1 ¬ ¼ n j z1,T ,V

ln

n b V V  ln  1 T 1 V  nT b V  nT b n1RT

or ln f1

§ P  P0 ln ¨ 1 1 ¨ © RT

ln

· ¸ ¸ ¹

ª w(nT 'a / RT ) º 1 « » wn1 ¬ ¼ n j z1,T ,V

n1 RT n b  T 1 V  nT b V  nT b § n b · n1RT exp ¨ T 1 ¸ V  nT b © V  nT b ¹

f1

Hence, f1

y1RT § b · exp ¨ 1 ¸ vb ©vb¹

22

Solutions Manual

5. a)

Starting with Eq. (3-51): For a pure component (ni

Pi

G ; ni

Pi0

nT ) :

Gi0 ni

Because

G U  PV  TS Pi0

Pi0  Tsi0  RT

(1)

P RT · V PV  Pi0  Tsi0  ¨  ¸ dV  RT ln n V n RT ni V © i i ¹

(2)

From Eq. (3-52), Pi

³

f§

From Eqs. (1) and (2), Pi  Pi0

f§

PV  ni RT P RT · V  ¨  ¸ dV  RT ln V ¹ ni RT ni V © ni

³

But, RT ln fi

Pi  Pi0

and V ni RT

zi P

Substitution gives § f· RT ln ¨ ¸ © P ¹i

b)

Starting with Eq. (3-53): For a pure component, yi

f§

P RT · ¨  ¸ dV  RT ln zi  RT ( zi  1) V ¹ V © ni

³

1. To use Eq. (3-53), we must calculate

FG wP IJ H wni K T ,V ,pure component i

Pressure P is a function of T, V, and ni and

23

Solutions Manual

§ wP · § wni ¨ ¸ ¨ © wni ¹V © wV

· § wV · ¸ ¨ wP ¸ ¹ni ¹P ©

1

FG wP IJ FG wV IJ FG wP IJ H wni K V H wni K P H wV K ni FG IJ OP H K PQ

LM MN

P P V wP   ni ni ni wV n i

But,

FG IJ H K

P V wP  ni ni wV n i

LM N

OP Q

1 w (PV ) ni wV n i

Then, f§

³

f

P 1 ni RT dV  d ( PV ) ni PV V ni

wP · ¨ ¸ w V © ni ¹T ,V ,n j z1

³

³

f

P PV dV  RT  ni V ni

³

f

P dV  RT ( z  1) n V i

³ Now Eq. (3-54) follows directly.

6.

The solubility of water in oil is described by f1

H (T ) x1

Henry’s constant can be evaluated at 1 bar where f1 Then, H (T )

f1 x1

1 35 u 10 4

286 bar

1 bar .

(t

140 D C)

To obtain f1 at 410 bar and 140qC, use the Steam Tables (e.g., Keenen and Keyes). Alternatively, get f at saturation (3.615 bar) and use the Poynting factor to correct to 410 bar. At 140qC,

24

Solutions Manual

RT ln f1

'g1o410 bar

'h1o410 bar  T 's1o410 bar

282 J g 1

RT ln f1

ln f1

(from Steam Tables)

(282) u (18) (8.31451) u (413)

1.48

Then, f1 = 4.4 bar at 410 bar and 140qC and x1

f1 H (T )

4.4 286

0.0154

7. To = 300 K

Tf = 300 K

Po = ?

Pf = 1 bar

Applying an energy balance to this process, 'h

hf  ho

0

This may be analyzed on a P-T plot. Assume 1 mole of gas passing through the valve. P

To ,Po

Tf ,Pf

I

III II T

A three-step process applies: I. Isothermal expansion to the ideal-gas state. II. Isobaric cooling of the ideal gas. III. Compression to the final pressure. For this process,

25

Solutions Manual

'h

'hI  'hII  'hIII

0

Since h = h(T,P),

FG wh IJ d P  FG wh IJ dT H w P K T H wT K P

dh

'h

T f Pf

³

dh

To Po

0

ª § wv · º «v  T ¨ ¸ » dP  Po ¬« © wT ¹ P ¼»T

³

o

³

Tf

To

c0p d T 

³

Pf

0

ª § wv · º «v  T ¨ ¸ » dP © wT ¹ P ¼»T ¬« f

But, RT 10 5  50  P T

v Then,

FG wv IJ H wT K P

R 10 5  P T2

and c 0p

'h

0

F 50  2 u 10 GH T o

5

yA c 0p,A  yBc 0p,B

I b P g  y e JK

0 A c p,A

o

jb

g FGH

 yBc 0p,B Tf  To  50 

I JK

2 u 10 5 Pf Tf

Substitution gives (616.7) u Po

(33.5 J mol 1 K 1 ) u (10 bar cm 3 J 1 ) u (200  300 K)  (950) u (1 bar )

Po = 55.9 bar

8.

From the Gibbs-Helmholtz equation:

w

FG g IJ HTK wT



h T2

26

Solutions Manual

or, alternately,

FG 'g IJ HTK F 1I wG J HTK

w

'h

h real  h ideal

(1)

Because, at constant temperature,

b g

d 'g

RTd ln

FG f IJ H PK

(2)

we may substitute Eq. (2) into Eq. (1) to obtain

FG f IJ H PK F 1I wG J HTK

w ln R

'h

From the empirical relation given,

ln

f P

0.067P 

30.7 0.416 P 2 P  0.0012P 2  T T

FG f IJ H PK F 1I wG J HTK

w ln R

30.7P  0.416 P 2

'h R

At P = 30 bar,

'h = (8.31451)u[(-30.7)u(30) + (0.416)u(30)2] 'h = -4545 J mol-1



9.

Consider mixing as a three-step process: (I) Expand isothermally to ideal-gas state. (II) Mix ideal gas. (III) Compress mixture isothermally. Starting with

du

LM FG wP IJ  POP dv MN H wT K v PQ

cv dT  T

27

Solutions Manual

FG wu IJ H wv K T Because, P

FG w P IJ H wT K

T

P v

RT a  , v  b v2

FG wu IJ H wv K T

a v2

Integration of this equation to the ideal-gas state (v 'u

z

f

v

a

v

2

dv

f) gives

a v

Therefore, 'uI

x1a1 x 2 a2  v1 v2

5914 J mol 1

[(1 bar)u(1 cm3) | 0.1 Joule]

'uII

'uIII



0

(because is the mixing of ideal gases)

amixt v mixt

[ x12 a11  2 x1x2 a11a22 u (1  0.1)  x22 a22 ] x1v1  x2 v 2

5550 J mol 1

' mix h

' mix u

'uI  'uII  'uIII

364 J mol 1

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

4

At 25 Å we can neglect repulsive forces. The attractive forces are London forces and induced dipolar forces; we neglect (small) quadrupolar forces. (There are no dipole-dipole forces since N2 is nonpolar.) Let 1 stand for N2 and 2 stand for NH3. London force: *12

F12



 d* dr

3 D 1D 2 I1I 2 2 r 6 I1  I 2 

9D 1D 2 I1I 2 r 7 I1  I 2

Since, D1

17.6 u 10 25 cm3

D2

22.6 u 10 25 cm3

I1 = 15.5 eV = 2.48 u 10 18 N m . u 10 18 N m I 2 = 11.5 eV = 184

then London F12

62.0 u 10 18 N

Induced dipole force:

29

30

Solutions Manual

*12



F12



D1P 22 r6

6D1P 22 r7

1 D 1 u 10 18 (erg cm 3 )1/ 2 1.47 D 1.47 u 10 18 (erg cm 3 )1/ 2

P2

ind F12

3.8 u 10 18 (erg cm 3 )1/ 2

Neglecting all forces due to quadrupoles (and higher poles),

2.

F tot

F London  F ind

F tot

65.8 u 10 18 N

From the Lennard-Jones model: V6

Attractive potential = 4H

Attractive force = 

d* dr

r6

24 H

* V6 r7

Assume force of form, Force = 

d* dr

FG H IJ V H kK r

(constant )

6

7

Using corresponding states:

FG H JI H kK

(constant ) u Tc D Tc

where D and E are universal constants.

V6

(constant u v c ) 2

E v c2

31

Solutions Manual

(V CH 4 ) 6 (rCH 4 ) 7

(constant ) (H / k ) CH 4 (constant ) (H / k ) B

force CH 4 force substance B

(V B ) 6 (rB ) 7

E (v cCH )2 D(TcCH )

8

2 u 10 force substance B

4

D (TcB )

4

u

(1u 10

7

cm)7

E(v cB ) 2 (2 u 10 7 cm)7

Force substance B = 4 u 10 10 dyne Force = 4 u 10 15 N

3. 8 u 10 16 erg

*AA

By the molecular theory of corresponding states: *ii Hi

*BB *AA Since H / k

FG H HH

B A

IJ ML f b2g PO K MN f b2g PQ

§ r · f¨ ¸ © Vi ¹

HB HA

(r

2V)

( f is a universal function)

0.77Tc (taking the generalized function f as the Lennard-Jones (12-6) poten-

tial), * BB

* AA

TcB TcA

(8 u 10 16 erg) u (180 K/ 120 K) *BB

12.0 u 10 16 erg 12.0 u 10 23 J

32

Solutions Manual

4.

For dipole-dipole interaction: *(dd )

PiP j (4 SH or 3 )

f (T i , T j , I)

with

f (Ti , T j , I)

2 cos Ti cos T j  sin Ti sin T j cos(Ii  I j )

For the relative orientation: j

i

i Ti Tj

j

Ti = 0º

Ii = 0º

Tj = 180º

Ij = 0º

Ij

PiP j

*(dd )

(4 SH o

r 3)

u [2 u 1 u (1)]

For Pi = Pj = 1.08 D = 3.603u10-30 C m and r = 0.5u10-9 m, ( 2) u (3.603 u 10

* ( dd )

(4 S) u (8.8542 u 10

-1.87 × 10

-21

12

30 2

)

) u (0.5 u 10

9 3

)

J

For the relative orientation: j

i

Ti = 0º

Tj = 90º

Ii = 0º

Ij = 0º

For the dipole-induced dipole interaction: *(ddi )



P i2D j 2(4 SH o ) 2 r 6

For the relative orientation:

(3 cos 2 T i  1) 

P 2j D i 2(4 SH o ) 2 r 6

(3 cos 2 T j  1)

2P i P j (4 SH or 3 )

33

Solutions Manual

j

i

*(ddi )

Ti = 0º

Tj = 90º

ª (3.603 u 1030 )2 u (2.60 u 1030 ) º u (4) » 2u « 10 9 6 ¬« 2 u (1.1124 u 10 ) u (0.5 u 10 ) ¼» 7.77 u 1023 J | 8.0 u 1023 J

For the relative orientation: j

i

Ti = 0º

*(ddi )



Tj = 90º

(3.603 u 1030 )2 u (2.60 u 1030 )

u (4  1)

2 u (1.1124 u 1010 ) u (0.5 u109 )6

4.85 u 1023 J | 4.8 u 1023 J

5.

The energy required to remove the molecule from the solution is 1

E

Hr

3.5

a3 a

FG Hr  1 IJ P 2 H 2Hr  1K

3.0 u 10-8 cm

P

2D

(See, for example, C. J. E. Böttcher, 1952, The Theory of Electric Polarization, Elsevier) E = 4.61u10-21 J/molecule = 2777 J mol-1

6. a) The critical temperatures and critical volumes of N2 and CO are very similar, more similar than those for N2 and argon (see Table J-4 of App. J). Therefore, we expect N2/CO mixtures to follow Amagat’s law more closely than N2/Ar mixtures.

34

Solutions Manual

b) Using a harmonic oscillator model for CO, F = Kx, where F is the force, x is the displacement (vibration) of nuclei from equilibrium position and K is the force constant. This constant may be measured by relating it to characteristic frequency X through: X

1 2S

b

K mC  mO mC m O

g

where mC and mO are, respectively, the masses of carbon and oxygen atoms. Infrared spectrum will show strong absorption at X. Argon has only translational degrees of freedom while CO has, in addition, rotational and vibrational degrees of freedom. Therefore, the specific heat of CO is larger than that of argon.

7.

Electron affinity is the energy released when an electron is added to a neutral atom (or molecule). Ionization potential is the energy required to remove an electron from a neutral atom (or molecule). Lewis acid = electron acceptor (high electron affinity). Lewis base = electron donor (low ionization potential). Aromatics are better Lewis bases than paraffin. To extract aromatics from paraffins we want a good Lewis acid. SO2 is a better a Lewis acid than ammonia.

8.

From Debye’s equation: ª H 1º v« r » ¬ Hr  2 ¼  

Total polarization

4 S N AD 3  



§ P2 · 4 SNA ¨ ¸ ¨ ¸ 3 © 3kT ¹

Static polarization independent of T

Measurement of molar volume, v , and relative permitivity, H r , in a dilute solution as a function of T, allows P to be determined (plot total polarization versus 1/T; slope gives P).

35

Solutions Manual

9.

We compare the attractive part of the LJ potential (r >> V) with the London formula. The attractive LJ potential is

FG V11 IJ 6 HrK F V I6 4H 22 G 22 J HrK F V I6 4H12 G 12 J HrK

*11

4 H11

*22

*12

We assume that V12

1/ 2(V11  V22 ) . The London formula is

*11



*22





*12

FG H

3D12 I1 4r 6 3D 22 I 2 4r 6

3 D 1D 2 2 r6

IJ FG I1I2 JI K H I1  I 2 K

Substitution gives

H12

Only when V11

1/ 2

(H11H22 )

V 22 and I1

ª º V11V22 » « «1 V V » 22 ¼» ¬« 2  11

6

ª º I1I2 » « «1 I I » ¬« 2  1 2 ¼»

I 2 do we obtain H12

(H11H22 )1/ 2

Notice that both correction factors (in brackets) are equal to or less than unity. Thus, in general, H12 d (H11H22 )1/ 2

10.

See Pimentel and McClellan, The Hydrogen Bond, Freeman (1960). Phenol has a higher boiling point and a higher energy of vaporization than other substituted benzenes such as toluene or chlorobenzene. Phenol is more soluble in water than other substituted benzenes. Distribution experiments show that phenol is strongly associated when dissolved in

36

Solutions Manual

nonpolar solvents like CCl4. Infrared spectra show absorption at a frequency corresponding to the –OH " H hydrogen bond.

11.

J acetone CCl ! J acetone CHCl because acetone can hydrogen-bond with chloroform but not 4 3 with carbon tetrachloride.

12. a) CHCl3

Chloroform is the best solvent due to hydrogen bonding which is not present in pure chloroform or in the polyether (PPD). Chlorobenzene is the next best solvent due to its high polarizability and it is a Lewis acid while PPD is a Lewis base.

Cyclohexane is worst due to its low polarizability.

n-butanol is probably a poor solvent for PPD. Although it can hydrogenbond with PPD, this requires breaking the H-bonding network between n-butanol molecules. t-butanol is probably better. Steric hindrance prevents it from forming Hbonding networks; therefore, it readily exchanges one H-bond for another when mixed with PPD. The lower boiling point of t-butanol supports the view that it exhibits weaker hydrogen bonding with itself than does n-butanol.

b) Cellulose nitrate (nitrocellulose) has two polar groups: ONO2 and OH. For maximum solubility, we want one solvent that can “hook up” with the ONO2 group (e.g., an aromatic hydrocarbon) and another one for the OH group (e.g., an alcohol or a ketone). c)

Using the result of Problem 5,

37

Solutions Manual

1

E

a3

FG Hr  1 IJ P 2 H 2Hr  1K

At 20qC, the dielectric constants are Hr (CCl 4 )

Hr (C8 H18 ) 1.948

2.238

Thus, H r (CCl 4 ) H r (C 8 H18 )

117 .

It takes more energy to evaporate HCN from CCl4 than from octane.

13.

At 170qC and 25 bar:

z H2 is above 1 H2

zamine is well below 1

1

z HCl is slightly below 1

z HCl

0

a)

1

yamine

A mixture of amine and H2 is expected to exhibit positive deviations from Amagat’s law.

b) Since amine and HCl can complex, mixtures will exhibit negative deviations from Amagat’s law. c) z argon

1 z z HCl

0

yargon

1

Solutions Manual

38

The strong dipole-dipole attractive forces between HCl molecules cause z HCl  1 , while argon is nearly ideal. Addition of argon to HCl greatly reduces the attractive forces experienced by the HCl molecules, and the mixture rapidly approaches ideality with addition of argon. Addition of HCl to Ar causes induced dipole attractive forces to arise in argon, but these forces are much smaller than the dipole-dipole forces lost upon addition of Ar to HCl. Thus the curve is convex upwards.

14. a) Acetylene has acidic hydrogen atoms while ethane does not. Acetylene can therefore complex with DMF, explaining its higher solubility. No complexing occurs with octane. b) At the lower pressure (3 bar), the gas-phase is nearly ideal. There are few interactions between benzene and methane (or hydrogen). Therefore, benzene feels equally “comfortable” in both gases. However, at 40 bar there are many more interactions between benzene and methane (or hydrogen) in the gas phase. Now benzene does care about the nature of its surroundings. Because methane has a larger polarizability than hydrogen, benzene feels more “comfortable” with methane than with hydrogen. Therefore, KB (in methane) > KB (in hydrogen). c) Under the same conditions, CO2 experiences stronger attractive forces with methane than with hydrogen due to differences in polarizability. This means that CO2 is more “comfortable” in methane than in H2 and therefore has a lower fugacity that explains the condensation in H2 but not in CH4. d) It is appropriate to look at this from a corresponding-states viewpoint. At 100°C, for ethane TR | 12 . , for helium, TR | 80 . At lower values of TR (near unity) the molecules have an average thermal (kinetic) energy on the order of H (because TR is on the order of kT / H ). The colliding molecules (and molecules near one another, of which there will be many at 50 bar) can therefore be significantly affected by the attractive portion of the potential, leading to z < 1. At higher TR , the molecules have such high thermal energies that they are not significantly affected by the attractive part. The molecules look like hard spheres to one another, and only the repulsive part of the potential is important. This leads to z > 1. e) Chlorobenzene would probably be best although cyclohexane might be good too because both are polar and thus can interact favorably with the polar segment of poly(vinyl chloride). Ethanol is not good because it hydrogen bonds with itself and n-heptane will be poor because it is nonpolar. f)

i) Dipole. ii) Octopole. iii) Quadrupole. iv) Octopole.

39

Solutions Manual

g) Lowering the temperature lowers the vapor pressure of heptane and that tends to lower solvent losses due to evaporation. However, at 0qC and at 600 psia, the gas phase is strongly nonideal, becoming more nonideal as temperature falls. As the temperature falls, the solubility of heptane in high-pressure ethane and propane rises due to increased attraction between heptane and ethane on propane. In this case, the effect of increased gas-phase nonideality is more important than the effect of decreased vapor pressure.

15. a)

They are listed in Page 106 of the textbook: 1. 2. 3. 4.

Q can be factored so that Qint is independent of density. Classical (rather than quantum) statistical mechanics is applicable. *total ¦ (*Pairs ) (pairwise additivity). * / H F (r / V ) (universal functionality).

b) In general, assumption 4 is violated. But if we fix the core size to be a fixed fraction of the collision diameter, then Kihara potential is a 2-parameter (V, H) potential that satisfies corresponding states. c) Hydrogen (at least at low temperatures) has a de Broglie wavelength large enough so that quantum effects must be considered and therefore assumption 2 is violated. Assumption 1 is probably pretty good for H2; assumption 4 is violated slightly. All substances violate assumption 3, but H2 isn’t very polarizable so it might be closer than the average substance to pairwise additivity. d)

Corresponding states (and thermodynamics in general) can only give us functions such as

c p  c 0p . Values of c 0p (for isolated molecules) cannot be computed by these methods, because the contributions to c 0p (rotation, vibration, translational kinetic energy) appear in Qint and the kinetic energy factor, not in the configuration integral.

16.

Let D represent the phase inside the droplet and E the surrounding phase. Schematically we have for the initial state and for the final (equilibrium) state:

40

Solutions Manual

D

NO3

NO3

+

Na

K

NO3

NO3

D

+

Na

K+

+

+

K+

Na

2

Lys

Lys2

E

E

Equilibrium state

Initial state

Because the molar mass of lysozyme is above the membrane’s cut-off point, lysozyme cannot diffuse across the membrane. Let: G represent the change in K+ concentration in D;  M represent the change in Na+ concentration in E. The final concentrations (f) of all the species in D and E are: In D:

In E:

c fD

Lys 2

c fE

Lys2

0

c 0E

c fD

Lys2

K

c 0D  G K

c fE K

c fD  c fE 

G

Na

c fD  NO

M

Na

c 0D   G  M NO 3

3

c 0E  M Na

c fE

NO 3

c 0E   G  M NO3

The equilibrium equations for the two nitrates are PE   PE  K NO

P D  P D  K NO 3

3

(1) PE   PE  Na NO3

PD   PD  Na NO3

Similar to the derivation in the text (pages 102-103), Eq. (1) yields cE cE  K NO 3

c D c D  K NO 3

(2) E

E

c c  Na NO

cD

3

cD Na NO 3

where, for clarity, superscript f has been removed from all the concentrations. Substituting the definitions of G and M gives:

FH c0D  GKI FG c0D   G  MIJ K H NO3 K FH c0E  MIK FG c0E   G  MIJ Na H NO3 K where

FG H

(G ) c 0E   G  M NO 3

FG H

IJ K

(M) c 0D   G  M NO 3

IJ K

(3)

41

Solutions Manual

c 0D K

0.01 mol 1 kg water

c0D -

NO3

9.970 u 10 3 mol L-1

(using the mass density of water at 25D C: 0.997 g cm-3 )

and

c 0E  Na

c 0D -

NO3

0.01 mol L1

Solving for G and M gives

4.985 u 10 3 mol L1

G

(4) 3

M

1

5.000 u 10 mol L

Because both solutions are dilute, we can replace the activities of the solvent by the corresponding mole fractions. The osmotic pressure is thus given by [cf. Eq. (4-50) of the text] E RT x s ln vs x sD

S

(5)

where xs is the mole fraction of the solvent (water) given by xsD

xwD

D D D 1  ( xNO  x  x ) 3

K

Na

 xE + K

 xE  Na

(6) xsE

E xw

E 1  ( xNO  3

E  xLys )

Because solutions are dilute, we expand the logarithmic terms in Eq. (5), making the approximation ln(1  A) |  A : RT vw

S

E E E D D D ª ( x E º «¬ NO3  xK   xNa   xLys )  ( xNO3  xK   xNa  ) »¼

Again, because solutions are very dilute, ci c | i c c ¦i w

xi

i

with v w cw | 1

Therefore, with these simplifying assumptions, the osmotic pressure is given by S

D D D RT [cNO  c  c 3

K

Na 

E E E  cNO  c  c 3

K

Na 

E  cLys ]

Using the relationships with the original concentrations, we have

42

Solutions Manual

0E RT [(c 0D   G  M )  (c 0D  G)  (M)  (c 0E  G M )  (c 0E   M)  (cLys )]

S

K

NO3

Na

NO3

Because c 0D 

c 0D

c 0E 

c 0E 

K

NO3

Na

NO3

we obtain S

0E RT (2c 0D  2c 0E   cLys  4G  4 M ) K

(7)

Na

The lysozyme concentration is 0E cLys

2g 1L

(2 / 14,000) mol 1L

1.429 u 10 4 mol L-1

Substitution of values in Eq. (7) gives the osmotic pressure S

(8314.51 Pa L mol -1 K -1 ) u (298 K) u [(2 u 9.97 u 10 3 )  (2 u 0.01)  (1.429 u 10 4 )  (4 u 4.985 u 10 3 )  (4 u 5.000 u 10 3 ) (mol L1 )] 354 Pa

17.

Because only water can diffuse across the membrane, we apply directly Eq. (4-41) derived in the text:

 ln aw

 ln( xw J w )

Sv pure w

(1)

RT

where subscript w indicates water. Since the aqueous solution in part D is dilute in the sense of Raoult’s law, J w | 1 . This reduces Eq. (1) to:  ln xw or equivalently,

 ln(1  xA  xA2 ) | ( xA  xA2 )

Sv pure w

RT

(2)

43

Solutions Manual

RT ( x A  xA 2 )

S

At T = 300 K, v pure w

v pure w

0.018069 mol L1.

Mole fractions xA and xA2 can be calculated from the dimerization constant and the mass balance on protein A: K

vw

Mw dw

aA2

10 5

2 aA

18.015 0.997

(5 / 5,000) mol A (1000 / 18.069) mol water

|

xA2

(4)

2 xA

18.069 cm3 mol 1

181 . u 10 5

x A  2 x A2

(5)

Solving Eqs. (4) and (5) simultaneously gives 7.34 u 10 6

xA xA2

5.38 u 10 6

Substituting these mole fractions in Eq. (3), we obtain S

(0.0831451) u (300) u (5.38 u 10 6  7.34 u 10 6 ) (0.018069) 0.01756 bar

1756 Pa

18. a)

From Eq. (4-45): S c2

RT  RTB*c2  " M2

Ploting S / c2 (with S in pascal and c2 in g L-1) as a function of c2, we obtain the protein’s molecular weight from the intercept and the second virial osmotic coefficient from the slope.

44

Solutions Manual

50

S /c2, Pa L g -1

47.5 45 42.5 40 37.5 35 0

20

40 -1 BSA Concentration, g L

60

From a least-square fitting we obtain: Intercept = RT/M2 = 35.25 Pa L g-1 = 35.25 Pa m3 kg-1 from which we obtain

M2

(29815 . ) u (8.314) 35.25

RT 35.25

70.321 kg mol 1

70,321 g mol -1

Slope = 0.196 = RTB*. Therefore, B* = 7.92u10-8 L mol g-2. The protein’s specific volume is given by the ratio molecular volume/molecular mass. The mass per particle is m

M2 NA

70,321 6.022 u 1023

1.17 u 10 19 g molecule 1

Because protein molecule is considered spherical, the actual volume of the particle is 1/4 of the excluded volume. Therefore the actual volume of the spherical particle is 118 . u 10 24 4

2.95 u 10 25 m3 molecule 1

which corresponds to a molecular radius of 4.13u10-9 m or 4.13 nm. For the specific volume: 2.95 u 10 25 117 . u 10 19

2.52 u 10 6 m3 g 1

2.52 cm3 g 1

b) Comparison of this value with the nonsolvated value of 0.75 cm3 g-1, indicates that the particle is hydrated.

45

Solutions Manual

c) Plotting S / c2 (with S in pascal and c2 in g L-1) as a function of c2 for the data at pH = 7.00, we obtain: 60 57.5

S /c2, Pa L g -1

55 52.5

50 47.5

45 42.5

40 15

25

35 45 BSA Concentration, g L-1

55

Slope = 0.3317, which is steeper that at pH = 5.34, originating a larger second virial osmotic coefficient: B* = 1.338u10-7 L mol g-2 = 13.38u10-8 L mol g-2. At pH = 7.00, the protein is charged. The charged protein particles require counterions so electroneutrality is obtained. The counterions form an ion atmosphere around a central protein particle and therefore this particles and its surrounding ion atmosphere have a larger excluded volume than the uncharged particle. It is the difference between the value of B* at pH = 7.00 and that at pH = 5.37 for the uncharged molecule gives the contribution of the charge to B*: (13.38  7.92) u10-8 L mol g-2 =

1000 z 2 4 M22U1mMX

In this equation, we take the solution mass density U1 | Uwater | 1 g cm 3 . Moreover, M2 = 70,321 g mol-1, and mMX | 0.15 mol kg-1: z2

4 u (5.46 u 10 8 ) u (70,321) 2 u (10 . ) u (0.15 u 10 3 ) | 162 1000

or z r13. Because pH is higher than the protein’s isoelectric point, the BSA must be negatively charged. Hence, z = -13. 19. a)

From Eq. (4-45) and according to the data: S c  c0

RT  RTB* (c  c0 )  " M2

46

Solutions Manual

Ploting S / (c  c0 ) (with S in pascal and c2 in g L-1) as a function of c  c0, we obtain the solute’s molecular weight from the intercept and the second virial osmotic coefficient from the slope. 18.0

S/(c-c0), Pa L g

-1

17.5

17.0

16.5

16.0

30

40

50

(c-c 0), g L

60

-1

From a least-square fitting we obtain: Intercept = RT/M2 = 14.658 Pa L g-1 = 14.658 Pa m3 kg-1 from which we obtain M2

RT 14.659

(298.15) u (8.314) 14.659

169.109 kg mol1

169,109 g mol -1

Slope = 0.053495 = RTB*. Therefore, B* = 2.16u10-8 L mol g-2. b) The number of molecules in the aggregate is obtained by comparison of the molecular weight of the original ether (M = 390 g mol-1) with that obtained in a):

Number of molecules in the aggregate =

169,109 | 434 390

Assuming that the colloidal particles are spherical, we obtain the molar volume of the aggregates from the value of the second virial osmotic coefficient B*. It can be shown (see, e.g., Principles of Colloid and Surface Chemistry, 1997, P.C. Hiemenz, R. Rajagopalan, 3rd. Ed., Marcel Dekker) that B* is related to the excluded volume Vex through B*

NA V ex 2 M22

Solutions Manual

47

From a), B* = 2.16u10-8 L mol g-2 = 2.16u10-5 m3 mol kg-2 and M2 = 169,109 g mol-1 = 169.109 kg mol-1. The above equation gives Vex that is 4 times the actual volume of the particles (we assume that the aggregates are spherical). Calculation gives for the aggregate’s volume, 5.13u10-25 m3 (or 30.90 dm3 mol-1) with a radius of 4.97u10-9 m or 5 nm.

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

5

Initial pressure Pi: - for n1 moles of gas 1 at constant T and V: PiV n1RT

B 1  n1 11 V

or

Pi

n1RT n12 RTB11  V V2

Final pressure Pf: - after addition of n2 moles of gas 2 at same T and V:

Pf where (n1  n2 ) 2 B

(n1  n2 ) RT (n1  n2 )2 RTB  V V2

n12 B11  2n1n2 B12  n22 B22

Pressure change 'P :

'P

Pf  Pi

n2 RT RT  (2n1n2 B12  n22 B22 ) V V2

Solving for B12:

B12

· 1 § V 2 'P  n2V  n22 B22 ¸ ¨ ¸ 2n1n2 ¨© RT ¹

49

50

Solutions Manual

2.

For precipitation to occur,

s

V fCO ! fCO 2

s

To obtain fCO , 2

s

ln

fCO

60 1 (27.6) dP (83.1451) u (173) 0.1392

³

2

(0.1392)

s

fCO

2

0.156 bar (at 60 bar, 173 K)

2

Next, find the vapor mole fraction of CO2 that is in equilibrium with the solid at the specified P and T: V fCO

yCO2

2

M CO2 P

Using the virial equation for the vapor, ln M VCO

2

2 ( yCO2 BCO2  yH2 BH2  CO2 )  ln z v

Because yCO2  1 , we may make the approximations

z

z H2

v

and

v H2

z H2 RT P

From data for H2 (see App. C) at –100qC,

8.8 cm3 mol 1

BH2 which indicates that z H2 From correlations:

1.

460 cm3 mol 1

BCO2 BCO2  H2

321 . cm3 mol 1

At equilibrium V fCO

and then

2

s

fCO

2

51

Solutions Manual

s

yCO2

fCO

2

­ 2P ª ½ yCO2 BCO2  (1  yCO2 ) BH2 CO2 º¼ ¾ P exp ® ¬ ¯ RT ¿

yCO2

0.00344

Because yCO2  0.01 at equilibrium, CO2 precipitates. To find out how much, assume solid is pure CO2. Let n be the number of moles of CO2 left in the gas phase. From the mass balance and, as basis, 1 mole of mixture, 0.00344

n

n n  0.99

0.003417

The number of moles precipitating is

0.01  0.003417 0.0066 moles CO2

3. To = 313 K

Tf = ?

Po = 70 bar

Pf = 1 bar

V L . Condensation will occur in the outlet if fCO ! fCO 2

2

First it is necessary to find the outlet temperature, assuming no condensation. JouleThomson throttling is an isenthalpic process that may be analyzed for 1 mole of gas through a 3step process: I, III: Isothermal pressure changes.

P

To ,Po

II: Isobaric temperature change.

I Tf ,Pf III P=0

II T

52

Solutions Manual

Then 'htotal

0

z LMMN 0

'hI

Po

(1)

FG wv IJ OP dP H wT K P PQ

(2)

v  To Tf

³T

'hII

z LMMN Pf

'hIII

'hI  'hII  'hIII

0

c p,mixt dT

(3)

o

v  Tf

FG wv IJ OP dP H wT K P PQ

(4)

Assuming that the volumetric properties of the gaseous mixture are given by the virial equation of state truncated after the second term, v

RT  Bmixt P

then,

FG H

FG wv IJ H wT K P

IJ K

dBmixt R  P dT P

where Bmixt is the second virial coefficient of the mixture. Because y12 B11  2 y1 y2 B12  y22 B22

Bmixt

2 = CO2)

(1 = CH4; dBmixt dT

y12

(5)

dB11 dB dB  2 y1 y2 12  y22 22 dT dT dT

(6)

If c (0 ) 

B

c (1) c (2)  T T2

then dB dT



c (1) T

2



2c (2) T3

(7)

Assume c 0p, mixt

then,

yCH 4 c 0p,CH  yCO2 c 0p,CO 4 2

(8)

53

Solutions Manual

0

0

ª

§ dBmixt dT

³P «¬ Bmixt  To ¨© o

0

§ 2c(1) 3c(2) (0) ¨ cmixt  mixt  mixt ¨ To To2 ©

Tf Pf ª · º § dBmixt c p,mixt dT  « Bmixt  Tf ¨ ¸ » dP  To 0 ¬ ¹P ¼ © dT

³

³

· º ¸ » dP ¹P ¼

· § 2c(1) 3c(2) (0) ¸ ( Po )  c p,mixt (Tf  To )  ¨ cmixt  mixt  mixt ¸ ¨ Tf Tf2 ¹ ©

· ¸ ( Pf ) ¸ ¹

From data: As y1

0.3 , c p,mixt

0.7 , y2

36.22 J K 1 mol1 according to Eq. (8).

From Eqs. (5), (6) and (7),

Substitution in Eq. (9) gives Tf

(0 ) cmixt

41849 .

(1) cmixt

18683

( 2) cmixt

34.12 u 10 5

278.4 K .

Second, the fugacities of liquid and vapor phases may be calculated.

L fCO

ª P vL º CO2 s Ms exp « » xCO2 J CO2 PCO d P s 2 CO2 « PCO » RT 2 »¼ ¬«

³

2

This equation may be simplified assuming that xCO2 , J CO2 , M sCO equal to unity. 2 s 3 1 L  39.8 bar and v At 278 K, P 49.0 cm mol . CO2

CO 2

L fCO

2

(39.8) u exp

LM (49.0) u (1  39.8) OP . ) u (278) Q N (831451

36.6 bar

Fugacity of vapor is calculated from V fCO

2

M CO2 yCO2 P

with ln MCO2

P ª 2( yCO BCO  yCH BCO CH )  Bmixt º ¬ ¼ RT 2 2 4 2 4

ln MCO2

>2 u (0.3) u (139)  2 u (0.7) u (77)  (69)@

P RT

(9)

54

Solutions Manual

M CO2

0.995 | 1

V fCO

0.3 bar

Then 2

Because V L fCO  fCO 2

2

no condensation occurs.

4.

The Stockmayer potential is *

4H

LMF V I 12 F V I 6 OP P 2 MNGH r JK  GH ( r JK PQ  r 3 g(T1, T2 , I2  I1 )

where P is the dipole moment.

T2

T1

I

We can write the potential in dimensionless form: * H

f

F r , P2 I GH V HV 3 JK

where f is a universal function.

Therefore, we can write the compressibility factor z in terms of the reduced quantities: z

f (T , P , P )

with ~ T

kT H

55

Solutions Manual

P

~ P

PV3 H

P HV 3

5. a) For acetylene: Tc 308.3 K , Z 0.184 . At 0qC, TR (see, e.g., AIChE J., 21: 510 [1975]):

0.886 | 0.90 . Using Lee-Kesler charts

's (0 ) / R

3.993

zV(0 )

0.78

z L(0 )

0.10

's (1) / R

3.856

zV(1)

0.11

z L(1)

0.04

' vap s

(8.31451) u [3.993  (0.184) u (3.856)] 391 . J K 1 mol 1 ' vap h T ' vap s (273) u (39.1) 10.67 kJ mol 1

' vapu

' vap h  RT (zV  z L ) 10.67  (8.31451) u (273) u (0.76  0.092) u 10 3 ' vapu

915 . kJ mol 1

b) C 4 H10 : Tc

At 461 K:

425.2 K

N 2 : Tc

126.2 K

Pc

38.0 bar

Pc

33.7 bar

Z

0.193

Z

0.04

vc

255 cm3 mol-1

vc

89.5 cm3 mol-1

TR

TR

1084 .

3.65

Using the Pitzer-Tsonopoulos equation (see Sec. 5.7):

BC 4 H10

267 cm3 mol-1

BN2

For B12 : Z 12

1 (Z 1  Z 2 ) 2

0.1165

15.5 cm3 mol-1

56

Solutions Manual

(Tc1Tc2 )1 2

Tc12 zc12

Pc12

Then, B12

( TR12

2316 . K

0.291  0.08Z12

0.2817

zc12 RTc12

34.3 bar

1 1/ 3 (v  v1c/ 3 ) 3 2 8 c1

216 . cm3 mol-1

Bmixt

177.2 cm3 mol 1

y12 B11  2 y1y2 B12  y22 B22

c) CH 4 (1): Tc

190.6 K

N 2 (2): Tc

126.2 K

H 2 (3): Tc

Pc

46.0 bar

Pc

33.7 bar

Pc

13.0 bar

Z

0.008

Z

Z

0.22

vc

99.0 cm3 mol 1

vc

0.040

89.5 cm3 mol 1

vc

33.2 K

65.0 cm3 mol 1

At 200 K:

TR

158 .

TR

6.02

TR

105 .

At 100 bar:

PR

2.97

PR

2.17

PR

7.69

Using the mixing rules suggested by Lee and Kesler: 1 ¦ ¦ x j x k (v1c/j3  v1c/k3 )3 8 j k

v c,mixt

Tc,mixt

1 ¦ ¦ x j x k (v1c/j3  v1c/k3 )3 (Tc j Tck )1/ 2 8v c j k Zmixt

¦ x jZ j # 0 j

Pc,mixt

(0.291  0.08Z mixt ) RTc,mixt / v c,mixt

v c,mixt

84.1 cm 3 mol 1

Tc,mixt

111.38 K ( TR

180 . )

Pc,mixt

31.47 bar ( PR

318 . )

1 hmixture  (h1  h2  h3 ). 3 Using the Lee-Kesler charts,

Enthalpy of mixing = h E

1990 . )

57

Solutions Manual

1 922  (5385  1383  0) 3

hE

1334 J mol -1

6.

L/G P t

L

5

G

40 bar 25 qC

40% N2 50% H2 10% C3 H8

From the mass balance for C 3 H 8 : yinG

yout G  x out (L  [ yin  yout ]G ) yout G

0.10 x C3

0.01898

0.05 yinG

0.005  5.005 xC3

mole fraction of C3H8 in effluent oil.

To find the driving force, note that

Pi

PC3

yC3 P total

(0.10) u (40)

Pi*

fC*3 / M *C3

PC*3

4 bar

where

fC*3

x C3 H

(0.01898) u (53.3) 1012 . bar

and ln M *C3

2P

z * RT

[ yC3 BC3  yN2 BC3  N2  yH2 BC3  H2 ]  ln z *

Obtain virial coefficients from one of the generalized correlations:

58

Solutions Manual

400.8 cm3 mol-1

BC3

We may estimate z * * yC 3

To find * yC 3

,

* yN , 2

BC3  N2

73.5 cm3 mol-1

BC3  H2

3.5 cm3 mol-1

0.95 (feed value). * * and yH , we know that ( yN / y* ) = 4/5. As first guess, assume 2 2 H2

0.10. Then we calculate,

M *C3

1012 . (40) u (0.855)

* 0.855 and yC 3

0.0292

This gives *  y* yN H2 2

0.9708

or

* yN 2

0.4308 , y*H2

0.540

With these y * ’s, calculate M *C3 again:

M *C3

0.924

and

* yC 3

0.027

That is close enough. Thus, PC*3

1012 . 0.924

Now we must check the assumption z * sumption is close enough.

bar 1095 . 0.95 was correct. Using the virial equation, the as-

Driving force = (PC3  PC*3 )

7.

Since fiV

(4.00  1095 . )

fiL , s v f ( P  Psolv ) xi Hi,solv exp i RT

yi Mi P s As Psolv

0,

yi xi Using the virial equation,

Hi exp(vif P / RT ) Mi P

2.91 bar

59

Solutions Manual

ln M1

2 ( y1B11  y2 B12 )  ln z mixt v

ln M 2

2 ( y2 B22  y1B12 )  ln z mixt v

Because Pv RT v2 

B RT RT v  mixt P P

0

3.23 cm3 mol-1

Bmixt v

B 1  mixt v

524 cm3 mol-1 , z mixt

10067 .

Thus, M1

y1 x1

M2

0.8316 (100) u exp

0.9845

LM (60) u (50) OP . )Q N (313) u (8314

(50) u (0.8316)

2.70

and y2 x2 D 2,1

8.

2131 .

FG y2 IJ FG x1 IJ H x2 K H y1 K

7.89

For methane (1) and methanol (2), we may write f 1V

f 1L

f 2V

f 2L

Neglecting the Poynting corrections [Note: the Poynting Correction is 1.035. Including this, we get y2 = 0.00268], y1M1P

x1H1,2

60

Solutions Manual

x 2 J 2 P2s M s2

y2 M 2 P

Because x2 = 1, assume J 2 1 . Use virial equation to get fugacity coefficients:

M s2 Assuming y1

1 , y2

exp

B22 P2s RT

(4068) u (0.0401) (8314 . ) u (273)

exp

0.993

0 as first estimate, M1

M2

0.954

0.783

Thus, x1

0.0187

(1  x1 )P2s M s2

y2

Using now y2

M 1P H1,2

0.00250

M2P

0.00250 , get M1

0.954

M2

0.770

0.01867

y2

0.00255

and x1

This calculation is important to determine solvent losses in natural gas absorbers using methanol as solvent.

9.

2(monomer)

dimer

The equilibrium constant is

Ka

ad

fd / fd0

(am )2

( fm / fm0 )2

2 fd §¨ fm0 fm2 ¨© fd0

· ¸ ¸ ¹

where ad is the activity of the dimer, and am is the activity of the monomer. 2

The quantity f m0 / fd0 is a constant that depends on T, but not on P or y. Then,

61

Solutions Manual

k f m2

fd

where k is a constant.

10.

CH3

CH3 CH

CH3

O

CH CH3

CH3CH2

Di-isopropyl ether

O

CH2CH2CH2CH3

Ethyl butyl ether

HCl can associate with the ether’s non-bonded electron pairs. However, the di-isopropyl ether will offer some steric hindrance. The cross-coefficient, B12 , is a measure of association. Both virial coefficients will be negative; B12 for ethyl butyl ether/HCl will be more negative.

11.

Let D be the fraction of molecules that are dimerized at equilibrium. 2A 1 D

nT

A2 D/2

1 D 

yA

2

yA

D 2

1

D 2

D/2 1 D / 2 1 D 1 D / 2

By assuming the vapor to be an ideal gas, we may write K

PA2

yA2 P

( PA )

2

( yA P)

(D / 2)(1  D / 2)

2

(1  D)2 P

At the saturation pressure, P = 2.026 bar and yA

0.493

D

0.6726

62

Solutions Manual

Then fAV

fAL

(0.493) u (2.026)

yA P

0.999 bar

The pressure effect on fugacity is given by

FG w ln f IJ H wP K

vi RT

i

T

Assuming the liquid to be incompressible in the range Ps to 50 bar, ln

fA (50 bar) fA (2.026 bar)

v A 'P RT

fA (50 bar) 1.1 bar

12. a)

The Redlich-Kwong equation is z

FG H

Pv RT

v 1 a  1 . 5 v  b RT vb

If z is expanded in powers of 1/ v : 1

z

B C  " v v2

This gives

b

B

b2 

C

a RT 1.5 ab RT 1.5

But, B'

C'

B / RT

C  B2 (RT )2

Substitution gives B'

FG H

1 a b RT RT 1.5

IJ K

IJ K

63

Solutions Manual

a

C'

3 3.5

R T

b)

FG 3b  a IJ H RT K 1.5

Using an equation that gives fugacities from volumetric data, we obtain ln M1

P RT

ª y2 a  2 y1a1  y22 a2  2 y22 (a1a2 )1/ 2 º « b1  1 1 » RT 1.5 «¬ »¼

Evaluate a and b using critical data: Ethylene (1):

Tc = 282.4 K

Pc = 50.4 bar

Nitrogen (2):

Tc = 126.2 K

Pc = 33.7 bar

a1 = 7.86u107 bar cm6 K1/2 mol-2 a2 = 1.57u107 b1 = 40.4 cm3 mol-1 Substitution gives M1

0.845 ,

y1M1P

f1

13.

8.44 bar

Using the virial equation,

P

RT RTBmixt  v v2

with y12 B11  2 y1 y2 B12  y22 B22

Bmixt

(1)

For maximum pressure,

FG wP IJ H wy K T 1

,v

Substituting Eq. (1) into Eq. (2) gives ( y2 § wBmixt · ¨ ¸ © wy1 ¹T

At maximum,

FG H

IJ K

RT wBmixt wy1 T v2

0

(2)

1  y1 ) :

2 y1 B11  2 y2 B12  2 y1 B22  2 B22

0

64

Solutions Manual

B22 B11  B22  B12

y1

(3)

Using the correlations, B11 B22

126.7 cm3 mol-1 (ethylene) 12.5 cm3 mol-1 (argon) 45.9 cm3 mol-1

B12

Substitution in Eq. (3) gives y1 = 0.134

14.

Consider a 3-step process: Mixed ideal gases

Pure ideal gases Tf

Tf

P = 20.7 bar I

Mixed gases Tf

Pure gases Ti

The overall enthalpy change is zero: 'H I  'H II  'H III 'H I

where

0

n1'H1  n2 'H 2

'H1

c 0p1 (Tf  Ti ) 

H0  H (RTc1 ) RTc1

'H 2

c 0p2 (Tf  Ti ) 

H0  H (RTc2 ) RTc2

H0  H is evaluated using Lee-Kesler Tables. RTc

(1 = hydrogen; 2 = ethylene)

65

Solutions Manual

For heat capacities, we can estimate

'H II

0

c 0p1

28.6 J K-1 mol-1

c 0p2

43.7 J K-1 mol-1

(because we are mixing ideal gas)

F H  H I RT GH RTc JK c 0

'HIII



, mixt

,mixt

evaluated at TR = Tf / Tc,mixt, PR = 20.7/Pc,mixt

Find Tf by trial and error: Tf

15.

247 K

At equilibrium,

fAV yA M A P

s

fA

(1  xCO2 )PAs M sA exp

z

s

P

vA

s PA

RT

dP

Assuming xCO2 | 0 , ( y A P)MA

PAs MsA

s

ª v ( P  Ps ) º A » exp « A RT « » ¬ ¼

But MsA

§ B Ps exp ¨ AA A ¨ RT ©

· ¸ 1.00 ¸ ¹

Because yA  yCO2 , yCO2 # 1

MA yA P PAs

or

P º ª exp «(2 BA CO2  BCO2 ) RT »¼ ¬

s

ª v ( P  P s ) (2 BA CO  BCO )P º 2 2 A »  exp « A RT RT « » ¬ ¼

66

Solutions Manual

s

s

ª s y Pº « PA v A  RT ln As » /(v A  2 BA CO2  BCO2 ) PA ¼» ¬«

P

Substitution gives P = 68.7 bar 19 . u 10 4  1 and assumption yA | 0 is correct.

For this pressure, yA

16.

Let 1 = ethylene and 2 = naphthalene. As, at equilibrium,

s

f2V

f2 and as

s

f2

P2s M s2 exp f2V

we may write

P22

RT

dP

y2 M 2 P

s

Using ideal-gas law: M s2

s

As v 2

128.174 1.145

M V2

1

111.94 cm 3mol 1 ,

y2

b)

v2

(1  x1 )P2s M s2 exp[v 2 (P  P2s ) / RT ] / M V2 P

y2 a)

z

s

P

s

ª v (P  Ps ) º 2 » / P 1.1u 10 5 P2s exp « 2 RT « » ¬ ¼

Using VDW constants: a a1 a2

27 R 2 Tc2 / 64 Pc

4.62 u106 bar cm6 mol-2 4.03 u107 bar cm6 mol-2

b

RTc / 8 Pc

b1

58.23 cm3 mol-1

b2

192.05 cm3 mol-1

67

Solutions Manual

b1  a1 / RT

B11

122.2 cm3 mol-1

1382 cm3 mol-1

B22

Because y2 << y1, y1 # 1 and Bmixt # B11 B Ps exp 22 2 # 1 RT

M2s

ln MV2

>2( y1B12  y2 B22 )  Bmixt @

b12  a12 / RT

B12

b12

with b12

12514 . cm3 mol-1

and

P P # (2 B12  B11 ) RT RT

1 / 2(b1  b2 ) and a12

(a1a2 )

a12 = 1.36u107 bar cm6 mol-2

Then B12 = -406 cm3 mol-1. MV2 y2

P º ª exp «(2 B12  B11 ) RT »¼ ¬

0.446

(2.80u10-4) u exp [(111.9u30)/(83.1451u308)]/(0.446u30) y2 = 2.4u10-5

17.

s

Water will condense if f HV O ! f H O . 2 2 Thus, the maximum moisture content, yH2O , is given by

f HV

2O

yH2O M VH O P 2

s

fH

x H2O PHs O 2

2O

exp

z

P

Ps H 2O

s

vH O 2 dP RT

Assuming that the condensate is pure (solid) water,

x H2O Then

1

and

M sH

2O

1

68

Solutions Manual

( yH2O MVH

2O

) u (30 bar)

ª 1.95 · º § (18 / 0.92) u ¨ 30  « (1.95 torr) 750.06 ¸¹ » © » exp « (750.06 torr/bar) « (83.1451) u (263.15) » « » ¬ ¼ y H 2O

8.90 u 10 5 M VH

2O

Let 1 = N2, 2 = O2, and 3 = H2O. To get M VH

2O

use the virial equation of state: ln MVH O 2

ln M3

[2( y1 B13  y2 B23  y3 B33 )  Bmixt ]

P RT

with Bmixt

¦ ¦ yi y j Bij i

j

Assume y3 << y1 where y1 = 0.80 and y2 = 0.20. Then, Bmixt

y12 B11  y22 B22  2 y1 y2 B12

Substitution in the equation for ln M 3 gives M 3 As y H 2O

21.8 cm3 mol-1

0.871.

8.90 u 10 5 M VH

2O

then

yH 2O

18.

1.0 u 10 4

The Joule-Thompson coefficient is defined as

PH {

FG wT IJ H wP K

(1) H

Applying the triple-product rule with T, P and H, we have

69

Solutions Manual

FG wT IJ FG wP IJ FG wH IJ H wP K H wH K H wT K H

T

1

(2)

P

Because § wH · ¨ wT ¸ { c p © ¹P

and 1 FG wH IJ H wP K T FG wP IJ H wH K T

combining Eqs. (1) and (2) gives

PH



1 § wH · c p ¨© wP ¸¹T

(3)

TdS  VdP , we have

From the fundamental equation dH

FG wH IJ H wP K

T T

FG wS IJ H wP K

V

(4)

T

However, we also have Maxwell’s relation:

FG wS IJ H wP K

 T

FG wV JI H wT K

P

Therefore,

FG wH IJ H wP K

T T

FG wV IJ H wT K

V

(5)

P

Substituting Eq. (5) into Eq. (3) gives PH

º 1 ª § wV · V» « T ¨ ¸ c p ¬ © wT ¹ P ¼



(6)

or, in terms of molar volume (v) and molar heat capacity at constant pressure (c p ) ,

PH



LM FG IJ  vOP MN H K P QP

wv 1 T cp wT

(7)

Because in this specific problem we want P H of the hydrogen-ethane mixture to be zero, Eq. (7) yields

v T

FG wv IJ H wT K

(8) P

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Solutions Manual

From the truncated-virial equation of state Pv RT

1

B v

(9)

we have

FG wT IJ FG P IJ FG v IJ FG 2  v IJ FG P IJ v(v  2B) H wv K H R K H v  B K H v  B K H R K (v  B) 2

P

(10)

Substituting the equation of state Eq. (9) and Eq. (10) into Eq. (8) yields

FG H

RT B 1 v P

IJ T LM R (v  B) OP K MN P v(v  2B) QP 2

(11)

or equivalently,

B

0

(12)

where B is the second virial coefficient of the hydrogen-ethane mixture at 300 K. Using the McGlashan and Potter equation [Eq. (5-52)], B vc

we obtain B11 B12

0.430  0.866

FG T IJ 1  0.694FG T IJ 2 H Tc K H Tc K

11.4 cm 3 mol 1 for hydrogen, and B22

(13)

173.3 cm 3 mol 1 for ethane, and

B21 11.4 cm3 mol 1 for the cross term, respectively. Applying Bmixt

¦ ¦ yi y j Bij i

and the material balance y1  y2

j

1, , we have

Bmixt ( cm 3 mol 1 ) 11.4 y12  22.8 y1 (1  y1 )  173.3(1  y1 ) 2

(14)

Equations (12) and (14) yield y1 0.73. Consequently, if we start out with 1 mol of H2, the amount of ethane that must be added to have a zero P H is 0.37 mol.

19.

Because methane does not significantly dissolve in liquid water at moderate pressures, the equation of equilibrium is f2V

L f pure 2

(1)

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Solutions Manual

or equivalently, P2s M s2 exp

y2 M 2 P

LM v2L eP  P2s j OP MM RT PP N Q

(2)

where subscript 2 denotes water. The fugacity coefficient from the volume-explicit virial equation of state is given by Eq. (533): ln M 2

2( y1B12  y2 B22 )  Bmixt

P RT

(3)

with Bmixt

y12 B11  2 y1 y2 B12  y22 B22

(4)

B22 P2s

(5)

Similarly, we also have ln M s2

(i)

RT

At the inlet (60ºC, 20 bar):

Substitute Eqs. (3), (4), and (5) into Eq. (2). Solving Eq. (2) using v 2L P2s

18 cm 3 mol -1 and

149 mmHg , we obtain y2i

(ii)

0.011 (superscript i denotes inlet).

At the outlet (25ºC, 40 bar):

Similarly, with v 2L

18 cm 3 mol -1 and P2s y2o

24 mmHg , we obtain

0.000951 (superscript o denotes outlet).

Because the gas phase is primarily methane, the amount of water that must be removed per mol methane is

mol water removed mol methane

20.

y2i  y2o | 0.01

Assuming negligible changes in potential and kinetic energies, the first law of thermodynamics for a steady-state flow process is

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Solutions Manual

'H

Q  Ws

(1)

where 'H is the change in enthalpy, and Q and Ws are, respectively, the heat and the shaft work done on the system (by the surroundings). Because we are given the initial and final state and enthalpy is a state function, 'H is fixed in this problem. Consequently, minimum Ws corresponds to maximum Q . Maximum Q occurs when the process is reversible, or equivalently Q T'S . Hence, Eq. (1) can be rewritten as Ws

'H  T'S

(2)

where Ws is now the minimum amount of work required for the process. To calculate 'H and 'S from the volume-explicit virial equation of state PV nT RT

1  BP

we take an isothermal reversible path from the initial to the final state. Expressions for enthalpy and entropy are given by Eqs. (3-9) and (3-10) in the text:

z LM z MN P

H

0

MMLV  T FGH wwVT IJK N

0 1 1

P,nT

 n2 h20 (3)

FG IJ OP H K PQ

P

S

OP PQ dP  n h

nT R wV  dP  R(n1 ln y1P  n2 ln y2 P)  n1s10  n2 s20 P wT P,n 0 T

Substituting

§ wV · ¨ wT ¸ © ¹ P,nT

nT R (1  BP) P

from the virial equation of state gives H

n1h10  n2 h20

S

nT RBP  R(n1 ln y1P  n2 ln y2 P )  n1s10

(4)  n2 s20

Applying Eq. (4) to this specific problem, we have

'H

0

'S

RP(nT Bmixt  n1B11  n2 B22 )  R(n1 ln y1  n2 ln y2 )

(5)

At 298 K, second virial coefficients are

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Solutions Manual

419 . cm3 mol -1

B11 B12

122.2 cm3 mol -1

B22

66.0 cm 3 mol -1

(6)

Hence, at 298 K,

i

Taking a basis of nT at 298 K

74.0 cm3 mol -1

¦ ¦ yi y j Bij

Bmixt

j

1 mol of the mixture initially, then n1

'H

n2

0.5 mol, Eq. (5) yields

0 (7)

T'S

248.5 kJ

Substituting Eq. (7) into Eq. (2), the minimum amount of work required for this process is 248.5 kJ mol 1 of initial mixture.

21.

The second virial coefficient for a square-well potential is given by Eq. (5-39) in the text

F GH

b0 R 3 1 

B

R3  1 R3

exp

H kT

I JK

(1)

with b0

Substituting H / k

469 K , V

2 SN AV 3 3

0.429 nm and R B(423 K)

(2) 0.337V gives

302 cm3 mol -1

Because, for a pure component, f P

ln M

ln

f

P exp

BP RT

fugacity is given by

FG BP IJ H RT K

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Solutions Manual

Hence, at 150ºC and 30 atm,

LM (30 atm) u (301.8 cm mol ) OP MN (82.06 atm cm mol ) u b423 Kg PQ 3

f

(30 atm) u exp

1

1

3

38.94 atm | 39 atm To obtain the standard enthalpy and entropy of dimerization of methyl chloride we assume a small degree of dimerization. In this case, the relation between the second virial coefficient and the dimerization constant is given by Eq. (5-113): b

B Applying P 0

RTK P0

1 mol L1 = 10 -3 mol cm 3 is the standard state) gives

RTc 0 (where c 0

103 (cm3 mol 1 )K

Bb

Because we only have a weak dimerization, b

2 SN AV 3 3

b0

(3)

and the second virial coefficient is essentially that of pure methyl chloride [Eq. (1)]. Combining Eqs. (1) and (3) gives

FG 2 SN V 3 IJ LMR3 FG1  R3  1 exp H IJ  1OP H3 K MN H R3 kT K PQ

10 3 (cm 3 mol 1 )K

A

The following table shows K (T ) calculated from Eq. (4). T (K)

K

100 200 300 400 500

10.74 0.939 0.376 0.222 0.155

Because the standard enthalpy and entropy of dimerization obey [Eq. (5-114)]  R ln K

'h 0  's 0 T

plotting R ln K as a function of 1/T gives 'h 0 as the slope, and 's 0 as the intercept. Results are 'h0 's 0

4.34 kJ mol 1 23.2 J K 1 mol 1

(4)

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Solutions Manual

22. a)

Substitution of given *(r) in Eq. (5-19) gives B

f n ª V º 2SN A « (1  ef / kT ) r 2 dr  (1  e A / kTr ) r 2 dr » V ¬ 0 ¼

³

³

f n ª V º 2SN A « r 2 dr  (1  e A / kTr ) r 2 dr » V ¬ 0 ¼

³

³

(1)

f ª V3 º n 2SN A «  (1  e A / kTr ) r 2 dr » V ¬« 3 ¼»

³

At high temperatures, A/kTrn is small. Because

ex | 1 x 

x2 " 2!

when x is small,

n

we expand the exponencial e A/ kTr : A

n

e A/ kTr | 1 

kTr n

"

Substitution of this approximation into Eq. (1) gives: f

³V (1  e

A / kTr n

) r 2 dr

f§

A · 2

³V ¨©1  1  kTr n ¸¹ r dr  A f 2 n (r )dr kT V

³

A § V3 n · ¨ ¸ kT ¨© 3  n ¸¹

 A f n 2 (r )r dr kT V

³

 A ª r 3 n º « » kT «¬ 3  n »¼

 AV3 n kT (n  3)

We substitute now this result in Eq. (1):

B

LM MN

2 V 3 n SN AV 3  2SAN A 3 kT (n  3)

Constant n is large (i.e., n > 3):

OP PQ

f

(2) V

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Solutions Manual

B

LM MN

2 V3 SN AV 3  2SAN A 3 kT (n  3)V n

OP PQ

LM N

OP Q

2 2 3A SN AV 3  SN AV 3 3 3 kT (n  3)V n

(3)

b) From Eq. (3) we see that it is the attractive part of the potential that causes negative B and is responsible for the temperature dependence of B [the first term on the right hand side of Eq. (3) is independent of temperature].

23.

Substitution of the square-well potential [Eq. (5-39)] into Eq. (5-17) gives B

2SN A

³

ª 2SN A « ¬

f

(1  e*(r ) / kT ) r 2 dr

0

³

V

(1  ef / kT )r 2 dr 

0

ª V3 2SN A «  ¬« 3

³

R'

V

³

R'

V

(1  eH / kT )r 2 dr 

f

º (1  e0 / kT )r 2 dr » R' ¼

³

º (1  eH / kT )r 2 dr  0 » ¼»

ª V 3 § R '3 V 3 · º 2SN A «  ¨  ¸ (1  eH / kT ) » 3 ¸¹ «¬ 3 ¨© 3 »¼ 2 2 SN AV3  SN A (1  eH / kT )( R '3  V3 ) 3 3

In the equation above, R' = RV = 1.55V. For argon, V = 0.2989 nm = 0.2989u10-9 m, = 141.06 K, and R' = 1.55V = 4.633u10-10 m. The above equation gives for T = 273.15 K,

B

H/k

3.368 u 10 5  (6.202 u 10 5 ) 2.834 u 10 5 m 3 mol 1 | 28 cm 3 mol -1

The calculated value compares relatively well with the experimental B for argon at the same temperature: Bexp = -22.08 cm3 mol-1.

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Solutions Manual

24.

Substitution of Sutherland potential [Eq. (5-37)] in Eq. (5-19) with N = NA gives

B(T )

ª 2SN A « ¬

³

V

(1  ef / kT ) r 2 dr 

0

³

f

V

6 º (1  eK / kTr ) r 2 dr » ¼

(1) 2 SN AV3  3 § K exp ¨ © kTr 6

³

f

V

6

(1  eK / kTr ) r 2 dr

· K K2 K3 K4    " ¸ | 1 kTr 6 2(kT )2 r12 6(kT )3 r18 24(kT )4 r 24 ¹

[ ex | 1  x 

(2)

x2 x3 x 4   "] 2! 3! 4!

We now have to replace the approximate result [Eq. (2)] in Eq. (1) and perform the necessary integrations. The result is:

B(T )

2SN A K 2SN A K 2 2SN A K 3 2SN A K 4 2 SN AV 3     " 3 3kTV 3 18(kT ) 2 V 9 90(kT )3 V15 504(kT ) 4 V 21

This equation is best solved using an appropriate computer software such as Mathematica, TKSolver, MathCad, etc. Making the necessary programming we obtain at 373 K, B(methane) = -20 cm3 mol-1 B(n-pentane) = -634 cm3 mol-1

In both cases, the agreement with experiment is very good.

25.

The equation of equilibrium for helium is f1L

f1R

where superscripts L and R stand for left and right compartments, respectively. Equivalently,

(1)

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Solutions Manual

y1L M1L P L

y1R M1R P R

(2)

Equation (5-33) of the text gives the fugacity coefficients in both compartments from the volume-explicit virial equation of state: ln M1

2( y1B11  y2 B12 )  Bmixt

P RT

(3)

with Bmixt

y12 B11  2 y1 y2 B12  y22 B22

(4)

Further, we also have material balances y1L  y2L

1

(5) y1R  y3R

1

Applying Eqs. (3), (4), and (5) to both compartments yields ln M1R

{2 ª¬ y L B 1

L L L º ª L 2 11  (1  y1 ) B12 ¼  ¬( y1 ) B11  2 y1 (1  y1 ) B12

PL  (1  y1L )2 B22 º ¼ RT

}

(6) ln M1L

{2 ª¬ y R B 1

R R R º ª R 2 11  (1  y1 ) B13 ¼ ¬( y1 ) B11  2 y1 (1  y1 ) B13

PR  (1  y1R )2 B33 º ¼ RT

}

Total mole balance on helium gives n1L  n1R

0.02 mol

(7)

Combining with mass balances on ethane and nitrogen gives

y1L

n1L 0.99  n1L (8)

y1R

n1R

0.02  n1L

0.99  n1R

.  n1L 101

Substituting Eqs. (6) and (8) into Eq. (2) yields n1L

0.013 mol

Combining this result and Eq. (8) gives y1L

0.013

y1R

0.007

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Solutions Manual

26.

Because the equilibrium constant is independent of pressure, the more probable reaction is the one that satisfies this condition. (1)

Assuming reaction (a) is more probable:

With this scheme, concentration of (HF) 6 is negligible compared to those of ( HF) and (HF) 4 . The equilibrium constant K (a ) is K (a )

y( HF ) 4 y(4HF ) P 3

y( HF ) 4 1  y( HF ) 4

4

(1) P3

Total mass balance for ( HF) gives y( HF) 4 nT (4 u M HF )  1  y( HF) 4 nT M HF

V u U HF

(2)

where V is the total volume; U HF and M HF are the mass density and the molar mass of hydrogen fluoride, respectively; nT is the total number of moles that can be calculated by assuming that the gas phase is ideal: nT

PV RT

(3)

Substituting Eqs. (2) and (3) into Eq. (1) yields

FG U RT  1IJ HPuM K LM1  (1 / 3)F U RT  1I OP GH P u M JK PQ MN (1 / 3)

K (a )

HF

HF

(4)

4

HF

P

3

HF

Applying Eq. (4) at the two pressures, 1.42 and 2.84 bar, we obtain for K (a ) : P (bar)

UHF (g/L)

K(a)

1.42 2.84

1.40 5.45

0.0595 0.453

Because K (a ) depends on pressure, reaction (a) cannot be the more probable one. Next we need to check for the pressure independence of K ( b ) . (2)

Assuming reaction (b) is more probable:

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Solutions Manual

In this case, concentration of (HF) 4 is negligible compared to those of ( HF) and (HF) 6 . The equilibrium constant K ( b ) is K( b)

y( HF )6 y(6HF ) P 5

y( HF )6 1  y( HF )6

6

(5) P5

Mass balance for HF in this case is: y( HF )6 nT (6 u M HF )  1  y( HF )6 nT M HF

V u U HF

(6)

where all terms are defined in Eq. (2). Substitution of Eqs. (3) and (6) into Eq. (5) gives

FG U RT  1IJ HPuM K MMNL1  (1 / 5)FGH PUu MRT  1IJK OPQP (1 / 5)

K ( b)

HF

HF

(7)

6

HF

P5

HF

Corresponding values of K ( b ) at 1.42 and 2.84 bar are: P (bar)

UHF (g/L)

K(b)

1.42 2.84

1.40 5.45

0.017 0.017

Because K ( b ) is independent of pressure, reaction (b) is the more probable.

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

The three equations of equilibrium (in addition to T L y1M1V P

x1J 1 f1L

y2M V2 P

x2 J 2 f2L

y3M V3 P

x3 J 3 f3L

T V and P L

6

P V ) are

with (assuming the liquid incompressible) f1L

v L (P  P1s ) P1s M1s exp 1 RT L

L

We write similar expressions for f 2 and f 3 . For M V we may write RT ln M1V

P(2 y1B11  2 y2 B12  2 y3 B13  Bmixt )

and similar expressions for M V2 and M V3 . In these equations, Bmixt

y12 B11  y22 B22  y32 B33  2 y1 y2 B12  2 y1 y3 B13  2 y2 y3 B23

81

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Solutions Manual

2.

Given g E

Ax1 x 2 with P1s / P2s

1.649 , and assuming ideal vapor,

y1P

J1

x1P1s y2 P

J2

At azeotrope, x1

x 2 P2s

y1: ln J 1

ln

ln J 2

ln

P P1s P P2s

or

ln

J1 J2

ln

P2s P1s

0.5

From the g E expression, ln J 1

A 2 x RT 2

ln J 2

A 2 x RT 1

and

Then J ln 1 J2

A RT

A 2 ( x  x12 ) RT 2

0.5 x22

 x12

1 4 x2  2

or x2

1 RT  2 4A

Because 0 d x 2 d 1 ,  1 2

1 RT 1 d d 2 4A 2

Thus, if | A| t RT , an azeotrope exists.

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Solutions Manual

3.

From the plot P-x-y we can see the unusual behavior of this system: 1. There is a double azeotrope 2. Liquid and vapor curves are very close to each other.

75

Pressure, kPa

74

73

72

0

Liquid (

)

Vapor (

)

0.2

0.4

0.6

0.8

1.0

x1, y1

4.

Neglecting vapor phase non-idealities, P

x1J 1P1s  x 2 J 2 P2s

(1)

At the maximum, § wP · ¨ ¸ © wx1 ¹T

0

§ wJ · § wJ · J1P1s  x1P1s ¨ 1 ¸  x2 P2s ¨ 2 ¸  J 2 P2s w x © wx2 ¹T © 1 ¹T

From the Gibbs-Duhem equation (at constant T and low pressure),

(2)

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Solutions Manual

§ w ln J1 · § w ln J 2 · x1 ¨ ¸  x2 ¨ ¸ © wx1 ¹T © wx1 ¹T

0

or x1 § wJ1 · ¨ ¸ J1 © wx1 ¹T

x2 § wJ 2 ¨ J 2 © wx2

· ¸ ¹T

(3)

Substituting Eq. (3) into Eq. (2) and simplifying, § x wJ · ( J1P1s  J 2 P2s ) ¨ 1  1 1 ¸ © J1 wx1 ¹

0

There are two possibilities: (1)

J 1P1s  J 2 P2s

0

J 2 P2s

1

Then

J 1P1s D

( y2 / x2 ) ( y1 / x1 )

D

1 corresponds to an azeotrope

(2)

x wJ 1 1 1 J 1 wx1

0

The solution to this differential equation is x1J 1 constant . To find the constant, use the boundary condition J 1 1 when x1 Hence J 1 x1 1. †

x1J 1P1s if J 1 x1 1, then y1 must be 1. Hence, the curve P-x goes through a maximum at x1 trivial one).

As y1P

5.

1.

1. This is also an azeotrope (but a

Given gE

A12 x1 x 2  A13 x1 x 3  A14 x1 x 4  A23 x2 x3  A24 x 2 x 4  A34 x3 x 4

where

†

This may not be immediately obvious. But J1 x1 is the activity, and the activity of component 1 cannot reach unity for any x1 less

than one because the solution will split into two phases of lower activity. See Fig. 6-25 in the text.

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Solutions Manual

x1

n1 / nT

x2

x2 / nT

# with nT n1  n2  n3  n4 the total number of moles. Because

F wn g I GH wn JK E

T

RT ln J 1

1

P,T ,n2 ,n3 ,n4

we find RT ln J1

A12 x22  A13 x32  A14 x42  x2 x3 ( A12  A13  A23 )  x2 x4 ( A12  A14  A24 )  x3 x4 ( A13  A14  A34 )

6.

Calculate T-y giving pressure and for x = 0.1, 0.2, …, 0.9  bubble-point calculation. We have to solve the equilibrium equations: M i yi P

J i xi Pis

(1)

Because total pressure is low (below atmospheric) we assume vapor phase as ideal: M i | 1 . The activity coefficients are obtained from the equation for GE given in the data. Using Eq. (6-47) of the text we obtain ln J 1

21 . x 22

ln J 2

21 . x12

(2)

As the pressure is fixed, temperature varies along with x1 (and y1) and is bounded by the saturation temperatures of the two components. These can be easily obtained from the vaporpressure equations. They are given in the form, ln P s

A

B T C

(3)

from which we obtain the saturation temperature

Ts

B A  ln P s

C

(4)

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Solutions Manual

For Ps = 30 kPa, we obtain T1s

387.26 K for cyclohexanone (1) and T2s

415.59 K for

phenol (2). To obtain the T-x1-y1 diagram we assign values for the liquid mole fraction x1. Total pressure is P

J 1 x1P1s  J 2 x2 P2s

P1s

P

or (5)

P2s

J 1 x1  J 1 x1 s P1

To start the calculation we make an initial estimate of the temperature: x1T1s  x 2T2s

T

(6)

0.5 u 387.26  0.5 u 415.59

For example, let us fix x1 = 0.5: T

401.42 K

With this temperature we obtain P1s and P2s from Eq. (3), the pure-component vapor pressure equations: P1s J1

47.243 kPa and P2s

exp(21 . u 0.52 )

0.592 and J 2 P1s

Next we recalculate

17.918 kPa Because we fixed x1, Eqs. (2) give

0.592 .

73.482 kPa from Eq. (5), which in turn gives a new temperature,

T = 416.34 K, from the pure cyclohexanone vapor pressure equation. The sequence of calculations is now repeated for this new temperature (we assume here that activity coefficients are independent of temperature), yielding: P2s

30.786 kPa; P1s

T1s

415.34 K

71.426 kPa

[from Eq.(5)]

[from Eq.(6)]

x x x

P1s

71.552 kPa;

T1s 415.40 K;

P2s

29.798 kPa

After these values, the change in temperature is small and therefore additional iterations leads to no significant further change in the remaining values. We can now calculate the vapor phase mole fraction from y1

x1J 1P1s P

(0.5) u (0.592) u (71552 . ) 30

0.706

The whole process is repeated for a new liquid mole fraction. The following figure shows the computed T-x1-y1 diagram for this system at 30 kPa.

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Solutions Manual

430

V

Temperature/K

420

410

L+V 400

L 390

380 0.0

0.2

0.4

0.6

0.8

1.0

Cyclohexanone Mole Fraction

Similarly, with the data calculated we can easily draw the corresponding y1-x1 diagram, shown in the figure below.

Vapor Mole Fraction Cyclohexanone

1

0 0.0

0.2

0.4

0.6

0.8

1.0

Liquid Mole Fracton Cyclohexanone

As both figures show, this system has an azeotrope at T az | 421 K and for the composition x1az | 0.3 .

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Solutions Manual

7.

Assume g E

Ax1 x 2 , where A is a function of temperature. Then, RT ln J 1f

RT ln J 2f

A

10 T  273

A RT

But ln J 1f

0.15 

Because

At x1

x2

w( g E / T ) wT

h E

w( g E / T ) wT

10 Rx1 x 2

h E

(T  273) 2

T2

T2

0.5 and T = 333K, hE

641 J mol 1

' mix h

8. a) From the equation for H w we can obtain the infinite dilution partial molar enthalpy of water in sulfuric acid solutions at 293 K and 1 bar as:

Hwf

b)

lim Hw

xw o 0

lim Hw

x A o1

-41.44

kJ mol-1

The mixing process is schematically shown below.

A

HW

HA

W

W

Cooling coils

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Solutions Manual

Taking the liquid in the vessel as the system, a first law balance gives for this flow process: dQ  dW  H Adn A  H w dnw

dU

where work W is dW pressure. Then,

 PdV , done on environment by the rising liquid level, under constant

d (U  PV ) HA

dH

dQ  H Adn A  H w dnw

Integrating between initial state (empty vessel) and final state (full vessel), because H A (T , P) of the pure acid and H w H w (T , P) of the pure water are constant, we obtain H

But H

Q  n A H A  nw H w

or

Q

H  n A H A  nw H w

n A H A  nw Hw , and the equation above becomes Q

n A (H A  H A )  nw (H w  H w )

n A 'H A  nw 'H w

'H

(1)

where nA = 1 mol and nw = 2 mol in the final state. In Eq. (1), the quantity (H w  H w ) is given by the equation given in the data, because the reference state in that equation has been chosen to be pure water at system T and P: Hw  Hw



134 x 2A (1  0.7983x A ) 2

kJ mol -1

(2)

We need now to calculate the quantity (H A  H A ) , knowing (Hw  Hw ) . This can be done by using the Gibbs-Duhem equation. At constant T and P: x Ad H A  x w d H w

Ÿ

0

dHA



xw d Hw xA



1 xA d Hw xA

(3)

Differentiating Eq. (2), at constant T and P, we obtain: d Hw

(134 kJ mol -1 ) u

2 x A (1  0.7983x A ) 2  2(0.7983) x 2A (1  0.7983x A ) (1  0.7983x A ) 4

dx A

(4) 268 x A (1  0.7983x A )

3

dx A

(kJ mol 1 )

Therefore, from Eqs. (3) and (4), dH A

268(1  x A ) (1  0.7983x A ) 3

dx A

(kJ mol 1 )

Integrating Eq. (5) between composition xA and composition xA = 1 (pure acid) gives

(5)

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Solutions Manual

H A  H A (x A

1)

z

HA  HA

xA

(1  0.7983x A ) 3

1

LM 335.71(x  0.1263) OP MN (1  0.7983x ) PQ A

268(1  x A )

dx A (6)

xA

2

1

74.51(1  x A ) 2 (1  0.7983x A ) 2

(kJ mol) 1

We can now calculate the heat load Q in Eq. (1). Setting n = nA + nw = 3 mol, and using Eqs. (2) and (6) in (1), Q

ª 74.51x A (1  x A )2 134 x 2A (1  x A ) º n«  » (1  0.7983x A )2 »¼ «¬ (1  0.7983x A )2 ª 74.51x A (1  x A )(1  x A  1.7983x A ) º n« » (1  0.7983x A )2 ¬« ¼» ª 74.51x A (1  x A ) º n« » ¬ 1  0.7983x A ¼

(7)

(kJ mol 1 )

Substitution of n = 3 mol and xA = 1/3 gives the desired heat load:

Q

39.23 kJ

Q is negative because heat is removed from the system.

9. a) Yes, it’s possible. Slight positive deviations merely mean that the physical interaction between SO2 and C4H8 makes a larger contribution to the excess Gibbs energy than does the chemical interaction. b) E gSO

2  isobutene

E ! gSO

2 n



butene  2

because the tendency to complex (which tends to make gE negative) is stronger with n-butene-2. Steric hindrance in isobutene is larger than in n-butene-2.

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Solutions Manual

10.

The suggested procedure is to integrate numerically a suitable form of the Gibbs-Duhem equation. At low pressures, we may write the Gibbs-Duhem equation:

FG IJ  x2 FG wJ 2 IJ H K T J 2 H wx 2 K T

x1 wJ 1 J 1 wx1

0

By assuming ideal-gas behavior, J1

y1P

P1

x1P1s

x1P1s

wJ 1 wx1

1 x1P1s

FG wP1 IJ  P1 H wx1 K x12 P1s

Similarly, wJ 2 wx1



1

x2 P2s

FG wP2 IJ  P2 H wx2 K x22 P2s

Substituting we find x1 wP1 P1 wx1

Because P

P1  P2 , dP

x2 wP2 P2 wx 2

dP1  dP2 , then wP2 wx2

wP [ 1x P ] wx2 1 2 1 x1P2

'P2 'x 2

'P 1 x 2 P1 'x 2 1 x1P2

In different form:

For P-x data, we choose a small 'x2 (say 0.05) and integrate to find 'P2 and thus P2 . We obtain P1 by difference: P1 P  P2 . This method is described by Boissanas, quoted in Prigogine and Defay, Chemical Thermodynamics, page 346. It gives good agreement with experimental partial-pressure data for this particular system.

11.

For a binary system, the Wilson equation gives

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Solutions Manual

ln J1

ª /12 º / 21  ln( x1  x2 /12 )  x2 «  » ¬ x1  /12 x2 x2  x1/ 21 ¼

(1)

ln J 2

ª /12 º / 21  ln( x2  x1/ 21 )  x1 «  » ¬ x1  x2 /12 x2  x1/ 21 ¼

(2)

At infinite dilution these equations become

For J 1f

12.0, J 2f

ln J 1f

 ln /12  (1  / 21 )

(3)

ln J 2f

 ln / 21  (/12  1)

(4)

3.89, solve Eqs. (3) and (4) to find, / 21

0.6185

/ 12

0.1220

Assuming ideal-gas behavior and neglecting Poynting correction, we may write:

P

y1P

x1J 1P1s

(5)

y2 P

x2 J 2 P2s

(6)

x1J 1P1s  x 2 J 2 P2s

From Perry’s, the saturation pressures at 45qC are: P1s P2s

0.188 bar 0.0958 bar

To construct the P-x-y diagram: 1. Choose x1 (or x2) 2. Calculate y1 (or y2) from Eq. (5) and using Eqs. (1) and (2) 3. Calculate P from Eq. (7).

12.

The solution procedure would be: 1. Find P1s and P2s at each T. 2. At this low pressure, assume ideal-gas behavior and neglect Poynting correction:

(7)

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Solutions Manual

y1P

x1J1P1s

y2 P

x2 J 2 P2s

For J’s use the Wilson equation with two parameters: /12 and / 21 . Assume that (O 11  O 12 ) and (O 22  O 12 ) are independent of temperature. /12 and / 21 are, however, temperature-dependent as given by Eqs. (6-107) and (6-108).

3. Assume value of (O 11  O 12 ) and (O 22  O 12 ) and calculate the total pressure: Pcalc

x1J 1P1s  x2 J 2 P2s

4. Repeat; assuming new values. Keep repeating until Pcalc is very close to 0.5 bar for every point; that is until n

¦ (Pcalc  P)2

is a minimum

i 1

where n is the number of data points.

13. a)

2-Butanone: H H

H

H

C

C

C

C

H

O

H

H

H

Cyclohexane:

(CH2)6 6 groups CH2: R = 0.6744; Q = 0.540 Molecule

Group

Number

R

Q

2-Butanone

CH3CO CH3 CH2 CH2

1 1 1 6

1.6724 0.9011 0.6744 0.6744

1.488 0.848 0.540 0.540

Cyclohexane

b) We use UNIFAC activity coefficient equations to calculate J1 and J2 for the equimolar mixture at 75ºC (for a detailed example of a similar UNIFAC calculation see Chapter 8 of The Prop-

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Solutions Manual

erties of Gases and Liquids by R.C. Reid, J.M. Prausnitz, B. E Poling (4th. Ed., McGraw-Hill, 1988). We obtain: ln J 1

ln J 1comb  ln J 1res

0.01228  0.2595

0.27238

Ÿ

J1

1.31

ln J 2

ln J 2comb  ln J 2res

0.01415  0.3420

0.35615

Ÿ

J2

1.43

c) Using UNIFAC we can calculate the activity coefficients as a function of composition at 75ºC. Total pressure is calculated from x1J 1P1s  x 2 J 2 P2s

P

and the vapor-phase composition from x1J 1P1s

y1

Using P1s

Antoine

vapor

pressure

0.8695 bar and for cyclohexane

P2s

P

equations

at

75ºC,

we

obtain

for

2-butanone

0.8651 bar .

The following figures show the calculated results in the form of P-x1-y1 and y1-x1 diagrams. 1.3

L 1.2

Pressure/bar

1.1

L+V

L+V

1.0

V

0.9 0.8 0.7 0.6 0.0

0.2

0.4

0.6

2-Butanone Mole Fraction

0.8

1.0

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Solutions Manual

Vapor Mole Fraction 2-Butanone

1

0 0.0

0.2

0.4

0.6

0.8

1.0

Liquid Mole Fracton 2-Butanone

The table below shows the calculated activity coefficients from UNIFAC, vapor composition and total pressure. x1

J1

J2

y1

P/bar

0 0.2 0.4 0.5 0.6 0.8 1.0

4.38 2.27 1.51 1.32 1.18 1.04 1.00

1.00 1.07 1.27 1.42 1.62 2.19 3.10

0 0.351 0.447 0.485 0.528 0.660 1.0

0.8511 1.124 1.175 1.178 1.169 1.096 0.8695

Comparison of the calculated J’s in this table with those given in the data, indicate that the latter are actually UNIFAC predictions and not experimental data. In the tables, at x1 = 0 and x1 =1 the activity coefficients listed are, respectively, J 1f and J 2f . UNIFAC predicts J 1f which compares well with the experimental ebulliometry data at 77.6ºC,

14.

The UNIQUAC equation is gE

E E gcombinatorial  gresidual

J 1f

3.70.

4.38 ,

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Solutions Manual

E gcomb RT

E gres RT

) ) T T z x1 ln 1  x2 ln 2  ( x1q1 ln 1  x2 q2 ln 2 ) )1 )2 x1 x2 2

 x1q1 ln(T1  T2 W21 )  x2 q2 ln(T2  T1W12 )

)1

x1r1 x1r1  x2 r2

T1

x1q1 x1q1  x2 q2

W12

a exp( 12 ) T

W21

a exp( 21 ) T

The condition for instability of a binary liquid mixture is

F w 2 ' mix g I  0 GH wx 2 JK P,T

(1)

where ' mix g is the molar change in Gibbs energy upon mixing, or § w 2 gE ¨ ¨ wx 2 © 1

· § 1 1 ·  RT ¨  ¸  0 ¸ ¸ © x1 x2 ¹ ¹ P,T

Incipient instability occurs at § w 2 ' mix g · ¨ ¸ ¨ wx 2 ¸ © ¹ P,T

0

§ w 3' mix g · ¨ ¸ ¨ wx 3 ¸ © ¹ P,T

0

and

Given x1 , x 2 and all parameters, we could determine if Eq. (1) is satisfied. However, the procedure is long and tedious. It is easier to graph ' mix g over the composition range and to look for inflection points. For the data given, phase separation occurs at -40qC.

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Solutions Manual

xn-Hexane 0

0.4

0.2

0.6

0.8

0

-100

' mix g (J mol-1)

-200

-300

-400

-500 223.15 K 233.15 K 243.15 K 253.15 K 263.15 K

15.

If the two curves cross ' mix g / RT is zero because ' mix g

' mix h  T ' mix s

This is not possible, because 'mixg must always be negative for two liquids to be miscible.

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Solutions Manual

16.

To relate J if to Hi, j :

fi

Ji

xi fi0

then, J if

§ f lim ¨ i xi o0 ¨ x f 0 © i i

· ¸¸ ¹

§ f · lim ¨ i ¸ 0 fi © xi ¹ 1

Hi, j fi0

Assuming fi0 # Pis , J if

Hi, j

J 1f

H1,2

2 107 .

1869 .

J 2f

H 2,1

16 . 133 .

1203 .

Pis

then, P1s

P2s

Using the van Laar equations, ln J1

Ac § Ac x1 · ¨1  c ¸ B x2 ¹ ©

ln J 2

2

Bc § B c x2 · ¨1  c ¸ A x1 ¹ ©

2

we get ln J 1f

Ac

0.625

ln J 2f

Bc

0.185

To solve for vapor composition (assuming ideal vapor and neglecting Poynting corrections), y1P

x1J1P1s

y2 P

x2 J 2 P2s

P

y1P  y2 P

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Solutions Manual

At x1

x2

0.5, ln J1

ln J 2

0.625

§ 0.625 · ¨ 1  0.185 ¸ © ¹

2

0.185

§ 0.185 · ¨ 1  0.625 ¸ © ¹

2

0.033

Ÿ

J1

1.033

0.110

Ÿ

J2

1.116

Therefore,

17.

y1P

(0.5) u (1.033) u (1.07)

0.55 bar

y2 P

(0.5) u (1.116) u (1.33)

0.74 bar

P

0.55  0.74 1.29 bar

y1

0.55 / 1.29

y2

0.574

0.426

To estimate the vapor-phase composition, assume ideal vapor: yi P

xi J i Pis

(i=1, 2, 3)

Then, P

y1P  y2 P  y3 P

To find the activity coefficients, assume that g E / RT is given by a sum of Margules terms [Eq. (6-149)]. Then, ln J 1

A12 c x 22  A13 c x32  ( A12 c  A13 c  A23 c ) x2 x3

(1)

ln J 2

A12 c x12  A23 c x32  ( A12 c  A23 c  A13 c ) x1 x 3

(2)

ln J 3

A13 c x12  A23 c x 22  ( A13 c  A23 c  A12 c ) x1 x2

(3)

We can find A12 c , A13 c , A23 c from binary data.

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Solutions Manual

From (1-2) binary: A RT

ln J 1f A12 c

c At 300K, A12

§ 320 · (0.262) u ¨ ¸ © 300 ¹

ln(13 . )

0.262

at 320K

0.262

0.280 (assuming regular solution).

From (1-3) binary: At azeotrope x1 y1P

x1J1P1s

y3 P

x3 J 3 P3s

J1

ln J1

At x3

0.5,

A RT

c A13

y1 , x3

P / P1s

J3

y3

1.126

A 3 x2 RT

0.475.

From (2-3) binary: A 2 x RT 3

ln J 2

At incipient instability,

A RT c

2

or A R

With x1

x2

2T c

A RT

§ A ¨ © RT c

A23

1.80

·§ Tc ¸ ¨¨ ¹© T

· ¸ ¸ ¹

§ 270 · (2) u ¨ ¸ 1.80 © 300 ¹

x3 , from Eqs. (1), (2) and (3):

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Solutions Manual

ln J 1

0.0322

Ÿ

J1

0.968

ln J 2

0.409

Ÿ

J1

1506 .

ln J 3

0.475

Ÿ

J1

1607 .

y1P

(1 / 3) u (0.968) u (0.533)

0.172 bar

y2 P

(1 / 3) u (1506 . ) u (0.400)

0.201 bar

y3 P

(1 / 3) u (1607 . ) u (0.533)

0.286 bar

P = 0.172 + 0.201 + 0.286 = 0.659 bar Then,

18.

y1

0.261

y2

0.305

y3

0.434

Using the 3-suffix Margules equation, g E / RT

x1 x2 [ A  B( x1  x2 )]

we obtain ln J 1

( A  3B) x 22  4 Bx23

ln J 2

( A  3B) x12  4 Bx13

At infinite dilution, ln J 1f

A B

ln J 2f

A B

which gives A = 1.89

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Solutions Manual

B = 0.34 For instability to occur,

F w2g I GH wx 2 JK 1 E

 RT P,T

FG 1  1 IJ  0 H x1 x2 K

Rewriting g E as gE wg E wx1

RT [( A  B) x1  ( A  3B) x12  2Bx13 ]

RT [( A  B)  2( A  3B) x1  6 Bx12 ]

w2g E wx12

RT [2( A  3B)  12Bx1 ]

Thus, the condition for instability (at constant T) is: ª 1 1 º RT « 2 A  6 B  12 Bx1   »0 x1 1  x1 ¼ ¬

Finding the zeros of the function in brackets, x1

0.421 and

x1

0.352 in the range

Thus, instability at T is in the range

0.352 < x1 < 0.421

19. a)

At the azeotrope

FG wP IJ H wx A K T With g E / RT of the form g E / RT

0

Ax A x B , ln J A

Ax B2

ln J B

Ax 2A

0 < x1 < 1.

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Solutions Manual

Assuming an ideal vapor phase,

§ wP · ¨ ¸ © wx A ¹

x A J A PAs

yB P

x B J B PBs

P

x A J A PAs  x B J B PBs

P

x A exp( Ax B2 )PAs  x B exp( Ax 2A )PBs

PAs exp( Ax B2 )(1  2 x A x B A)  PBs exp( Ax 2A )(1  2 x B x A A)

PAs exp( Ax B2 ) Ax B2

yA P

PBs exp( Ax 2A )

ln( PBs / PAs )  Ax 2A

At 30qC,

Then, x A

PAs

0.235 bar;

PBs

0.658 bar;

A = 0.415

PAs

0.539 bar;

PBs

0.658 bar;

A = 0.415

PAs

1.119 bar;

PBs

1.367 bar;

A = 0.330

0.30.

At 50qC,

Then x A

0.26.

At 70qC,

Then x A

b)

0.20.

Assuming ideal vapor,

At azeotrope, x A

y A , xB

JA

y A P / x A PAs

JB

y A P / x B PBs

yB . Then

0

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Solutions Manual

JA

P / PAs

JB

P / PBs

Taking the ratio JA / JB ln( J A / J B )

PBs / PAs ln PBs  ln PAs 4050 4050  11.92  T T

12.12 

0.20 Because ln J A

Ax B2

ln( J A / J B )

A

ln J B

and

1 5  10 x A

A( x B2  x 2A )

Ax 2A , 0.2

because 0 < x A < 1

If |A| > 0.2 there is an azeotrope. The pure component boiling points are: tbA

67qC

tbB

61qC

In the range 61qC < t < 67qC, A is always larger than 0.2. Therefore, the azeotrope exists.

c) The enthalpy of mixing equation cannot be totally consistent since the expression for g E is quadratic in mole fraction and the expression for ' mix h is cubic. However, they may be close. To check this, we use the Gibbs-Helmholtz equation: wg E / RT wT

wg E / RT wT

h E RT 2

gE RT

A(T ) x A x B

x A xB

wA # x A x B (0.00425) wT

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Solutions Manual

Because h E

' mix h,

' mix h (1) RT

(323) u (0.00425) x A x B

1373 . x A xB

The other data indicate ' mix h (2) RT

(1020 .  0.112 x A ) x A x B

Looking at selected values: xA

'mixh(1)/RT

'mixh(2)/RT

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.123 0.220 0.288 0.330 0.343 0.330 0.288 0.220 0.123

0.093 0.167 0.221 0.256 0.269 0.261 0.231 0.178 0.101

The above shows the degree of inconsistency of the two sets of data.

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

7

Using regular-solution theory and data for A in CS2 we find the solubility parameter for A. Then, we predict vapor-liquid equilibria for the A/toluene system. Let B refer to toluene and C refer to CS2. From regular-solution theory,

RT ln J A

b

v A ) 2C G A  G C

g

2

(for A in CS2)

Further, assuming ideal vapor phase, we have yA P

JA

PA

xA J A PAs

8 (0.5) u (13.3)

1203 .

or ln J A

0.185

Then, 1/ 2

G A  GC

ª RT ln J A º r« » ¬ vA ¼

1 )C

with vA

200 cm3 mol-1 GA  GC

and

)C

0.234

r 6.30 (J cm 3 )1/ 2

107

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Solutions Manual

For liquid hydrocarbons, G is approximately 12-18 (J cm-3)1/2. Therefore, we take the smaller value. For A in toluene,

RT ln J A

b

2 vA)B GA  GB

g

2

or JA

RT ln J B

118 .

b

v B) 2A G A  G B

g

2

or JB

137 .

For ideal vapor, P

PA  PB

x A J A PAs  xB J B PBs

25.3 kPa

Hence,

2.

yA

0.31

yB

0.69

Excess properties ( h E , s E ) are defined in reference to an ideal (in the sense of Raoult’s law) mixture of pure components. The partial molar quantities h E and s E are the contributions to these excess properties per differential amount added to the solution. The “pure” acetic acid is highly dimerized, so as the first bits go into solution thse dimers must be broken up. This will require energy ( h1E ! 0 ) and will increase the entropy more than is accounted for by the ideal mixing term ( s1E ! 0 ). As x1 gets larger, some dimer will begin to exist in the solution, so these effects will diminish. Therefore, at small x1, the curves should look something like this:

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Solutions Manual

s1

h1

0

3.

0.5

0

x1

0.5

x1

The K factors for hexane (1) and benzene (2) (neglecting Poynting corrections and assuming ideal-vapor phase) are: J 1P1s

K1

P J 2 P2s

K2

P

where x1J 1P1s  x 2 J 2 P2s

P

Using regular-solution theory,

RT ln J 1 RT ln J 2

b v ) bG

g G g

v1) 22 G1  G 2 2

2 1

1

2

2

2

The volume fractions are )1

0.389

)2

132 .

J2

0.611

From the above equations, J1

1.08

P = (0.3)u(1.32)u(0.533) + (0.7)u(1.08)u(0.380) = 0.498 bar K1

KC6H14

(1.32) u (0.533) 0.498

K2

KC6H6

(1.08) u (0.38) 0.498

1.41

0.82

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Solutions Manual

4.

Ether and pentachloroethane hydrogen bond with each other (but not with themselves). Then, g E  0 . x

0

1

0

gE

5.

The relative volatility of A and B is D A,B

( yA / x A ) ( yB / x B )

Assuming ideal vapor phase and neglecting Poynting corrections,

At the azeotrope, x A

yA P

x A J A PAs

yB P

x B J B PBs

yA and

D A,B

1

J A PAs J B PBs

From regular-solution theory,

b bG

RT ln J A

2 vA)B GA  GB

RT ln J B

v B) 2A

A

 GB

g g

2

2

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Solutions Manual

Because v A

v B , J A J B , then PAs For the ternary mixture,

PBs .

v i ( G i  G )2

RT ln J i

G

¦ )i Gi i

As ) A Then

)B

0.2 and ) C

0.6 , G

ln J A

(100) u (14.3  17.2)2 (8.31451) u (300)

Ÿ

JA

1.40

ln J B

(100) u (16.4  17.2)2 (8.31451) u (300)

Ÿ

JB

1.03

J A PAs

D A,B

6.

17.2 (J cm 3 )1/ 2 .

J B PBs

JA JB

1.40 1.03

1.36

Assuming ideal vapor phase and neglecting Poynting corrections,

P

y1P

x1J 1P1s

y2 P

x2 J 2 P2s

x1J 1P1s  x 2 J 2 P2s

Using regular-solution theory,

Because v1

As v1

v2

RT ln J1

v1) 22 (G1  G2 )2

RT ln J 2

v 2 )12 (G1  G2 )2

v 2 , we can rewrite [Eqs. (7-25) and (7-26)]:

160 cm3 mol-1,

RT ln J1

v1x22 (G1  G2 )2

RT ln J 2

v 2 x12 (G1  G2 )2

(1)

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Solutions Manual

J1

exp(0.616x22 )

J2

exp(0.616 x12 )

Substitution in Eq. (1) gives P

0.533exp(0.616 x22 )  0.800 exp(0.616 x12 )

At the azeotrope,

FG wP IJ H wx1 K T

0

Thus, after differentiation, 0.616 x 22

0.40547  0.16 x12

Solving for x1, x1

7.

0.171

Neglecting vapor-phase non-idealities and Poynting corrections, the total pressure, P, is P

x A J A PAs  x B J B PBs

Because the two fluids are similar in size, simple and nonpolar, we can assume that J’s are given by two-suffix Margules equations:

As xA

xB

ln J A

A 2 x RT B

ln J B

A 2 x RT A

0.5, P

0.667

LM FG A IJ OP u b0.427  0.493g N H 4RT K Q

0.5 u exp

A = 4696 J mol-1 From Eq. (6-144) of the text, Tc

A 2R

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Solutions Manual

Tc

282 K

If one considers the effect of non-randomness (based on the quasichemical approximation), Eq. (7-110) gives A 2.23R

Tc

(assuming that

A 2R

w 2k

253 K

constant and that the temperature dependence is given by ln J propor-

tional to 1/T). Thus, random mixing predicts a value higher than that given by quasichemical theory. The observed consolute temperature is likely to be lower than both.

8. V

yi = ?

Let: 1 = benzene z1= z2= 0.5

2 = n-butane

L

There are three unknowns: x1 , y1, and V / F . To solve for them, we use two equilibrium equations and one mass balance. Assuming ideal vapor and neglecting Poynting corrections: y1P

(1  y1 )P

z1

x1J 1P1s

(1  x1 )J 2 P2s

FG V IJ y1  FG1  V IJ x1 H FK H FK

Using regular-solution theory,

RT ln Ji

vi )2j (Gi  G j )2

xi = ?

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Solutions Manual

with v1 = 92 cm3 mol-1 and v 2 = 106 cm3 mol-1. Then, J1

exp(0.828) 22 )

J2

exp(0.950)12 )

Substitution gives y1 (1  y1 )

0.368x1 exp(0.828) 22 )

(1)

4.76(1  x1 ) exp(0.950)12 )

(2)

FG V IJ y1  FG1  V IJ x1 H FK H FK

0.5

(3)

) 1 and ) 2 are related to x1 and x2 by Eqs. (7-25) and (7-26). To solve Eqs. (1), (2), and (3) for x1, y1, and V / F, assume first that J i x1

y1

0.856

0.315

V F

1. This gives,

0.658

A second approximation ( J i z 1 ) gives

x1

9.

xC6H6

0.94

y1

0.35

V F

0.741

As derived in Sec. 7.2 of the text, the regular-solution equations can be written in the van Laar form gE

Ax1 x 2 A x1  x 2 B

(1)

where parameters A and B are related to pure-component liquid molar volume and solubility parameters as follows [Eqs. (7-38) and (7-39)]: A v A (G A  G B ) 2 (2) B

v B (G A  G B ) 2

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Solutions Manual

Substituting the given liquid molar volumes and solubility parameters, we obtain A

(120 cm3 mol 1 ) u [(18  12) 2 J cm 3 ]

4320 J mol 1 (3)

B

(180

cm3 mol 1 ) u [(18  12) 2 J

cm 3 ]

6480 J

mol 1

As discussed in Section 6-12, the temperature and composition at the consolute point are found from solving:

FG w ln aA IJ FG w2 ln aA IJ H w xA K T ,P H w xA2 K T ,P

0

(4)

Upon substitution of Eq. (1) into Eq. (4), the results are given in Eq. (6-146) in the text: c xA

[( A / B) 2  1  ( A / B)]1/ 2  ( A / B) 1  ( A / B)

(5) Tc

c c 2 xA (1  x A )( A2 / B) c c 3 R[( A / B) xA  (1  x A )]

where superscript c denotes consolute. Substituting Eq. (3) into Eq. (5), we finally obtain

10.

c xA

0.646

Tc

328 K

For each phase we choose the standard-state fugacity for cyclohexane as its pure subcooled liquid at 25ºC. The equation of equilibrium is x 3(1) J (31)

x 3(2) J (32)

(1)

where subscript 3 denotes cyclohexane and superscripts (1) and (2) denote, respectively, carbon disulfide phase and perfluoro-n-heptane phase. Rearrangement of Eq. (1) gives x (1) K{ 3 x 3(2)

J (32) J (31)

(2)

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Solutions Manual

Because phase (1) contains only carbon disulfide and a trace amount of cyclohexane, whereas phase (2) contains only perfluoro-n-heptane and a trace amount of cylohexane, J (31) and J (32) are essentially activity coefficients at the infinite-dilution limit of cyclohexane.

Hence, we can write

K{

x3(1) x 3(2)

LM J (32) OP f MN J (31) PQ

(3)

where superscript f denotes the infinite-dilution limit of cyclohexane. From the regular-solution theory [Eq. (7-37)], J (31) º ln ª J (1) ¬ 3 ¼

f

and J (32)

f

v3 (G1  G3 )2 RT

f

v3 ( G 2  G 3 )2 RT

f

are given by:

(4) º ln ª J (2) ¬ 3 ¼

Substituting the pure-component liquid molar volumes and solubility parameters, we obtain º ln ª J (1) ¬ 3 ¼

f

(109 cm3 mol-1 )

u ª(20.5  16.8)2 J cm-3 º ¼ (8.314 J mol-1 K-1 ) u (298 K) ¬

0.602 (10)

º ln ª J (2) ¬ 3 ¼

f

3

-1

(109 cm mol )

u ª(12.3  16.8)2 J cm-3 º ¼ (8.314 J mol-1 K-1 ) u (298 K) ¬

0.891

Substituting Eq. (5) into Eq. (3), we have

K

11.

1.34

From the definition of the solubility parameter, G, G2

'ucomplete vaporization vL

[Complete vaporization means going from saturated liquid to ideal gas at constant T.] Then,

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Solutions Manual

F h0  h L I RT  RT (1  z L ) GH RTc JK c

G2

z L RT Ps

LM h0  h L OP MM RTL c  1  1L PPPRs z MM z TR PP N Q

2

G Pc

Because

h0  h L RTc

f (TR , Z )

and zV

zL

f (TR , Z)

PRs

f (TR , Z)

f (TR , Z)

for TR d 1,

G2 Pc

b g

b g b

f (0) TR  Z f (1) TR  higher terms

g

[Reference: Lyckman et al., 1965, Chem. Eng. Sci., 20: 703].

12. a) Pure methanol is hydrogen-bonded to dimers, trimers, etc. In dilute solution (in iso-octante), methanol is a monomer. For an order-of-magnitude estimate, we can assume that, to make a monomer, approximately one hydrogen bond must be broken. Thus h E = 12 kJ mol-1. b) From solubility parameters we get (roughly) an endothermic heat of 263 J mol-1. The molar specific heat is (roughly) 125 J K-1 mol-1 . Thus, 't | 2D C. c)

We want a Lewis acid that can hook on to the double bond in hexene. Good Solvents are: Dimethyl sulfoxide Sulfur Dioxide Acetonitrile

U| V| W

Strong Lewis acids

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Solutions Manual

Poor Solvents are: Ammonia

UV W

Aniline

Weak Lewis acids

It is also important that the solvent should not be something that prefers self-interaction than that with hexene molecules. Strongly hydrogen-bonded liquids (e.g. water and most alcohols) would therefore be poor solvents.

13.

Let (HA) be the acid. In ionized form, H+ + A-

HA

Hexane (HA)

H

(HA)

A- + H+

W

Water

Equilibrium constants are defined as

K1

K2

bHAg bHAg

W

(H )(H ) (HA)W

H

(A )2 (HA)W

In hexane: CH = (HA)H In water:

CW = (HA)W + (A-) = (HA)W +

b g

K 2 HA

b g

W

b g

K1 HA H  K1K 2 HA H CW

Thus,

K1C H  K1K 2C H

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Solutions Manual

CH

FG1  K H

1

CW

1

K1K2

CH CW

IJ K

with

1

a

K1K2 and K1

b

14. Benzene AB

1/3 AT

AW Water

Assume constant distribution coefficient and “reaction” equilibrium. AB AW

K1

AT

b AB g3

K2

In water: CW

AW

In benzene: CB CB

CB CW

AB  3 AT

3 K1C W  3K 2 K13C W

2 K1  3K 2 K13C W

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Solutions Manual

Thus, plot

CB 2 . as a function of CW CW

Slope is 3K 2 K13 and intercept is K1.

15. a) Pure CH2Cl2 and acetone do not hydrogen bond themselves but some hydrogen bonding is likely to occur between dissimilar pairs, which explains the negative deviations from Raoult’s law observed for this system. Pure methanol is highly hydrogen bonded. However, in dilute solutions of CH2Cl2, methanol exists primarily as monomer. Hydrogen bonding between methanol and CH2Cl2 is likely to be weak. (Note that at infinite dilution, activity coefficient J 1f indicates the effect on a molecule 1 when surrounded by molecules of the other component).

b) Nitroethane has a large dipole moment. Both n-hexane and benzene are non-polar but, due to S electrons, benzene is more polarizable. Therefore, we expect nitroethane/benzene interactions to be stronger than those for nitroethane/n-hexane. Thus Jnitroethane in n-hexane is larger than that in benzene. c) Both CHCl3 and methanol are polar and slightly acidic. Although methanol has a slightly higher dipole moment, CHCl3 is likely to solvate the coal tar more easily because methanol tends to form strong hydrogen bonds with itself.

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

8

Given

FG1  1IJ  F H rK

ln *1f

For r >>1, ln *1f

1 F

If F = 0.44, ln *1f

*1f

Ÿ

1.44

4.22

As defined, *1f

a1 )1

a1 10 4

4.22

Ÿ

a1

4.22 u 10 4

Because

a1 P1

P1 P1s

(4.22 u 10 4 ) u (4.49)

For a non-volatile polymer, P2

0.0019 bar

0. Therefore, P # P1

0.0019 bar

121

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Solutions Manual

2.

We can use the data for the Henry’s-law region to evaluate the Flory interaction parameter, F, and then predict results at higher concentration. Let: 1 = solvent 2 = polymer wi = weight fraction ) i = volume fraction Then, w1 U1

v1 v1  v 2

)1

w1 w2  U1 U2

In the Henry’s law region, w1 o 0, a1 )1

w2 o 1

f1

w1H1,2

)1 f10

)1 f10

But, as w1 o 0, )1

w1U 2 / U1

)2

1

Then, a1 )1

If f10

U1H1,2 U2 f10

P1s ,

a1 )1

f1

U1H1,2

)1 f10

U2 P1s

(0.783) u (18.3) (111 . ) u (3340 / 760)

From Flory-Huggins theory (r is large), a ln 1 )1

) 2  F) 22

If )1 o 1 ,

ln(2.94) 1  F

Ÿ

F

0.078

2.94

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Solutions Manual

At higher concentrations (w1 = 0.5), assume F z F(w1 ) :

)2

(0.5) u (1/1.11) (0.5) u (1/1.11)  (0.5) u (1/ 0.783)

()1

0.414

0.586)

Then, a ln 1 )1 a1 )1 P | f1

0.414  0.078 u (0.414) 2

153 .

a1

or

a1 f10 | a1P1s

0.898

(0.898) u (3340)

3000 torr

3000 torr | 3.9 bar

P

3. a)

The generalized van der Waals partition is given by [Eq. (8-39)] Q T , V , N

1 § Vf · ¨ ¸ N ! ©¨ / 3 ¹¸

N

ª

Eo · º ¸» © 2kT ¹ ¼

>qext (V )@N >qint (T )@N «exp §¨  ¬

N

(1)

Following the discussion on pages 442 and 443 of the textbook, we further have Vf /3 Eo 2

qext

§ Vf · ¨¨ 3 ¸¸ ©/ ¹

rc

rsK 2v

(2) Vf

v

ª§ v ·1/ 3 º Wr v* «¨ ¸  1» «© v* ¹ » ¬ ¼ V Nr

Substituting Eq. (2) into Eq. (1) yields

3

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Solutions Manual

 ln N ! Nrc ln / 3  Nrc ln(W r v* )

ln Q

(3) ª§ v ·1/ 3 º NrsK  3Nrc ln «¨ ¸  1»  N ln qint  * 2vkT «© v ¹ » ¬ ¼

Because the first, second, third and fifth terms on the right-hand side of Eq. (3) are only functions of temperature, the equation of state is given by 1 § w ln Q · ¨ ¸ Nr © w v ¹ N ,T

§ w ln Q · ¨ wV ¸ © ¹ N ,T

P kT

(4) 1 ª 3Nrc § 1 v 2 / 3 · NrsK § 1 « ¨ ¸ ¨ Nr « (v 1/ 3  1) ¨ 3 v*1/ 3 ¸ 2kT © v 2 © ¹ ¬

·º ¸» ¹ »¼

We can rewrite Eq. (4) as v 1/ 3

P v T

 1/ 3

v

1  T v 1 

(5)

where the reduced properties are defined by T

T

T* P

P

*

P

2v*ckT sK 2

2v * P sK

(6)

v

v

v*

Equation (5) is the Flory equation of state [Eq. (8-45) of the textbook].

b)

The configurational partition function [Eqs. (8-82) and (8-83] Q

§ E · QC exp ¨  ¸ © kT ¹

(7) QC

(constant) N

/ v* ) N

(V 1 N 0 ! N ! (V / v* ) N (r 1)

where

V

v*

N0  r N

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Solutions Manual

z E =  H Nr 2

ª rN º « » «¬ (V / v* ) »¼

(8)

Combining Eqs. (7) and (8) gives ln Q

ln(constant) 

§V · V ln ¨ ¸  v* © v * ¹ v* V

§V ·  ln N 0 !  ln N !  N (r  1) ln ¨ ¸ © v* ¹

(9)

§V ·  H* (v*r N )2 ¨ ¸ V 2 © v* ¹

where H*

z H 2 kT

(10) ª§ V · º ln N 0 ! ln «¨ ¸  r N » ! ¬© v* ¹ ¼ ª§ V · º ª§ V · º ª§ V · º «¨ * ¸  r N » ln «¨ * ¸  r N »  «¨ * ¸  r N » ¬© v ¹ ¼ ¬© v ¹ ¼ ¬© v ¹ ¼

The equation of state is thus given by P kT

§ w ln Q · ¨ wV ¸ © ¹ N ,T § V · § V · v* (1/ v* ) 1 ln ¨ ¸  ¨ ¸  V v* v* © v * ¹ © v* ¹ 1

(11) 

1 v*

1 §V · §V · ln ¨ rN ¸¨ rN ¸  V * * * ¹ ©v ¹ ©v rN v * v

 N (r  1)

We can rewrite

1/ v*

§ V 2 ·  H* (v*r N )2 ¨  ¸ ¨ v* ¸ V / v* © ¹ 1/ v*

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Solutions Manual

§ PV · § 1 · ¨ kT ¸ ¨ r N ¸ © ¹© ¹

P v T V § · ¨ ¸ * §V · §V · v ¸ v ln ¨ ¸  v ln ¨  r N ¸  (v  1) ¨ ¨ V rN ¸ © v* ¹ © v* ¹ ¨ * ¸ ©v ¹ 1 * rN  v  1   H r V / v*

(12)

1 ª § 1 ·º 1  1  v ln ¨ 1  ¸ »  r «¬ © v ¹ ¼ T v

where T

P

v =

T T*

T z H / 2k

P

P

P*

z H / 2v*

(13)

v v*

Equation (12) is the Sanchez-Lacombe lattice-fluid equation of state [Eq. (8-84) of the textbook].

4.

The Flory-Huggins equation for the activity of the solvent [Eq. (8-11)] is ln a1

2 § 1· ln(1  )*2 )  ¨ 1  ¸ )*2  F)*2 r © ¹

Conditions for incipient instability give [analogous to Eqs. (6-141) and (6-142)]: § w ln a1 · ¨ ¸ ¨ w)* ¸ © 1 ¹ P,T § 2 · ¨ w ln a1 ¸ ¨¨ *2 ¸¸ © w)1 ¹ P,T

Equivalently, we have

0

0

(1)

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Solutions Manual

§ w ln a1 · ¨ ¸ ¨ w )* ¸ 2 ¹ P,T ©

0

(2) § 2 · ¨ w ln a1 ¸ ¨¨ * 2 ¸¸ © w) 2 ¹ P,T

0

because )1*  )*2

1

Substituting Eq. (1) into Eq. (2), we obtain 1 c 1  )*2

c § 1·  ¨ 1  ¸  2Fc )*2 © r¹

§ ¨ 1 ¨¨ *c © 1  )2

0

2

· ¸  2F c ¸¸ ¹

0

where superscript c stands for critical. Hence, we obtain )*2

Fc

5.

c

1 1  r1/ 2 1§ 1 · ¨ 1  1/ 2 ¸ 2© r ¹

2

The Flory-Huggins equation for the activity coefficient of HMDS (1) [Eq. (8-12)] with F is ln J1

0

ª § 1· º § 1· ln «1  ¨ 1  ¸ )*2 »  ¨ 1  ¸ )*2 ¬ © r¹ ¼ © r¹

Using data in Table 8-5 of the text, calculated molecular characteristic volumes V * , ratios of molecular segments r and activity coefficients of HMDS (at )*2 lowing table:

0.8 ) are given in the fol-

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Solutions Manual

Substance

V*

v*sp Mn

r

(cm3 mol-1)

HMDS PDMS PDMS PDMS PDMS PDMS PDMS PDMS

3 10 20 100 350 1000 f

V2* / V1*

 3.21 5.13 8.09 21.61 34.07 40.70 f

162.30 521.53 832.89 1313.8 3507.8 5530.1 6604.8 f

ln J1 ()*2

0.8)

 -0.249 -0.389 -0.528 -0.677 -0.722 -0.735 -0.809

As we increase the molecular weight of PDMS, ln J1 becomes more negative. J1 is smaller than unity and increasingly deviates from unity as the molecular weight of PDMS is increased. This example illustrates the effect of differences in molecular sizes of HMDS and various PDMS with F 0 (Fig. 8-3).

6.

The flux of gas i through the membrane is given by, Eq. (8-118) Ji

Di G G G SiF PiF  SiP PiP GM





(1)

Because solubility coefficients for both O2 and N2 in the feed and permeate are assumed to be equal and the permeate pressure is vacuum, Eq. (1) reduces to

Ji

Di SiG PiF GM (2)

Di SiG GM

yi PF

where yi and PF ( PF 2 u 105 Pa) denote, respectively, the mole fraction of component i and the total pressure of the feed. For the feed mixture (air) we have

yO2 # 0.21 (3)

yN2 # 0.79 Substituting Eq. (3) and the given data for membrane thickness, solubility and diffusion coefficients into Eq. (2), the corresponding fluxes of O2 and N2 are

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Solutions Manual

J O2

0.148 u 10 5

m3 m-2 s-1

J N2

0.197 u 10 5

m3 m-2 s-1

The separation factor is defined by [Eq. (8-121)] G DO2 SO

2

D O2 / N 2

G DN2 SN 2

2.79

Although DO2 / N2 ! 1 , the net flux of N2 is larger than that of O2 due to the difference in partial pressures in the feed.

7.

If the feed pressure were low, we could use Eq. (8-112) to calculate J1 , the flux of carbon dioxide, and J2 , the flux of methane. Equation (8-113) then gives the composition (y) of the permeate. However, because the pressure of the feed is high, we must allow for the effect of pressure on nonideality of the gas phase. Equation (8-111) is J1

D1 M c1F  c1MP GM





(1)

where c1M is the concentration of carbon dioxide in the membrane; subscripts F and P refer to feed and permeate. To find c1MF , we use the equilibrium relation

( y1PM1 )F

ª § v1P · º M « H1c1 exp ¨ ¸» © RT ¹ ¼ F ¬

(2)

where H1 and v1 are Henry’s constant and partial molar volume for carbon dioxide in the membrane, both at 300 K and 100 bar. Fugacity coefficient M1 is given by the virial equation of state, truncated after the second virial coefficient [Eq. (5-33)]:

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Solutions Manual

§ 2 · P ¨2 yi B1i  Bmixt ¸ ¨ ¸ RT © i 1 ¹

¦

ln M1

(3) 2

2

¦¦ yi y j Bij

Bmixt

i 1j 1

Substituting the given temperature, pressure and second virial coefficients into Eq. (3), we obtain M1

0.729

Substituting Eq. (4) and all other given data into Eq. (2) yields 0.314 mol L-1

c1MF

(4)

To find c1MP we use the equilibrium relation ( y1PM1 )P

( H1c1M )P

(5)

y1P 19

mol L1

(6)

where PP 1 bar and M1P 1. Hence, Eq. (5) reduces to c1MP

The quantity y1P is an unknown in this problem. Substituting Eqs. (4) and (6) into Eq. (1) gives

§ mol · J1 ¨ ¸ ¨ cm2 s ¸ © ¹

y 5 u 106 § · 0.314 u 103  1P u 10 3 ¸ ¨ 0.1 © 19 ¹

(7)

Applying the same procedure for methane (2), we obtain § mol · J2 ¨ ¸ ¨ cm2 s ¸ © ¹

y 50 u 10 6 § · 0.155 u 10 3  2 P u 10 3 ¸ ¨ 0.1 © 50 ¹

(8)

The (steady-state) mole fraction of carbon dioxide in the permeate is given by [Eq. (8-113)] y1P

J1 J1  J2

(9)

Further, the mass conservation gives y1P  y2 P

1

(10)

Substituting Eqs. (7), (8) and (10) into Eq. (9) gives y1 P = 0.168 for carbon dioxide in the permeate. Therefore, for methane in the permeate, y2 P = 0.832.

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Solutions Manual

8. a)

Flux of water through the membrane is given by Eq. (8-128): Jw

ª v w ( PF  PP ) º ½ § permeability · ­ L L ¨ thickness ¸ u ® xwF  xwP exp «  »¾ RT © ¹ ¯ ¬ ¼¿

(1)

L where xwP 1 (pure water in the permeate); PP Pws 0.0312 atm. To calculate concentration of water in the feed, we use

( Pw )F

( xwL Pws )F

(2)

with Pws

0.0312 atm (1  0.0184) u (0.0312) atm

PwF

Therefore, we obtain L xwF

0.9816

Because the permeate is pure water, we obtain vw | vw

18.015 g mol 1 0.997 g cm 3

18.069 cm 3 mol 1

The feed pressure is thus given by 7.2 u 10 4 g cm 2 s1

§ 2.6 u 10 5 g cm cm 2 s1 · ¨ ¸ ¨ ¸ 10 u 10 4 cm © ¹ ª (18.069 cm3 mol 1 ) u ( PF  0.0312 atm) º °½ °­ u ®0.9816  (1) u exp «  »¾ 1 1 °¯ ¬« (82.06 atm L K mol ) u (298.15 K) ¼» ¿°

Therefore, the feed pressure is PF = 63.93 atm

b) 1u 106 gallons/day

3785.4 m3 /day

0.0438 m3 /s

43.67 kg/s

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Solutions Manual

Flux

7.2 u 10 4 g cm 2 s1

43.67 u 103 g s1 ( A cm 2 )

where A is the membrane area needed. Solving Eq. (3) for the area, we get A

6.07 u 10 7 cm 2

6.07 u 10 3 m 2

6.5 u 10 4 ft 2

(3)

S O L U T I O N S

T O

P R O B L E M S C H A P T E R

1.

9

The solubility product is the equilibrium constant for the reaction Ag+ + ClAqueous solution

AgCl Solid defined as K SP

(a  )(a  ) Cl Ag

being the standard states the pure solid AgCl and the ideal dilute 1-molal aqueous solution for each ion.

a)

Let S be the solubility of AgCl in pure water, in a  Ag K SP

J

Ag

S

(a  )(a  ) Cl Ag

mol AgCl (a molality). kg water

a  Cl (J

Ag

J S Cl

)(J  ) S 2 Cl

(J r S ) 2

(1)

Because the solution is very dilute, J r | 1 and S | K SP and is of the order of 10-5 molal and therefore the ionic strength is also very low: I | 131 . u 10 5 molal. Therefore we may apply the Debye-Hückel limiting law. I

1 [m u (1) 2  m u (1) 2 ] 2

m

S

Using Eq. (9-50a) of the text,

133

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Solutions Manual

log J (rm)

0.510 u (1) u (1) I 1/ 2

0.510 S 1/ 2

(2)

We now replace J r given by Eq. (2) into Eq. (1) and solve for the solubility S: 1/ 2

[(10 0.510 S

172 . u 10 10

)S ]2

S = 1.31 ×10-5 mol kg-1

( J r | 10 . )

b) With the addition of NaCl the ionic strength increases and we need to evaluate J r because now the solution is not very dilute and therefore we don't have J r | 10 . . Let S be the new solubility of AgCl in this aqueous solution that contains NaCl. The molalities are m  Ag

S

m  Cl

S  0.01

The total ionic strength (due almost exclusively to NaCl because S is small) is I

1 [S u (1) 2  S u (1) 2  0.01 u (1) 2  0.01 u (1) 2 ] | 0.01 mol kg 1 2

We use in this case the extended limiting law [Eq. (9-52)] with AJ

ln J r



1174 . u (0.01)1/ 2 1  (0.01)1/ 2

Ÿ

Jr

1174 . kg1/2 mol 1/2 :

0.90

As in a), K SP

(a

Ag

)(aCl  )

(J

Ag

)( J Cl  )(m

Ag

)(mCl )

( J r )2 (S )(S  0.01) 1.72 u 10 10

or substituting J r

0.90 ,

S (S  0.01)

212 . u 10 10

Because S is small and S << 0.01, we obtain S | 2.12 ×10-8 mol kg-1. The addition of NaCl reduces the AgCl solubility from 1.31u10-5 mol kg-1 [as calculated in a) for pure water] to 2.12u10-8 mol kg-1 (in a 0.01 molal NaCl aqueous solution). This is the common ion (“salting out”) effect.

c)

Similarly, let S be the new AgCl solubility. Molalities are: m

Ag

S

m  Cl

S

Again, the ionic strength is almost exclusively due to NaNO3:

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Solutions Manual

I | 0.01 mol kg 1 K SP

(a

Ag

)(aCl  )

( J r )2 ( S )2

Jr

and

(J

Ag

0.90

)( J Cl  )(m

Ag

)(mCl )

1.72 u 10 10

S = 1.46 ×10-5 mol kg-1

Compared to the solubility of AgCl in pure water, the solubility of AgCl in a 0.01 molal NaNO3 aqueous solution increases by roughly 10%, because the higher ionic strength reduces the activity of Ag+ and Cl- ions and causes more AgCl to dissolve (“salting in” effect).

2.

The solubility product is the equilibrium constant for the reaction (PbI2 is a 1-2 electrolyte) Pb2+ + 2IAqueous solution

PbI2 Solid defined as K SP

(a 2 )(a  ) 2 Pb I

(J r ) 3 S 3

being the standard states the pure solid PbI2 and the ideal dilute 1-molal aqueous solution for each ion. In the above equation, S is the solubility of PbI2 in pure water, in

mol PbI 2 (a mokg water

lality). Because the solution is very dilute, J r | 1 and K SP

(S ) 3

(1.66 u 10 3 ) 3

4.57 u 10 9

For the solution with KI the ionic strength increases and we need to evaluate J r because now the solution is not very dilute and therefore we may not have J r | 10 . . Let S be the new solubility of PbI2 in this aqueous solution that contains KI. The molalities are m 2 Pb

S

m I

2S  0.01

m  K

0.01

The total ionic strength is I

1 [2S u (1) 2  S u (2) 2  0.01 u (1) 2  0.01 u (1) 2 ] 2

(3S  0.01) mol kg 1

We use in this case the extended limiting law [Eq. (9-52)] with AJ

1174 . kg1/2 mol 1/2 :

(1)

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Solutions Manual

ln J r



1174 . u (3S  0.01)1/ 2 1  (3S  0.01)1/ 2

(2)

Simultaneous solution of Eqs. (1) and (2) gives Jr S

3.

0.88 1.89 u 10 3 mol kg 1

The solubility product for PbI2 in an aqueous solution is 3log mr  3log J r

log K SP

(1)

where

With I

1 2 (2 m  2m) 2

m

mPb2 

m

mI

mr

ª m(2m)2 º ¬ ¼

m

(2)

2m 1/ 3

1.587m

3m Eq. (9-50a) gives log J r

0.510 u (2) u (3m)1/ 2

(3)

With m 1.66 u 10 3 mol kg-1 we obtain log K SP

7.953

(4)

For the solutions containing sodium chloride or potassium iodide saturated with PbI2, we have log K SP

7.953 3 u log mr  0.510 u (2) u I 1/ 2

For the NaCl solution, we write

(5)

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Solutions Manual

m Pb2

mr

I

m

m I

ª m u (2 m )2 º ¬ ¼

2m

1/3

mNa

m Cl

0 .0 1

1 .5 8 7 m

1 ª 2 2 m  2 m  2 u ( 0 . 0 1) º ¼ 2 ¬

3 m  0 .0 1

Substitution in Eq. (1) yields m

1.89 mol kg -1

For the KI solution, we write m Pb 2 

mr

I

m

m I

2 m  0.0 1

ª m u (2 m  0.0 1) 2 º ¬ ¼

mK 

0.0 1

1/ 3

1ª 2 (2 ) u m  2 m  2 u ( 0.0 1) º ¼ 2¬

3 m  0 .0 1

Substitution in Eq. (1) yields m

0.24 mol kg  1

For both systems, calculated and experimental values are in good agreement. The large decrease in PbI2 solubility in the KI solution follows because all iodide ions are included in mr . This reduction in solubility is called the common-ion effect.

4.

The dissociation of acetic acid is represented by CH3COOH

H+ + CH3COOH-

or schematically AH The dissociation constant is

H+ + A-

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Solutions Manual

K

(aH )(aH )

1.758 u 10 5

aAH

(m) Q (mQ  )(mQ  ) ( J r ) (m) mAH

2 (mH )(mA ) ( J (m) r )

J (m) AH

(m) mAH

J (m) AH

Designating by D the extent of ionization, and by m the stoichiometric molality of acetic acid, in dilute solutions we may assume a  a  D and aAH m  D. Further, the activity H A of undissociated acetic acid approaches its molality at infinite dilution. Assuming that the activity coefficients are unity (very dilute solutions) we have K

D2 mD

(D )(D ) mD

1758 . u 10 5

For a m = 10-3 molal aqueous solution, the equation above gives D The fraction of acetic acid ionized is: 124 . u 10 4

D m

5.

1.24 u 10 4 .

0.124

1 u 10 3

In SI units, the Debye length is defined by Eq. (9-47) of the text: N 1

F RTH o H I 1/2 GH 2d N 2 e2 I JK r

s

A

where ds is the density of the solvent in kg m-3. For water at 25ºC, ds = 997 kg m-3. The ionic strength is I | 0.001 mol kg-1 for the 0.001 M solution and I | 0. 1 mol kg-1 for the 0.1 M solution. Substitution of values gives the values for the Debye length N-1 (in nm) presented in the following table: Solution

Water

Methanol

0.001 M 0.1 M

9.6 0.96

6.1 0.61

We see that N-1 decreases ten times with a hundredfold increase in concentration. For the solutions at higher concentrations, shielding effects are more important and N-1 is low. Further, the Debye length increases with increasing dielectric constant: when Hr is large (as in water), the ionic atmosphere is weak and the coulombic interactions are strongly reduced.

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6. a)

The molality of NaCl in seawater is

mNaCl

§ 3.5 · ¨ ¸ mol NaCl © 58.5 ¹ (100  3.5) g water § 3.5 · ¨ 58.5 ¸ © ¹ u 1000 mol (100  3.5) kg 0.620 mol/kg water

The ionic strength is given by I

1 (m z2  m  z2 ) Cl Cl  2 Na  Na 

m

0.620 mol / kg water

which is a relatively high value. Also,

Q

Q Na  QCl

2

The molar volume of water at 25ºC is Mw dw

vw

18 g / mol 0.997 g / cm 3

18.05 cm 3 mol -1

To obtain the molal osmotic coefficient, I, we need an expression for J r in terms of the ionic strength I (= m). Since solution is not dilute, we use Bromley’s model:

ln J r

with, for NaCl at 25ºC, AJ



AJ I 1/ 2 1  I 1/ 2



(0.138  138 . B)I (1  15 . I )2

1.174 kg1/2 mol 1/2 and B

 2.303BI

0.0574 kg1/2 mol-1/2 .

From Eq. (9-11) of the text (reminding that in this case I = m),

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Solutions Manual

I 1

1



1 I I d ln J r I 0

³

A I I 1/ 2 I dI  (0.138  1.38B) I 0 2(1  I 1/ 2 )2 (1  1.5I )2

³

I (0.138  1.38 B) I I B dI  dI 2 2.303 0 I 0 (1  1.5I )

³

³

Performing the integrations, we obtain: I 1



Aª 1 º 1  I 1/ 2  2 ln(1  I 1/ 2 )  » I «¬ 1  I 1/ 2 ¼

º IB (0.138  1.38 B) ª 1  3I 1  ln(1  1.5I ) »  2.303 « 2 I 1.5 1.5 2 ¬« (1  1.5I ) ¼»

Substituting AJ = 1.174, B = 0.0574, and I = 0.62 we obtain I = 0.924.

b) From the expression that relates the osmotic pressure, S, to the molal osmotic coefficient, I, we obtain S

Q RTMw Im 1000 v w (2) u (8.314 J K 1 mol 1 ) u (298.15 K) u (18 g mol 1 ) (1000) u (18.05 u 10 6 )

u (0.924) u (0.620 mol kg1 )

2.83 u 106 Pascal =28.3 bar | 28 atm Linear interpolation from osmotic pressure data of aqueous NaCl listed in Perry gives for m = 0.62 mol kg-1, 28.0 atm, in good agreement with our calculated osmotic pressure.

7.

For K 2 SO4 , which is a 2-1 electrolyte, we have

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Solutions Manual

m

mK 

m

mSO2 

I

2m ; z

4

2

m ; z

1 (2m  22 m) 2

3m ; Q 

AJ z z I 1/ 2

ln J (m) r

1

1  BaI 1/ 2

2, Q 

 bI

Ÿ

1, Q

3

1/ 2 º ª 1 « (m) AJ z z (3m) »  ln J 3m « r 1  Ba(3m)(3m)1/ 2 » «¬ »¼

b

With

J (m) r

0.4 for m

0.12 mol kg-1; AJ

1.174 kg1/2 mol-1/2 ; B

0.33 kg1/2 mol1/ 2 Å-1; a

4Å

we obtain b

1 ª (1.174) u (2) u (0.36)1/ 2 º 1 « ln 0.4  » kg mol 0.36 «¬ 1  (0.33) u (4) u (0.36)1/ 2 »¼ 0.362 kg mol 1

For m 0.33 mol kg1 , we obtain J r 0.25 (the experimental value is 0.275). To use Eq. (9-25) for calculating the activity of water, we first need to calculate the osmotic coefficient, I . I 1

1.174 0.3615 u (3 u 0.33) (2) u (3 u 0.33)1/ 2 V( y)  3 2

where (0.33) u 4 u (3 u 0.33)1/ 2

y

1.32

and V( y 1.32)

I

0.618

3

1 º ª «1  1.32  2 ln(1  1.32)  1  1.32 » ¼ (1.32) ¬ 3

0.257

ª º 1000 (g/kg) « » u ln aw ol/kg) 18 (g 3 0.33 (m /mol) u u ¬ ¼ ln aw

0.011

Ÿ

aw

0.989

Pw 0.0314 bar Pw / Pwsat 0.989 Ÿ The vapor pressure has not changed much; it is only about 1% lower than that of pure water because m is still small.

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Solutions Manual

8. S RT

2 cBSA  cCI  BcBSA

concentration in mol L-1

c CI

counter ion

Because the charge on BSA is –20, there are 20 counter ions (protons) for each molecule of BSA. B is the osmotic second virial coefficient characterizing the BSA-BSA interaction in a 1 M aqueous NaCl medium. We neglect contributions from proton-proton and proton-BSA interactions, and also contributions from interactions with NaCl. Because the concentration (1 M) of NaCl is the same in both sides and because 1 M is much larger than the concentration of counter ions, we neglect any (tiny) charges that might occur in NaCl concentration due to interactions of Na+ and Cl- with counter ions or with BSA. S

224 13

cBSA

17.2 mmHg; T

44.6 66, 000

298 K; R

62.36 mmHg L mol 1 K 1

6.76 u 10 4 mol L1; cCI

(20) u (6.76 u 10 4 ) mol L1

Substitution gives B = -29040 L mol-1

9. a)

To calculate the activity coefficients of water we use the Gibbs-Duhem equation: d ln J w

x  s d ln J r xw

(1)

where subscripts w and s refer to water and salt, respectively. Integration of Eq. (1) between mole fractions xs = 0 (or xw = 1, for which Jw = 1) and xs, gives

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Solutions Manual

z

or

ln J w ( xs )

xs

 xs d ln J r  xs 1 0

ln J w ( x s )

z

xs

0

FG H

IJ K

 xs d ln J r dxs 1  xs d xs

(2)

The derivative in Eq. (2) can be obtained from the truncated Pitzer equation given: 8.766 x 1/ 2  124.598 x 3/2

ln J r

(3)

1  9x 1/ 2

We made I = xs, because being NaBr a 1-1 electrolyte, the solution ionic strength is 1 ¦ xi zi2 2 i

Ix

1 ( xs  xs ) 2

(4)

xs

Differentiation of ln Jr in Eq. (3) in order of x and substitution in Eq. (2) gives ln J w ( xs )

z

OP PQ

LM MN

 xs 4.383xs 1/ 2  186.897 xs 1/ 2  1121382 . xs dx s 1/ 2 2  x 1 (1  9 xs ) s 0 xs

(5)

The salt mole fractions are easily calculated from the given molalities (between m = 0 and m = 5 mol kg-1) from: xs

nNaBr nw  nNaBr

mNaBr 1000  mNaBr 18.015

mNaBr 55.51  mNaBr

(6)

The following figure shows the activity coefficients of water in different NaBr aqueous solutions at 25ºC, calculated using the Sympson rule to evaluate the integral in Eq. (5). As the figure shows, Jw | 1 until about m = 1.5 mol kg-1 and becomes less than one after that concentration. For example, Jw = 0.92 for m = 8 mol kg-1 and Jw = 0.88 for m =10 mol kg-1.

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Solutions Manual

1.00

Jw

0.95

0.90

0.01

0.10

1.00

10.00

Molality NaBr

b)

The mean ionic activity coefficients for NaBr aqueous solutions at 25ºC are calculated from

Debye-Hückel equation, ln J r  Ax I x , and from the Pitzer equation as given in this problem. The following figure compares both predictions. As expected, they agree only at very low salt concentrations.

Mean Ionic Actvity Coefficient

1.2

1.0

0.8

Pitzer

0.6

0.4

0.2 0.001

Debye-Huckel

0.01

0.1

1

10

Molality NaBr

c) Equation (4-44) gives the Van’t Hoff equation for the osmotic pressure: (valid for ideal, dilute solutions): SV

ns RT

where V is the total volume and ns the number of moles of the salt. Taking into consideration the nonideality of the liquid phase we write [Eq. (4-41)]:

(7)

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Solutions Manual

 ln aw

S vw RT

 ln(J w xw )

(8)

where v w is the molar volume of the pure solvent (water). We use Eq. (7) to calculate the osmotic pressure for the simplest case (Van’t Hoff equation) and Eq. (8) for the more correct calculation that takes into account the solution nonideality, with Jw obtained from Eq. (5). Assuming that NaBr is completely dissociated into Na+ and Br- in water, we rewrite Eq. (7) as S

2cs RT

with

cs

ns V

mol L1

(9)

We obtain the salt concentrations (molarities) from the given molalities (m2) from cs (mol L1 )

d ms 1  0.001Ms ms

where d is the mass density (in g cm-3) of the solution and Ms is the molar mass of NaBr (in g mol-1). To use Eq. (8) we take the molar volume of pure water as v w

18.015 cm3 mol 1 ,

1  2 xs with Jw given by Eq. (5). The following figures compares the results obtained from the Van’t Hoff equation [Eq. (9)] with the equation that takes into account the solution nonideality [Eq. (8)]. xw

700

Osmotic Pressue (atm)

600

Eq. (8)

500

400 Van't Hoff 300

200

100

0 0

2

4

6

8

10

12

Molality NaBr

As the figure shows, the solution behaves as an ideal solution (i.e. Van’t Hoff equation is valid) up to a concentration of about mNaBr = 2 mol kg-1. However, for more concentrated solutions (mNaBr > 2 mol kg-1), the effect of thermodynamic nonideality can not be neglected anymore.

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Solutions Manual

10.

For the dissociation reaction A+ + B-

AB

the equilibrium constant is

(aA )(aB )

K

5 u 103 mol kg1

aAB

The equilibrium equation is L fAB

V fAB

(1)

where f denotes fugacity. 30 bar kg mol 1 , yAB

Equation (1) is equivalent to (Henry’s constant H AB,w ª mAB HAB,w exp « «¬

³

P

Pwsat

º vf AB dP » RT »¼

MAB P

1)

(2)

Component AB in the vapor phase is in equilibrium with the undissociated AB dissolved in water. The total molality of AB (solubility) in water is:

mT

mAB  mA

If D is the fraction dissociated, we obtain

mA

D mT

and

mAB

(1  D )mT

The equilibrium constant then is K

5 u 10 3 mol kg1

J 2r D 2 mT 1 D

(3)

and Eq. (2) becomes ª v f ( P  Pwsat ) º (1  D)mT H AB,w exp « AB » RT «¬ »¼

MAB P

(2a)

where P  Pwsat # P because P 50 bar !! Pwsat . The fugacity coefficient at 50 bar is MAB

ª PBAB,AB º exp « » ¬ RT ¼ ª º (50 bar) u (200 cm3mol 1 ) exp « » 1 1 3 ¬« (83.14 cm bar K mol ) u (298 K) ¼»

With I

1 (m   mB ) 2 A

mA

0.668

D u mT , the mean ionic activity coefficient is [Eq. (9-52)]

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Solutions Manual

ln J r



AJ D mT 1  D mT

(4)

1.174 kg1/2 mol 1 / 2 .

where AJ

We have now three equations [Eqs. (2a), (3), (4)] and three unknowns mT , D, and J r . Solving these gives: 1.038 mol kg1

mT D

0.0871

Jr

0.762

Iteration procedure is: Start with D 0 in Eq. (2a) and calculate mT . Then calculate a first approximation for D with J r

1 using Eq. (3).

Use mT and D in Eq. (4) to obtain better J r , etc.

11.

In his theory of absolute reaction rates, Eyring states that reactants A and B form an activated complex (AB) as an intermediate state in the reaction

(AB)

A + B

o

Products

(1)

By assumption, reactants A and B are in equilibrium with the activated complex (AB), so that K

a(AB) aA aB

(2)

The reaction rate is proportional to the concentration of (AB), i.e., constant u c(AB) . Replacing the activities by the products of concentrations and activity coefficients in Eq. (2) we obtain c(AB)

J J KcA cB A B J (AB)

(3)

J J (constant)KcA cB A B J (AB)

(4)

Hence, Rate of reaction

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Solutions Manual

The rate of reaction can also be expressed in the usual manner by kcA cB , where k is the observed specific rate. Hence, from Eq. (4), we can write k

J J (constant)K A B J (AB)

J J ko A B J (AB)

(5)

Equivalently, we can write J J log ko  log A B J (AB)

log k

(6)

To calculate the second term on the right side of Eq. (6), we use Eq. (9-50a) J J log A B J (AB)

Substituting z(AB)

2 0.510 I u ( zA  zB2  z2(AB) )

(7)

zA  zB gives

J J log A B J (AB)

0.510 I u (2 zA zB )

(8)

Combining Eqs. (6) and (8) gives

log ko  1.02zA zB I

log k Therefore, a plot of log k versus

I is a straight line with slope 1.02zA zB .

Change of k

zAzB

Reaction

(9)

I

-2

Decreases with increasing I

II

0

Constant

When the inert salt NaCl is added, I changes. For mNaCl 0.01 and negligible molalities of reactants, we obtain m

mNa 

0.01

z

1 (10)

m

mCl-

0.01

z

1

and I

1 (0.01  0.01) 2

Combining Eqs. (9) and (11) gives

0.01 M

(11)

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Solutions Manual

k

zAzB

Reaction I

-2

-0.204

II

0

0

Some helpful relations for single-electrolyte solutions (i.e., one cation M and one anion X)

12.

Q

QM  QX

Q M zM  Q X zX mQ

QM QX  Q Q

or

1

or

Q M zM

0

mM  mX

¦ mi

Q X zX

mQ M  mQ X Qm

i

¦ mi

zi

m(Q M zM  Q X zX )

i

I

a)

1 ¦ mi zi2 2 i

1 2 2 ) m(Q M zM  Q X zX 2

Eq. (9-59) from Eq. (I-13): For a single electrolyte: c = 1 = M and a = 1 = X All )’s, <’s, and O’s are zero. Equations (I-14), (I-15), (9-61) and Eq. (I-13) give

(1)

(2)

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Solutions Manual

' zM zX f J  zM zX mM mX BMX

ln J r



§ · QM mX ¨ 2 BMX  ¦ mi zi cMX ¸ ¨ ¸ Q i © ¹



§ · QX mM ¨ 2 BMX  ¦ mi zi cMX ¸ ¨ ¸ Q i © ¹



mM mX 2Q M zM cMX Q

' where BMX and BMX are given by Eq. (I-19a) and Eq. (I-19b), respectively, and CMX is given by Eq. (I-18). Equations (I-16), (I-19), and (9-62) – (9-65) give

' 2BMX  I BM

J BMX

and CMX

2 J 1/ 2 C /(2 zM zX ) 3 MX

' can be summarized as: From Eq. (9-45), the terms with BMX or BMX

§ Q m  Q X mM · ' zM zX mM mX BMX  2¨ M X ¸ BMX Q © ¹ 2I 2 1 ' m Q MQ X BMX  (Q MQ X m  Q X Q M m)2 BMX mQ Q 2Q M Q X ' m( IBMX  2 BMX ) Q § 2Q Q m2 ¨ M X © Q

Summarizing the CMX terms gives:

· J ¸ BMX ¹

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Solutions Manual

ª§ Q M º2 QX mM mX 1 · J CM «¨ Q mX  Q mM ¸ (mM zM  mX zX )  Q 2Q M zM » 3 ¹ ¬© ¼ 2 zM zX 1/ 2 J ª§ Q M Q X º CMX Q Q m2 Q M Q X · 2Q M zM » m  X M m ¸ (Q M zM  Q X zX )m  «¨ Q Q ¹ «¬© Q »¼ 3 zM zX 1/ 2

ª 2(Q Q )3/ 2 º J m2 « M X » CMX Q ¬« ¼»

are identical to the 3 terms in Eq. (9-59).

The 3 terms above marked with b)

Eq. (9-60) from Eq. (I-10): As before c = 1 = M and a = 1 = X and all )’s, <’s, and O’s are zero. Equations (9-64) and (I-18) and Eq. (I-10) give I 1

 2I / ¦ mi f I  

(1)



2 I mM mX BMX mi ¦   

(2)



m m 2 I ¦ mi zi M X1/ 2 CMX m ¦ i 2 zM zX   

(3)

Term (1): 2 I / ¦ mi

1 2 2 2 m(Q M zM )  Q X zX 2 2 2 Q M zM  Q X zX Q

With Q M zM

Q X z X , it follows

(1)

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Solutions Manual

Term (1)

· Q QX 1§ zX  Q X zX M zM ¸ ¨ Q M zM Q© QM QX ¹ Q · §Q zM zX ¨ X  M ¸ Q ¹ © Q Q · §Q zM zX ¨ X  M ¸ Q ¹ © Q zM zX

Term (2): 2m2 Q M Q X Qm

2 mM mX ¦ mi

§ 2Q Q · m¨ M X ¸ © Q ¹

Term (3): 2 ¦ mi

¦ mi

Again using Q M zM

zi

mM mX 2 zM zX

1/ 2

Q M Q X m2 m(Q M zM  Q X zX ) 1/ 2 Qm z z M X

Q X zX gives 1/ 2

Q M zM zX

Q X zX zX

Ÿ

zM zX

1/ 2

§ QX · ¨ ¸ © QM ¹

zX

and

Q M zM  Q X zX

2Q X zX

Therefore, Term (3)

Q M Q X 2 2Q X zX m 2 1/ 2 Q Q1/ zX X QM

ª 2(Q Q )3/ 2 º M X » m2 « Q « » ¬ ¼

Comparison of the results for the terms (1), (2), and (3) with those in Eq. (9-60) shows that they are identical.

S O L U T I O N S

T O

P R O B L E M S

C H A P T E R

1.

1 0

Let 1 = methane, 2 = benzene, 3 = m-xylene, and 4 = hexane. Neglecting Poynting corrections and vapor non-idealities, y1 x1

K1

L J 1 f pure 1

M 1P

L From Fig. 10-13 of the text we obtain f pure at 366 K and 13.8 bar. 1

Because y1 # 1 we can use Lewis’ fugacity rule to obtain M1 by writing M1 We find J 1 from ln J1

v1 (G1  G)2 RT

In the first iteration, find G using x1 0. For a second estimate, first calculate y1 from y1 y2

( f / P) pure 1 .

x2 J 2 f20 P

,

1  y1  y3  y4 , where etc.

Then, x1

y1 K1

(other xi from relative amounts)

Recalculate G for second estimate of J 1 to find

153

154

Solutions Manual

K1

2.

KCH 4

34

Let 1 = argyle acetate and 2 = helium. Because we have two data points, we can use the Krichevsky-Kasarnovsky equation to evaluate the two parameters H 2,1 and v2f . ln

FG f2 IJ H x2 K

(Ps )

ln H 2,11 

v2f (P  P1s )

RT

Assume: 1. J *2

1

2. P1s  P Then, for helium, f2

y2M 2 P

From data given in App. C for helium, B22 (293 K) 12.1 cm3 mol 1

Using the virial equation and assume y2 ln M 2

1, B22 P RT

Find f 2 at different pressures: P (bar)

M2

f2 (bar)

25 75 150

1.012 1.038 1.077

25.3 77.8 161.5

At 25 bar, ln

f2 x2

ln(25.3 u 10 4 ) ln H 2,1 

ln(27 u 10 4 )

(25) u ( v2f ) (8314 . ) u (293)

ln H 2,1 

(75) u (v2f ) (8314 . ) u (293)

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Solutions Manual

These equations give vf 2

32.4 cm3 mol 1

and ln H 2,1

12.4

At 150 bar, ln f2  12.62

ln x2 x2

3.

x He

5.37 u 10 4

Over a small range of values for G, Hildebrand has shown that log x 2 is linear (approximately) in G1. A plot of log x 2 vs. G solvent gives x2 in liquid air. log x 2

G1

Because 1/ 2

Gi

§ ' vap ui · ¨¨ v ¸¸ i ¹ ©

we can calculate G’s from data given:

G CH 4

15.0 (J cm 3 )1/ 2

G CO

1.4 (J cm 3 )1/2

and

For air, assume a mixture of N2 and O2: G air

1/ 2

§ ' vap hi  RT · | ¨¨ ¸¸ vi © ¹

11.4 (J cm 3 )1/2

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Solutions Manual

From plot of log x 2 vs. G1 we find for G

11.4 (J cm 3 )1/2 ,

2.63 u 10 3

xH2

x2 (in air)

Note that in this case we hit one of our known points. In general, we must interpolate or extrapolate.

4. a) The equilibrium equation between the gaseous phase containing oxygen and the liquid phase saturated in dissolved oxygen is: fOG

fOL

2

(1)

2

At the low pressures of interest here, we assume the gas phase as an ideal gas mixture:

fOG

2

yO P

(2)

2

Moreover, because oxygen is sparingly soluble, i.e., oxygen is present at very low concentrations in the liquid phase, Henry's law holds: fOL

x O 2 kO 2

(3)

yO P

x O kO

(4)

2

From Eqs. (1), (2) and (3), 2

2

2

Equation (4) is the condition for phase equilibrium that characterizes the dissolution of a sparingly soluble gas. Under the given pressure and gaseous composition, Henry's law constant kO2 can be determined, once the solubility xO2 is known. This solubility is given here by the Bunsen coefficient D. Substituing t = 20ºC in the equation for D we obtain, D = 31.01u10-3

Ncm3 (O2 ) cm3 (H2 O)

where Ncm3 stands for normal cubic centimeters, i.e., cubic centimeters of gas measured at 0ºC and 1 atm. We convert normal cubic centimeters of gas to moles using the ideal gas law:

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Solutions Manual

(8.31451) u (293.15) 101325

RT P

v O2

0.022414 m3 mol 1 22414 cm3 mol 1

The molar density of water at 20ºC and 1 atm is 1

U H 2O

v H 2O

M H2O

0.9982 18.015

0.0554 mol cm 3

Under the conditions of the Bunsen experiment, we have: 3101 . u 10 3 22414

D v O2

Dissolved oxygen

13835 . u 10 6

mol O 2 cm3 H 2 O

The liquid phase is made exclusively of H2O and O2. Then the mole fraction of O2 in the liquid is

FDI GH vO2 JK F D I F 1 I GH vO2 JK GH v H2O JK

xO2

|

. 13835 u 10 6 0.0554

By the definition of D, yO2 From Eq. (4), kO 2

. 13835 u 10 6 . 13835 u 10 6  0.0554

2.497 u 10 5

mol O 2 mol O 2  mol H 2 O

10 . and P = 1 atm = 1.01325 bar.

yO 2 P

(10 . ) u (101325 . )

xO2

(2.497 u 10 5 )

4.058 u 10 4 bar

Here we want to determine xO2 given yO2 , P and kO2 , using again Eq. (4). The atmospheric air above the water phase is a mixture of water vapor, oxygen and other atmospheric gases (predominantly nitrogen). Assuming that the air is saturated in water vapor, b)

yH2O

PHs O 2 P

17.5 760

0.0230

The mole fraction of oxygen in the vapor phase is then

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Solutions Manual

PO2

yO2

P

(0.2095) u [(760)  (17.5)] (760)

0.02047

For the ambient pressure we obtain form Eq. (4), xO2

yO2 P

(0.2047) u (101325 . )

kO 2

(4.057 u 10 4 )

5112 . u 10 6

mol O 2 mol liquid

This solubility can now be expressed as mass of gas per volume of liquid, making the simplifying assumption that the liquid phase is pratically pure water:

Solubility=

(5.112 u 10 6 mol O2 / mol liquid) u (32.0 g O2 / mol O2 ) § · 1 (18.015 g H2 O/mol H2 O) u ¨ ¸ u (1 mol H2 O / mol liquid) ¨ 0.9982 g H O/cm3 H O ¸ 2 2 ¹ © 9.06 u 10 6 g O2 /cm3 H2 O = 9.06 mg dm 3

5.

9.06 ppm

The number of moles for each component is n1

180 18.015

10.0 mol

n2

420 84.16

5.0 mol

n3

28 28.012

10 . mol

The volume available for the vapor phase is V

3.0 

FG 420  180 IJ u 10 6 H 0.774 0.997 K

2.28 u 10 3 m 3

Therefore the pressure inside the vessel is (assuming vapor  formed almost exclusively by nitrogen whose second virial coefficient at 25ºC is zero  as ideal), P

n3 RT V

(1) u (8.314) u (29815 . ) (2.28 u 10 6 )

10.88 u 10 5 Pa

10.88 bar

Because the solubilities are very small, we use Henry's law to describe the fugacity of cyclohexane in gas phase: f3 | p3

y3 P

x3 H3,i .

Further, because we neglect mutual solubility of water and cyclohexane, we obtain for the solubility of nitrogen in water,

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Solutions Manual

x3,w

FG H3,1 IJ H p3 K

FG 86,000 IJ H 10.88 K

1

1

126 . u 10 4

and for the solubility of nitrogen in cyclohexane, x3,c

6.

FG H3,2 IJ H p3 K

FG 1,300 IJ H 10.88 K

1

1

8.37 u 10 3

Let 1 = N2 and 2 = H2. From Orentlicher’s correlation, ln

f2 x2

( Ps )

ln H2,11 

v f ( P  P1s ) A 2 ( x1  1)  2 RT RT

Assuming that the vapor is pure H2: f2

f2V

y2 M 2 P

(with y2

88 bar

1)

Because, P1s A

1 bar 7.1 L bar mol 1

H2,1

v 2f

467 bar 31.3 cm3 mol 1

by trial and error we find x2

7.

From Fig. 10-11 with G1

0.17

14.9 (J cm 3 )1/ 2 , at 25oC and at 1.01325 bar partial pressure, log x2

For t = 0oC, use Eq. (10-26):

3.1

Ÿ

x2

8 u 10 4

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Solutions Manual

x (T ) 's T ln 2 2 # 2 ln 2 x 2 (T1 ) R T1

But at 25oC and 1.01325 bar,  R ln x2

59.29 J mol 1 K 1

and from Fig. 10-7, s2L  s2G # 17 J mol 1 K 1

Then,

ln

17 J mol 1 K 1

s2L  s2G

's 2

x2

17 273 ln 8.31451 298

8 u 10 4

x2

6.7 u 10 4

(at 1 bar)

Assuming x2 (P2 ) x 2 (P1 )

( y2 P) 2 ( y2 P)1

we obtain at 0oC and 2 bar partial pressure, x2 = 1.34 ×10-3

8.

Let 1 = ethylene oxide and 2 = CH4. Then, f ln 2 x2

(Ps )

ln H 2,11 

v2f (P  P1s ) RT

From Tables 10-2 and 10-3 at 100C, H 2,1

v2f

At 10oC, P1s | 1 bar. Then,

621 bar

45 cm3 mol -1

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Solutions Manual

f ln 2 x2

6.5

f2 x2

665 bar

f2

M 2 y2 P

But

Assuming y2 | 1, B22 P RT

ln M 2

with B22

49 cm3 mol 1 (Table 10-3) M2 f2

0.949

(0.949) u (25)

xCH 4

x2

23.7 bar

f2 665

0.036

9. a)

Henry’s constant H2,1 is calculated from [Eq. (10-21)]: (Ps )

H2,11

P1s M 2L,f

where M 2L ,f is the fugacity coefficient of solute 2 in the liquid phase at infinite dilution (x2 = 0); M2 may be obtained from Eq. (12-64) of the text with bmixture | b1

v mixture | v1s P | P1s x2 | 0

Then, H2,1 is given by

162

Solutions Manual

ln H 2,1

ln



b2 a1b2 v1 RT   v1  b1 v1  b1 RTb1[v1 (v1  b1 )  b1 (v1  b1 )] a1b2

2

2 RTb12

ln

a12 (v1  2.414b1 ) (v  2.414b1 ) ln 1  (v1  0.414b1 ) RT 2b1 (v1  0.414b1 )

where a12

(a1a2 )1/ 2 (1  k12 )

k12

0.0867

Constants a1 and a2 are obtained from Eqs. (12-61) to (12-63) and constants b1 and b2 from Eq. (12-60). Substitution gives H 2,1

b)

360 bar

From Eq. (10-22),

LM F wP I M M GH wn JK MM FGH wwVP IJK N

2 T ,V ,n 1

v2f

T ,n1,n2

OP PP PP Q

n2 0

Using Peng-Robinson equation of state [Eq. (12-59)], we find

RT (b2  v1s  b1 ) (v1s  b1 ) 2

v2f

2a1b2 (v1s  b1 ) 2a12  s s  v1 (v1  b1 )  b1 (v1s  b1 ) [v1s (v1s  b1 )  b1 (v1s  b1 )]2

RT (v1s  b1 ) 2



2a1 (v1s  b1 ) [v1s (v1s  b1 )  b1 (v1s  b1 )]2

Substitution gives v 2f

c)

69.5 cm 3 mol 1

Margules parameter A can be found from Eq. (10-23): A



F GH

I JK

L RT w ln M 2 2 wx 2 P P1s ,T , x2 0

The result is:

A



R| S| T

a1b2 v'2b2 v1s (a12  a1 ) v' (b2  b1 ) RT b2 (b2  b1 )  v' b2   2 RTb1[v1s (v1s  b1 )  b1 (v1s  b1 )] (v1s  b1 ) 2 v1s  b1

163

Solutions Manual



{

}

{

{

RT b1 v1s (v1s  b1 )  b1 (v1s  b1 )







}

a1b2v1s (b2  b1 ) v1s (v1s  b1 )  b1 (v1s  b1 )  2b1 v' (v1s  b1 )  (v1s  b1 )(b2  b1 )

FG H

}

2

IJ K

v s  2.414b1 2b1 (a12  a1 )  a1 (b2  b1 ) 2a12 b2 ln 1  a1 b1 v1s  0.414b1 2 2 RTb12

LM a1 (a2  a12 )  2a12 (a12  a1 )  b2 (b2  b1) OP ln v1s  2.414b1 a12 2b12 2 RTb1 MN PQ v1s  0.414b1 a1

FG H

IJ K

v1s (b2  b1 )  b1v' a1 2a12 b2  RTb1 a1 b1 (v1s  2.414b1 )(v1s  0.414b1 )

U| V| W

with v'

FG wv IJ H wx2 K x2 0

v2f  v1s

Substitution gives

A 13,900 bar cm3 mol1

S O L U T I O N S

T O

P R O B L E M S

C H A P T E R

1.

Assuming fi

s

1 1

s

f pure i

and fiL

with Tt

Tm , 'c p

L xi f pure i

(i.e. J i

1)

0,

ln

L f pure i

s i f pure

ln

FG 1 IJ H xi K

' fus hi RT

LM1  T OP N Tm Q

Rearranging,

T

For i = benzene, ' fus hi

ª « § ' fus hi · « ¨ R ¸« © ¹« § 1 ln ¨ «¬ © xi

º » 1 » · ' fus hi »» ¸ ¹ RTm,i »¼

9843 J mol-1, xi = 0.95, Tm,i T

For i = naphthalene, ' fus hi

278.7 K , we obtain

275 K

19008 J mol-1, xi = 0.05, Tm,i

T

353.4 K , we obtain

241 K

165

166

Solutions Manual

We choose the higher temperature (i.e., benzene precipitates first). 353 K

T, K

279 K 275 K

Benzene

Naphthalene

At T = 275 K, a solid phase appears.

2.

We need activity aA at x A

sat xA / 2.

At saturation,

s

fAL

fA

s

f pure A

Because fAL sat  ln aA

Assuming, Tm

§ fL ln ¨ A ¨f © A

·

s ¸¸¹pure

sat Tt , we find aA

L aA f pure A

' fus h § T · 'c p § Tt  T · 'c p Tt ln  ¨1  ¸  RT © Tt ¹ R ¨© T ¸¹ R T 0.118 .

sat xA

0.05 , J A 2.36 . To find the activity at another composition, assume that

Because

RT ln J A

Using the above data, we find D Hence at xA

D(1  x A ) 2

2357 J mol 1 .

sat xA /2,

JA

2.47

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Solutions Manual

xA J A

aA

0.0618

Then T 1 T

Ÿ

8.04

T

0.89

Thus, 89% of sites are occupied.

3.

Klatt’s data are really at 00C, not -700C. This is above the freezing points of toluene and xylene and near that of benzene. Let HF be component 1 and the solute (2) be A (benzene), B (toluene), and C (m-xylene). The order of increasing substitution (basicity) is A, B, C. To simplify things, ignore the solubility of 1 in 2. Then: P

P

P

P

f pure 2

f2, in 1

x2 J 2 f20

But, f20 f pure 2 , so this reduces to: 1 x2 J 2 . Therefore, x2 is inversely proportional to J2. We might think that J2 depends only on the 1-2 interaction. On this basis, we expect J C  J B  J A , and thus x C ! x B ! x A . Klatt’s data show the reverse. There is, however, another factor: the strength of the 2-2 interactions. At 00C, U

PAs

| 0.036 bar ,

PBs

| 0.009 bar

and PCs

U

P

P

| 0.002 bar . This means that pure C “holds on” to its

molecules more tightly than B which in turn has a tighter grip than A. In other words, the more volatile solute (that has the weakest 2-2 interactions) exerts more “pressure” to enter the solvent phase. This is discussed in a qualitative manner by Hildebrand, 1949, J. Phys. Coll. Chem., 53: 973. It may be helpful to look at this from a lattice theory (interchange energy) perspective. Using the simplest form of this theory, we can say: w 2 x kT 1

ln J 2 1 2

z [*12  (*11  * 22 )]

w

where w is the interchange energy. With the attractive interaction, we expect *1A ! *1B ! *1C . This produces a higher w (and hence a higher J2 and lower x2) for the less-substituted molecule. But, if we look at the vapor pressures we see that the less-substituted molecules have less attractive 2-2 interactions. Hence, *AA ! *BB ! *CC . This produces a higher w (and hence a U

U

higher J2 and lower x2) for the more-substituted molecule. Sometimes, this effect is greater than that of the 1-2 interactions; that is apparently true in this case. U

U

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Solutions Manual

4.

Let 1 = naphthalene, 2 = iso-pentane, and 3 = CCl4. At saturation,

s

f1

s

Assuming, f1, pure

f1L

x1J 1 f1,L pure

s

f1 ,

§ ln ¨ ¨ ©

f1L · ¸ f1s ¸¹ pure

 ln a1

' fus h § T · ¨1  ¸ RT © Tt ¹

 ln x1J1

From the regular-solution theory, assuming x1 = 0 initially, we obtain G

14.9 (J cm-3)1/2

and

ln J 1 G1

v1 (G1  G )2 RT

20.3 (J cm-3)1/2 ln a1 x1

118 . 0.073

Now repeat the calculation using x1 = 0.073 and G

1.45

x2 x3

7 , to obtain 3

15.3 (J cm-3)1/2

and x1

One more iteration gives x1

5.

0.093

xnaphthalene | 0.10 .

The equilibrium equation for benzene (B) is

Partial pressure of B yB P

s xB J B Pliquid B

(1)

169

Solutions Manual

where xB ( xB zene;

s Pliquid B

0.10 ) and J B denote liquid-phase mole fraction and activity coefficient of ben-

is the vapor pressure of pure, subcooled liquid benzene at 260 K.

s To find Pliquid B , we use the approximation s Pliquid B

§ fL ¨ ¨f ©

·

s ¸¸¹pure B

s Psolid B

(2)

s s where Psolid B ( Psolid B 0.0125 bar at 260 K ) is the vapor pressure of pure solid benzene at 260 K; the fugacity ratio for pure benzene is calculated from Eq. (11-13) neglecting 'c p for benzene

§ fL ln ¨ ¨f ©

·

' fus h § Tm ·  1¸ RTm ¨© T ¹

s ¸¸¹pure B

Substituting ' fus h 30.45 cal g1 9944.07 J mol 1 , Tm J K-1 mol-1 into Eqs. (2) and (3), we obtain s Pliquid B

278.7 K , T

260 K and R = 8.314

(4)

1.362

s Psolid B

(3)

Hence, s Pliquid B

1.362 u (0.0125 bar)

0.0170 bar

(5)

To calculate J B in Eq. (1), we use Eq. (7-55): vB (GB  G )2 RT

ln J B

(6)

where 3

¦ ) i Gi

G

i 1

(7)

)i

xi vi 3

¦ x jv j j

Substituting T 260 K , R 8.314 J K 1 mol 1 and the given liquid-phase mole fractions, pure-component molar volumes and solubility parameters into Eqs. (6) and (7), we obtain JB

Combining xB

(8)

1.305

0.10 and Eqs. (1), (5) and (8) yields Partial pressure of B

yB P

0.0022 bar

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Solutions Manual

6.

This is similar to Problem 1, but includes activity coefficients. fi

Then, considering 'c p

s

s

0 and Tt

§ fL ln ¨ i ¨f © i

s

fi L

fi,pure

· ¸ ¸ ¹pure

xi J i fi,Lpure

Tm ,

' fus hi RT

§ T ¨¨ 1  © Tm,i

· ¸¸ ¹

§ 1 · ln ¨ ¸ © xi J i ¹

Using the regular-solution theory,

ln J i

vi (G1  G 2 )2 ) 2j

RT

Let 1 = benzene and 2 = n-heptane )2

ln J 1

0.935

(89) u (18.8  151 . ) 2 u (0.935) 2 (8.31451) u (T )

128 T

Then, 9843 § T · 1 RT ¨© 278.7 ¸¹

ª § 128 · º  ln «(0.1) u exp ¨ ¸» © T ¹¼ ¬

Solving for T,

T

200 K

Similarly, for n-heptane, )1

ln J 2

0.065

(148) u (18.8  15.1)2 u (0.065)2 (8.31451) u (T )

1.03 T

Then, 14067 § T · 1 ¨ RT © 182.6 ¸¹

ª § 1.03 · º  ln «(0.9) u exp ¨ ¸» © T ¹¼ ¬

Solving for T,

T

181 K

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Solutions Manual

As temperature decreases, benzene starts to precipitate at 200 K.

7.

Plotting the data we obtain:

1245

t ( C)

1200

1000 935

805 800

100

40

60

80

Mole % Cu2O

A compound, Cu2OP2O5, is formed with a congruent melting point at 1518 K. Eutectics occur at 1208 K and 1078 K.

8. a) According to the ideal solubility equation, Tm of the solvent has no influence on the solubility. Any difference would have to come from nonideality (i.e. activity coefficients). If we look at solubility parameters, we find that the solubility parameter of CS2 is closer to that of benzene. Therefore, we expect greater solubility in CS2. b) Let 1 = benzene, 2 = CS2 and 3 = n-octane. Assuming U

s

f1

and

s

f pure 1

f1L

L x1J 1 f pure 1

U

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Solutions Manual

'c p

0

Tt # Tm we have

' fus h § T ¨1  RT ¨© T f

 ln J1x1

· ¸ ¸ ¹

(1)

Using the regular-solution theory, ln J 1

v1 (G1  G ) 2 RT

(2)

with 3

G

¦ )iGi i 1

From Tables: Component

G (J cm-3)1/2

v (cm3 mol-1)

1 2 3

18.8 20.4 15.3

89 61 164

From Eq. (1) with x1 = 0.3, ln J 1

0.144

Then, from Eq. (2), ln J 1

0.144

v1 (G1  G ) RT

G 171 . (or 20.6; this value is probably meaningless since it is higher than G’s of pure components). 3

G

¦ )iGi i 1

)1

Solving for x2 and x3,

x1v1 , x1v1  x 2 v 2  x 3v 3

etc.

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Solutions Manual

9.

At x 2

x1

0.30

x2

xCS2

x3

0.38

0.32

0.25 ,

s

s

f2

f pure 2

x2 J 2

s

Because f pure 2

s

P2

(f

0.56

2.24

0.99 bar, L f pure 2

Assuming ln J 2

L x 2 J 2 f pure 2

s / f L )pure 2 J2

,sat

f2L

Ax12 , at x 2

177 . bar

0.25 , then J 2

2.24 , and

A 1.434 At x 2

0.05 , f2V

ln J 2

f2L

(1.434) u (0.95) 2

J2

P2

10.

PCO2

L x 2 J 2 f pure 2

3.65

(0.05) u (3.65) u (177 . )

0.323 bar

At 250 K, we need a standard-state fugacity for a hypothetical liquid.

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Solutions Manual

§ fL · ln ¨ A ¸ ¨ fs ¸ © A ¹pure

' fus h § T · ¨1  ¸ RT © Tm ¹ (13000) § 250 · u ¨1  (8.31451) u (250) © 300 ¸¹

Hence, fAL

fAs Because solid is pure,

s

s

fA

f pure A

2.84

s

PA ,sat

§ 35 · ¨ ¸ u (2.84) © 750.06 ¹

L fpure A

35 torr

0.1325 bar

For the A-CCl4 system, fAV

fAL

L x A J A f pure A

Neglecting vapor-phase non idealities and the Poynting correction factor, L x A J A f pure A

yA P

(5 / 750.06) (0.03) u (0.1325)

JA

1.677

Using the regular-solution theory, RT ln J A

v A (G A  G CCl 4 ) 2 ) 2CCl 4

Thus, (8.31451) u (250) u (ln1.677)

(GA  GCCl4 )2

(0.97) u (97) ª º (95) u « » ¬ (0.03) u (95)  (0.97) u (97) ¼

G A  G CCl 4 As for G CCl 4

3.4 (J cm-3)1/2

17.6 (J cm-3)1/2, P

P

GA

or

2

210 . (J cm-3)1/2 P

P

175

Solutions Manual

14.2 (J cm-3)1/2

GA

14.2 (J cm-3)1/2.

Because A is a branched hydrocarbon, we choose G A For the A-hexane system, L x A J A f pure A

yA P

ln J A

v A (G A  Ghex )2 2 ) hex RT

(0.99) u (132) ª º (95) u (14.2  14.9)2 u « » ¬ (0.01) u (95)  (0.99) u (132) ¼ (8.31451) u (250)

JA

102 .

Then, yA P

(0.01) u (102 . ) u (0.1325)

PA PA

11.

0.00135 bar

Let fAL

L x A f pure A

(J A

1)

fBL

L x B f pure B

(J B

1)

From Eq. (11-13) with T # Tm , 'c p # 0 , § ln ¨ ¨ ©

L · fpure A s A ¸¸ fpure ¹

' fus hA § T ¨¨ 1  RT © Tm,A

§ ln ¨ ¨ ©

L · fpure B ¸ s B¸ fpure ¹

' fus hB § T · ¨¨ 1  ¸ RT © Tm,B ¸¹

Because solids A and B are mutually insoluble,

s

fA

s

f pure A

· ¸¸ ¹

2

176

Solutions Manual

s

fB

At equilibrium,

s

f pure B

s

L x A f pure A

s

L xB f pure B

fA

fB

Then, § ln ¨ ¨ ©

L · fpure A ¸ fpure A ¸¹

s

ln

1 xB

§ 1 · ln ¨ ¸ © xA ¹

(8000) T · § u 1 (8.31451) u (T ) ¨© 293 ¸¹

(12000) T · § u ¨1  (8.31451) u (T ) © 278 ¸¹

with x B 1  x A . Solving the above equations, we obtain x A T 244 K.

0.516 (or 51.6 mol % A), xB

0.484 and

S O L U T I O N S

T O

P R O B L E M S

C H A P T E R

1 2

1. i)

ii) T = TU

T < TU

P

P L

L+L’

L+L’

U

L’

V L+

L

L’ V L+

V

L+V

V

x

n-Alkane

iii)

Water

TU < T < TCn

iv)

P

x

n-Alkane

Water

TCn < T < TCb

P L+L’

L’

L+G

L’

V L+

V

n-Alkane

V

x

Water

n-Alkane

x

Water

177

178

Solutions Manual

T = TCb

v)

TCb < T < TCH O 2

vi)

P

P G+G’

L+ G

V

x

n-Alkane

vii)

Water

x

n-Alkane

Water

T > TCH O 2

P G+G’

x

n-Alkane

Water

2.

T = T1

a)

P1 < PUCEP

b)

P

V

T L

L+V

L’

L’+ V

L+L’

L+ V

L

L+L’ L’

L’+V

V 1

x

2

1

x

2

179

Solutions Manual

P2 = PUCEP

P3 > PUCEP V

T

V

T

L+V

L+V

L’+ V

L L+L’

L

L’ L+L’

1

3.

x

2

x

1

Let A stand for alcohol. For alcohol distributed between phases ' and " ' ' xA JA

" " xA JA

Then, K

F xA' I G "J A o0 H xA K lim

x

F J "A I G'J A o0 H J A K lim

x

At 0qC and 1 bar, ln J 'A

2400 ' )2 (1  x A RT

ln J "A

320 " )2 (1  xA RT

The pressure correction to JA is

J A (P2 ) The temperature correction is:

J A (T2 ) Thus, we can write,

z z

J A (P1 ) exp

J A (T1 ) exp

P2

P1

vE dP RT

T2

h E

T1

RT 2

dT

2

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Solutions Manual

ln J 'A

[

100 2400 16  dP (8.31451) u (273) 1 (8.31451) u (273)

³

303

4800

³273 (8.31451) u (T 2 ) dT ]u (1  xA )



'

2

' 2 0.9178u (1  xA )

[

ln J"A

100 320 10 dP  (8.31451) u (273) 1 (8.31451) u (273)

³

303



600

³273 (8.31451) u (T 2 ) dT ]u (1  xA )

" 2

0.0712(1  x"A )2

This gives K

4.

lim

xA o 0

J "A

0.429

J 'A

For pure benzene, neglecting fugacity coefficients and assuming constant density of each phase with respect to pressure,

fBL PBs,L exp with v L

v L (P  PBs,L )

87.7 cm3 mol 1 and v

RT

s

s

fB PBs,

s exp v s (P  P s ) s, B

RT

77.4 cm 3 mol 1 , as obtained from density data.

ª (87.7) u (200  P s,L ) º B » PBs,L exp « (83.1451) T «¬ »¼

PBs,

s exp ª« (77.4) u (200  PBs,s ) º» «¬

(83.1451)T

»¼

Temperature T can be found from the intercept of the curves obtained by representing each side of the last equation as a function of temperature.

s

In an alternate way, we can express PBs,L and PBs, from vapor-pressure equations:

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10 (7.96221785/T ) 750.06

PBs,L

PBs,

s

10 (9.846  2310 /T ) 750.06

which, together with the last equation, can be solved for T: Tm(200 bar) = 284.4 K

5.

The Redlich-Kwong equation is: RT a  v  b T 1/ 2v(v  b)

P

with a

2a 2 zA AA  2zA zBaAB  zBaBB

¦ ¦ zi z j aij i

j

¦ zi bi

b

zA bA  zBbB

i

Assuming that aAB is given by the geometric rule, aAB

we get for zA

zB

(aAA aBB )1/ 2

0.5, a

4.35 u 10 8 bar (cm3 mol-1) K1/2

b = 91.5 cm3 mol-1 Substitution in the R-K equation gives for total pressures:

P

413 . bar

Because this result is absurd, we use Henry’s constant data to find aAB . For infinitely dilute solutions of A in B,

H A,B

(PM A ) xA

0

f PBs M B

s At infinite dilution, Ptotal # Ppure B which can be obtained from the R-K equation with the

appropriate constants [ a

4.53 u 10 8 bar (cm3 mol-1)2 K1/2 and b

82.8 cm3 mol-1].

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This gives, PBs

113 . bar

Therefore, H A,B

f MA

PBs

7.01 113 .

6.195

For the R-K equation, fugacity coefficients are given by: f ln M A

ln

LM N

OP Q

b abA 2aAB v vb vb b Pv  A  ln  ln   ln 3/2 3/2 2 v  b v  b RT b v v vb RT RT b

Using b # bB , a # aB and v # v B (infinite dilution of A). Solving for aAB , aAB

3.963 u 10 8 bar (cm3 mol-1)2 K1/2 P

Then, for the mixture, a

4.159 u 10 8 bar (cm3 mol-1) K1/2 b

91.5 cm3 mol-1

Calculating again the pressure we obtain,

P

6.

4.14 bar

Let 1 = C2H6 and 2 = C6H6. The K factor of component i is defined as Ki

yi xi

M iL (P, x ) M Vi (P, y)

with M

c (0)  c (1) P  c (2) zi

Thus, we need to solve for P and y1 (or y2). At equilibrium, y1M1V

x1M1L

(1)

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(1  y1 )M V2

(1  x1 )M 2L

(2)

From the given equations, we rewrite Eqs.(1) and (2) for x1 = 0.263: (12545 .  2.458 u 10 4 P  0.4091y1 ) y1

11699 .  0.008345P

[0.74265  7.0069 u 10 3 P  0.50456(1  y1 )](1  y1 )

0.24596  0.001874 P

The above equations can be solved (either graphically or numerically) for P and y1:

y1

P

59 atm

0.715

( y2

0.285)

Then,

[From Kay’s data: K1

7.

K1

y1 x1

0.715 0.263

2.72

K2

y2 x2

0.285 0.737

0.387

2.73 and K 2

0.41 ].

The stability criterion is [see Eq. (6-131) of text]:

F w2 g E I  RT F 1  1 I  0 GH wx 2 JK GH x1 x2 JK 1 T ,P We need an expression for gE valid at high pressures. Because

F wg E I GH wP JK T ,x

we write g E (T , P, x )

g E (T , P

1 atm, x ) 

(RT ) u (1877 . ) x1 x 2 

vE

z z

P E

1 P

1

v dP

x1 x 2 (4.026  0.233 ln P )dP

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Thus, g E (T , P, x )

(RT ) u (1877 . ) x1 x 2  (P  1) u (4.026) x1 x 2

 (P  1) u (0.233) x1 x2  0.233P (ln P ) x1 x 2 (42043  4.259P  0.233P ln P) x1 x 2

For gE of this form ( g E 6.12):

Ax1 x 2

Ax1 x 2 , where A is a constant), the stability criterion is (see Sec. A !2 RT

or

42043  4.259P  0.233P ln P ! (2) u (82.0578) u (273) Solving for P,

P 1046 atm (or 1060 bar) At pressures higher than 1060 bar, the system splits into two phases. To solve for the composition at a higher pressure, we use: x1' J 1'

(1  x1' )J '2

x1" J 1"

(1  x1" )J "2

where (42043  4.259P  0.233P ln P ) x 2j

RT ln J i

At 1500 atm (or 1520 bar) and 273 K, ln J i

2.0477 x 2j

Thus, x1' exp[2.0477(1  x1' ) 2 ]

(1  x1' ) exp[2.0477( x1' ) 2 ]

x1" exp[2.0477(1  x1" ) 2 ]

(1  x1" ) exp[2.0477( x1" ) 2 ]

Solving (either graphically or numerically), we obtain x1'

0.37

( x2'

0.63)

x1"

0.63

( x2"

0.37)

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Solutions Manual

8.

We want to relate hE to volumetric data. Relations given in Chapter 3 of the text may be used. We write hE at any pressure P relative to hE at 1 bar as:

z LMMN P

h E (P )  h E (1 bar)

vE  T

1

F wv E I OP dP GH wT JK P PQ

Thus, we need the above integrand as a function of pressure at 333 K. From volumetric data, using linear regression at each pressure between 323 K and 348 K,

F wv E I GH wT JK 1 bar

F wv E I GH wT JK 100 bar

0.0186

F wv E I GH wT JK 250 bar

0.0154

F wv E I GH wT JK 500 bar

0.01239

0.00963

Using linear interpolation, at 333 K, v E (1 bar) 1.091

v E (100 bar)

0.9638

v E (250 bar)

v E (500 bar)

0.6846

0.8284

If F (P)

vE  T

F wv E I GH wT JK P

then: P (bar)

1

100

250

500

F(P) (J bar mol-1)

-0.5102

-0.4164

-0.3297

-0.2522

Using a trapezoid-rule approximation,

³

360

F ( P)dP

128 J mol 1

1

Therefore, at 333K h E (360 bar)

h E (1 bar) 

360

³1

F ( P)dP

1445  128

h E (360 bar, 333K)

1317 J mol 1

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Solutions Manual

9.

For condensation to occur, L V fW ! fW

To find the temperature for condensation (at constant pressure and vapor composition), we solve the equilibrium relation L fW

V fW

The liquid phase is assumed to be pure water. Its fugacity is given by L fW

L f pure exp

z

P

s PW

vW dP RT

As a good approximation, let s s L # P s exp v W ( P  Pw ) fW W RT

(obtain data from Steam Tables)

Thus, we are neglecting M sW and we assume that (liquid) water is incompressible over the s and P (150 atm). pressure range between PW

The vapor phase is described by an equation of state. Therefore, V fW

yW M VW P

To obtain M VW , we use the Redlich-Kwong equation of state: P

RT a  v  b T 1/ 2v(v  b)

(1)

from which we obtain ln MW

ln

b v §vb·  W  (2¦ y j aWj )( RT 1.5b)1 ln ¨ ¸ vb vb © v ¹ j

(2) 

b · Pv § vb ¨ ln v  v  b ¸  ln RT ¹ b ©

abW RT

1.5 2

where v is the molar volume of the mixture and

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Solutions Manual

a

¦¦ yi y j aij i

b

j

¦ yi bi i

In these equations, aW and aCO2 are given as functions of temperature; the cross-coefficient is

aij

(ai (0) a j (0) )1/ 2  0.5R 2T 2.5 K

V Use the following procedure: With a trial-and-error procedure, we can calculate f W .

1. 2. 3. 4. 5.

Guess temperature. Calculate v from equation of state [Eq. (1)]. Use T and v (along with P and y) to calculate M W from Eq. (2). Calculate the fugacity of vapor. V with saturation pressure of water at that temperature. Compare f W

Typical results are: T (K)

MW

V fW (atm)

475 500 525 550

0.588 0.633 0.711 0.746

17.7 19.9 21.3 22.4

V and L as a function of temperature (see figure), we see that Plotting f W fW L fW

V fW

at T | 482 K. That is the temperature where condensation first occurs (dew-point temperature of the mixture).

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Solutions Manual

Fugacity of water, atm

100

L fW

75

50

V fW

25

0 400

450

500

550

600

Temperature, K

Fugacity of water in vapor phase and in liquid phase at P = 150 atm.

10. a)

For equilibrium between solid solute and solute dissolved in the supercritical fluid,

s

f2 (P, T ) or

s

d ln f2

f2f (P, T , y2 ) d ln f2f

(1)

where subscript 2 refers to solute and superscript f to fluid phase. Expanding Eq. (1) with respect to T, P and composition (see Sec. 12.4), we obtain (temperature is constant):

F w ln f2s I GG wT JJ H K P, y F w ln f2f I GG wT JJ H K P, y

0

0

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Solutions Manual

F w ln f2s I GH wx2 JK P,T

(pure solid solute)

0

F w ln f2f I F fI GG wP JJ dP  GG w lnwy2f2 JJ dy2 H K T ,y H K T ,P

F w ln f2s I dP GH wP JK T

(2)

But because

F w ln f2s I GH wP JK F w ln f I GG wP JJ H K

s

v2

RT

T

f 2

v2f RT T

Equation (2) becomes:

s

(v 2  v2f ) dP RT

F w ln f2f I GG wy2 JJ dy2 H K T ,Pdy

F w ln f2f I GG w ln y2 JJ d(ln y2 ) H K T ,P

(3)

Finally, because

f2f

y2M 2 P

Equation (3) becomes

FG w ln y IJ H wP K

s

2

b)

T

v 2  v2f RT w ln M 2 1 w ln y2 T ,P

FG H

(4)

Maxima (or minima) occur when

FG w ln y IJ H wP K 2

v 2s

IJ K

0 T

Because w ln M 2 / w ln y2 is always greater than –0.4, the above derivative is zero when

v2f . f It is necessary, then, to calculate v2 as a function of pressure.

Using Eq. (12-41),

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Solutions Manual

FG wP IJ H wn2 K T ,V ,n1  FG wP IJ H wV K T , all n

v2

and the Redlich-Kwong equation of state with the mixing rules,

¦¦ xi x j aij

a

i

j

¦ xi bi

b

i

we obtain

v2

b · RT § 1 2  v  b ¨© v  b ¸¹

2(¦ xi a2i )  ab2 /(v  b) i

v(v  b)T 1/ 2 RT a ª 2v  b º  2 1/ 2 « 2 2» ( v  b) T ¬« v  (v  b) ¼»

Assuming that the fluid phase is almost pure solvent, v, a and b are those for pure solvent 1. Cross parameter a12 is given by: a12

(a11a22 )1/ 2 (1  k12 )

Constants are: bar (cm3 mol 1 )2 K1/2

a11

0.7932 u 108

a22

0.11760 u 1010 bar (cm3 mol 1 )2 K1/2

a12

0.3264 u 109

b1

bar (cm3 mol 1 )2 K1/2

40.683 cm3 mol 1

b2

140.576 cm3 mol 1

Using volumetric data for ethylene at 318 K (IUPAC Tables), and because

s

v2

128174 . 1144 .

the maximum (and minimum) occurs ( v2

112 cm3 mol 1

s

v 2 ) at (see figure below)

minimum = 19 bar

maximum = 478 bar These values are in good agreement with results shown in Fig. 5-39 of the text.

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Solutions Manual

500

Solubility minimum

Solubility maximum

s

v 2 =112 0

10

v2 (cm3 mol -1)

19 bar -500

100

1000

10,000

P (bar)

478 bar

-1000

-1500

-2000

Partial molar volumes of naphthalene infinitely dilute in ethylene at 318 K calculated from Redlich-Kwong equation of state with k12 = 0.0182.