The results of the first paper of this series, and a generalization of a method due to Eyring, are used to obtain an expression for the free volume of a liquid, Vf=fb3g3/h3n3×[RT/ΔEv]3⁠, and an equation for the entropy of vaporization. ΔS=R[ln Vg−lnVl−ln β+3lnβ+3ln(ΔH/RT−1)]⁠. Here β=γfb3g3/h3n3⁠, where γ measures the interference in the liquid with the internal motions (rotations, vibrations) of the molecule, and f, b, g, h, n are quantities which depend on the geometry of the liquid and the energetic and dynamic interaction of the molecules. The rule of Barclay and Butler, that the 25°C value of ΔS for various pure liquids has a rough linear relationship to the corresponding ΔH of vaporization, is shown to imply a general tendency for a liquid to have a smaller β the larger its ΔH of vaporization. In many cases this means a smaller γ, resulting from increased interference with rotation of the molecules in the liquid. Pitzer's perfect liquid has a value β = 16, sensibly independent of ΔH. This is taken to mean that in such a liquid as benzene or carbon tetrachloride (β≈6) the interference with free rotation is considerable. For CS2 there is evidence that the intermolecular force field differs from ``normal,'' and the difference in potential function between liquid metals and normal liquids shows up strongly. Accepting the value β = 16 found for the ideal liquid as a norm, it is proposed to call R ln (16/β) for any liquid the hypothetical entropy defect (HED) and interpret it as the amount by which the entropy of the liquid (referred to the same substance as a perfect gas) is less than that of the ideal liquid in the ``corresponding'' state.

Topics
Entropy, Physics of gases, Thermodynamic states and processes, Liquid metals, Intermolecular forces, Heterocyclic compounds
1.
H. S.
Frank
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J. Chem. Phys.
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2.
I. M.
Barclay
 and
J. A. V.
Butler
,
Trans. Faraday Soc.
34
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1445
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1938
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3.
R. P.
Bell
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Trans. Faraday Soc.
33
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496
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1937
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4.
M. G.
Evans
 and
M.
Polyani
,
Trans. Faraday Soc.
32
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1333
(
1936
).
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5.
R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (The Macmillan Company, New York, 1940), 800, p. 319, discuss the commonly accepted distinction between normal and associated liquids. We shall use the words to refer to the substances to which these authors apply them, but shall find it necessary to reject their hypothesis that molecular rotations are unhindered in the liquids they consider typically normal.
6.
H.
Eyring
 and
J. O.
Hirschfelder
,
J. Phys. Chem.
41
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249
(
1937
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7.
Fowler and Guggenheim, reference 5, 802, p. 325.
8.
G.
Mie
,
Ann. d. Physik
11
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657
(
1903
).
9.
J. F.
Kincaid
 and
H.
Eyring
,
J. Chem. Phys.
5
,
587
(
1937
).
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10.
Fowler and Guggenheim, reference 5, 806, p. 332.
11.
J. H. Hildebrand, Solubility (New York, 1936), second edition, p. 99.
12.
F.
Simon
 and
F.
Kippert
,
Zeits. f. Pysik. Chemie
135
,
113
(
1928
).
Rice
 (
J. Am. Chem. Soc.
63
,
3
(
1941
)) has shown that this value is too high to be reasonably accounted for by a plausible inter‐atomic potential curve for solid argon,
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and has (
J. Chem. Phys.
12
,
289
(
1944
)) proposed a very different sort of curve to fit this datum. We shall not enter into a discussion of the relative plausibilities of the various assumptions which may be made to resolve the difficulty, but note that a smaller value than Simon and Kippert’s for (∂P/∂T)V of solid argon would also fit better into the picture we are presenting here, giving a smaller value for n, and therefore a smaller one for g and a larger one for (∂d/∂l)T.
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13.
Cf., for example,
J. E.
Lennard‐Jones
 and
A. F.
Devonshire
,
Proc. Roy. Soc. London
A163
,
53
(
1937
).
14.
J. H. Hildebrand, Solubility (New York, 1936), second edition, p. 102.
15.
K. S.
Pitzer
,
J. Chem. Phys.
7
,
583
(
1939
). Pitzer tabulates φl(= Vl/Vc), log π(π = Pvap/Pc), Vg/Vl, and ΔSvap, for variable values of ϑ(= T/Tc) for the ideal liquid in equilibrium with its saturated vapor. Corrections can be made for gas imperfection using the Berthelot equation. Then ΔSvap yields Vg/Vf, which, with the tabulated Vg/Vl gives Vf/Vl. Since this ΔS is an equilibrium quantity ΔS/R = ΔH/RT, so that β is easily obtained. It is noteworthy that ΔHvap contains Tc as a factor, and, if T is to be 25 °C, Tc cannot be higher than about 533 °K without making ϑ so low (<0.56) that the ideal liquid would freeze. On the other hand, this makes ΔH only some 5600 cal./mole. The fact that ΔH = 5600 is the highest value for which β can be calculated for an ideal liquid at 25 °C detracts somewhat from the comparison here made. The meaning of the calculation, however, is that β is sensibly independent of temperature for these liquids, which justifies using the value 16.0 as a “norm” for any liquid at any temperature. Incidentally, it appears that most ordinary liquids, with ΔH⩾6000 cal./mole, would be solids at room temperature if they behaved as ideal substances. The reason that they are liquids at room temperature, then, is that in melting they gain entropy from sources which are not available to the ideal liquid—orientations and bendings. It is very striking that hexamethyl ethane, which is so bunched up and symmetrical as to be of nearly “ideal” shape, melts at about 102 °C (G. Egloff, The Physical Constants of Hydrocarbons (Reinhold Publishing Corporation, New York, 1939), p. 57), compared with melting points well below 0 °C for all the other open chain hydrocarbons of C8 or below. The high melting points of benzene and cyclohexane also support this point of view.
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16.
Without entering here into a full discussion of the variation of β with T in real liquids, we may remark that (14) leads to the relation
∂ ln β∂ ln T = 1−∂ ln Vl∂ ln T−3(1−RTΔH)−1−ΔCPR(1−4RTΔH)(1−RTΔH)−1
. This shows that the more negative ΔCP, the more strongly β increases with T in qualitative agreement with our physical picture. Numerically ∂ ln β/∂ ln T at 25 °C is predicted to be positive for ΔCP more negative than about −7.0 at ΔH = 7200, or about −6.0 at ΔH = 10,800.
17.
Cf. note at end of reference 2.
18.
J. H.
Hildebrand
,
J. Chem. Phys.
7
,
233
(
1939
).
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19.
R. S.
Halford
,
J. Chem. Phys.
8
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496
(
1940
).
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20.
J. F.
Kincaid
 and
H.
Eyring
,
J. Chem. Phys.
6
,
620
(
1938
).
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1945
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