On the Heat Capacity of Non-Polar Solid Compounds

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334

PROC. N. A. S.

infinite momentum ko, then the totality of values mk as found in (8)-is nothing but the totality of values lim {[(ko + ki) + k] +

*km}) -

ko

> co

each diminished by ko, and that by our theorem is equal to the totality of values lim [ko + (ki + k2+ ... km)] ko

coa

each diminished by ko; i.e., the totality of the quantized projections of all the momenta resulting from the composition of ki, k2,... km. Similar remarks apply to r. * Using Sommerfeld's quantum numbers which are smaller by I than those of Lande. 1 H. N. Russell and F. A. Saunders, Astroph. J., 61, 38, 1925; S. Goudsmit, Zs. Phys., 32, 794, 1925; W. Heisenberg, Ibid., 32, 841, 1925. 2 H. N. Russell, Proc. Nat. Acad. Sci., 11, 314, 1925.

ON THE HEAT CAPACITY OF NON-POLAR SOLID COMPOUNDS By E. 0. SALANT* DEPARTMENT OF PHYSICS, JOHNS HOPKINS UNIVERSITY

Communicated April 16, 1926

In a previous paper,' the -writer attempted to apply the quantum theory of the heat capacity of solids to aliphatic compounds. By a series of approximations, there was obtained a set of equations giving rough agreement with measured heat capacities over a fair range of temperatures. By introducing some of the results of infra-red spectroscopy, the general equat.ions of Born will be extended in-the present paper to solids of nonpolar compounds. These will then be applied to some organic compounds, the rough approximations oft"'he previous work being discarded. The general theory of the heat capacities of crystalline solids, developed by Born,2 stated that the"first thr'ee frequencie's are characteristic of the crystal and their contribution to the heat capacity is expressed by Debye functions, and the remaining frequencies are characteristic of the atomic (nuclear) 'vibrations and their contribution to the heat capacity is expresse.d by Einstein functions. Nernst3 arrived at the division of heat capacity..into Debye and Einstein functions by considering the Debye functions; as characteristic of the molecule; this view was adversely, criticized at first because Nernst. used -it for polar substances, which -do. not unite to form molecules in the crystal. But X-ray analysis has -shown that molecules. of many-substances, for example, of organic compounds,

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335

do exist in the solid state, and so for these Nernst's viewpoint can quite justifiably be retained. For the molecular heat capacity at constant volume, C,, of a compound of n atoms per molecule, we have, then 3

3n

j=1

j=1

Cv = Dj D(vl/T) + 1i E'(vj/T)

(1)

where the three Debye functions are the contributions of the molecular vibrations and the 3n -3 Einstein functions are the contributions of the atomic vibrations. The work of Ellis particularly4 and of the writer5 has shown conclusively that various non-polar bonds have characteristic frequencies in the short infra-red region, which yary only slightly from compound to compound. Let us take, now, a compound AXJByCzDw and having b such bonds and in which the atoms A, B, etc., are not arranged to form any closed ring. We have, first, introducing the b bond frequencies: 3n-b

3

D'(v,/T) +

Cr = j=1

b

E E'(vjlT) + i=iE E'(vj/T) j=1

(2)

in which the last term is the contribution of the various bond frequencies. For compounds such as we have chosen, having no ring formation, we have b = n - 1 3n - b -3 = 2(n - 1). We may write, then, 3n-b 2(n-1) (2a) ,EE = E E'(vjlT). j=4

j=1

Our problem now is to distribute these 2(n - 1) frequencies among the atoms in the molecule. Suppose we take one particular atom, say of the species A, as at the center of vibration of the molecule; then its contributions have been expressed in the Debye functions and we have the vibrations of only x - 1 atoms of the species A to consider. This distribution becomes more exact, obviously, the more symmetrical the molecule and the greater the number of atoms contained in it. An attempt will first be made to understand the physical significance of the 2(n - 1) = 2 [(x -1) + y + z + w] frequencies. Take, for example, the molecule CC14 and take a particular Cl atom. It can vibrate in all iirections in space, of course. It is held firmly to the C atom by the bond, and the forces binding it do not vary much if other atoms are substituted for the remaining Cl atoms; this is shown by the heats of combustion of bonds6 and by the characteristic infra-red bond frequencies. On the other hand it is subject to weaker forces from the other atoms to which it is not

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PROC. N. A. S.

bonded, that is, to the other Cl atoms. Consequently the vibrations of this Cl atom along the line of the bond, its bond vibrations, may be expected to lie in a shorter region of the infra-red spectrum than will its vibrations resolved in the two directions perpendicular to the bond. Furthermore, these slower perpendicular vibrations will depend on the neighboring atoms to which this particular Cl atom is unbonded and so cannot be expected to be found unchanged from compound to compound. It is reasonable, therefore, to identify the 2(n - 1) frequencies with the vibrations of the atoms across the bond; and the bond frequencies represent the vibrations of the atoms along the lines of their bonds. Having, now, two unbonded vibrations for each atom (except the atom at the center of the molecule, already taken care of by the Debye functions), 2(n-1)

2(x-1)

E E= E ( j=1

j=1

2y

2z

2w

Z (/C)/T) + E ( (D) T) (2b) A)/IT) + j=1E (v(B)/T) + j=1 j=1

in which the indices refer, of course, to the atomic species. We will now introduce into our equations certain approximations which are necessary for comparison with measurement, but which probably do not introduce any appreciable inaccuracies. First, following Nernst, a mean vibration in all directions will be used for the molecule instead of three separate vibrations, 3

Z D'(vj/T) =

j=l

f'D(Vm/T)

(2c)

f'D being now the Debye atomic heat equation and v. the mean molecular vibration. Suppose we neglect, now, the variation of the unbonded vibrations of an atom with its position in the molecule; that is, for C C14, we assume that the unbounded Cl vibrations are all the same, or, in general, in an open chain carbon compound that the unbonded C vibrations are the same, etc., and suppose that instead of two separate vibrations we use mean vibrations. We have, then, 2(x-1)

j=1

( A)/T) E

=

2(x

-

1) E(v(A)/T)

2y! E'(74 I/T) = 2y E v I)/T)l j=1 2z

E>E(4C)/T) = 2z E(-v(C)/T) .=1

fj E'(P )/T) = 2z E'(vID)/T) j=1

F

(2d)

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Substituting equations 2a, 2b, 2c, 2d in (2) we have for the molecular heat capacity of AXBYCZDw

Cv

=

f'D(vm/T) + 2[(x- 1) E'(v(A)/T) + y E'(v(B)/T) + z E'(v(C)/T) b + w E'(v(D)/T)I + E E'(v(b)/T). (3) j=1

Here

fOm (O/T)4e6/Tdo

9R

f

(OmIT)3 Jo Ef

=

R(O/T)2e/lT T 2 (eGI

-

1)

(eWI

- 1)2 h k

1 J

where h is Planck's constant, k the gas constant per gram, R the gas constant per mol, T the temperature and v the frequency. The first term of (3) expresses, then, the contributions to the heat capacity of the molecular vibrations; the frequency is determined, on the strict theory, by the elastic constants of the crystal or from the heat capacity at some one very low temperature, according to the method of Hopf and Lechner.7 The last term expresses the contributions of the bond frequencies; these are determined by the characteristic bond infra-red measurements. The intermediate terms are the contributions of the vibrations of the atoms across their bonds; these are likewise determined, strictly, from infra-red measurements, but one difficulty is to be expected, namely, the distribution of measured frequencies among the 2(n - 1) vibrations. For if these are not constant from compound to compound, the assignment of bands that appear in the spectrum of one substance and not in those of another will not be at all clear. In a subsequent paper the selection of the appropriate frequencies will be discussed and with them the application of the equations (3) and(4) to the heat capacities of certain organic solids will be made. The general form of these equations will permit of application, also, to other non-polar compounds. * Johnston Scholar, Johns Hopkins University. 1 E. 0. Salant, Proc. Nat. A cad. Sci., 11, 227, 1925. 2 M. Born, Atomtheorie des festen Zustandes, Teubner, Berlin, p. 647. 3 W. Nernst, Theory of the Solid State, Univ. of London Press, 1914, §11. 4 J. W. Ellis, Physic. Rev., 23, 48, 1924; 27, 298, 1926. 5E. 0. Salant, Proc. Nat. Acad. Sci., 12, 74, 1926. 6 A. Eucken, Fundamentals of Physical Chemistry, McGraw-Hill, New York, 1925, §354. 7 L. Hopf and G. Lechner, Verhandl. deutsch. physikal. Ges., 16, 643, 1914.

PHYSIOLOGY: T. M. CARPENTER

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419

of the different effect of rectal injection upon metabolism are that the material is absorbed in such a manner that it goes directly into the systemic circulation and avoids the portal circulation and that differences observed in the results are due to the slowness of absorption. The first explanation seems untenable because all of the indications both from this work and from other researches are that the material passes high enough into the large intestine to be absorbed into the branches of the portal circulation and that consequently it must pass through the liver by means of the portal circulation. Experiments by others with small quantities of sugar given by mouth indicate that it was not the small quantity absorbed which produced the effects in question so that the slowness of absorption can hardly explain the results. From the results in this investigation it is believed that the material introduced by rectum passes into the portal circulation and through the liver and then throughout the body. As a hypothesis it is suggested that the liver, as well as the rest of the alimentary tract, under the conditions of the experiments is not in an active state so that the substance is metabolized throughout the body in a manner similar to the utilization of material in muscular work. Dextrose and alcohol are both substances which can be utilized in the body when given in enemata. A review of the earlier literature, the details of the experiments, a discussion of the practical applications and theoretical calculations, together with a suggested hypothesis, appear in Publication No. 369 of the Carnegie Institution of Washington. 1 Benedict and Dodge: Tentative Plan for a Proposed Investigation into the Physiological Action of Ethyl Alcohol in Man. Proposed Correlative Study of the Psychological Effects of Alcohol on Man. Privately printed, Boston, Massachusetts, 1913. See, also, Dodge and Benedict, Carnegie Inst. Wash. Pub., No. 232 (1915), pp. 266-275. 2 Higgins, Jour. Pharm. and Exper. Therap., 1917, 9, p. 441. 3 Benedict and Carpenter, Carnegie Inst. Wash. Pub., No. 261, 1918, pp. 240-246. 4 Benedict and Tompkins, Boston Med. and Surg. Jour., 1916, 174, pp. 857, 898 and 939.

Dodge and Benedict, Carnegie Inst. Wash. Pub., No. 232, 1915, p. 233. Miles, Carnegie Inst. Wash. Pub., No. 333, 1924, p. 225. 7Rosemann, Oppenheimer's Handbuch d. Biochem., 1911, 4 (1), p. 423. 5 6

ERRATA These PROCEEDINGS, pp. 335 ff., May, 1926. 3n

In equation (1),

3n

_ E E should read j=4.i .1 1 =

3n-b

in equation (2),

3n-b

should read

j=4 in equation (3), vb) should read (b), and on page 337 "k the gas constant per gram" should read "k the gas constant per

molecule."-E. 0. SALANT