T H E ENTROPY OE' VAPORIZATION OF UNASSOCIATED LIQUIDS BY JOHN CHIPMAN
Numerical relationships between the boiling point and heat of vaporization of unassociated liquids have proven useful in predicting heats of vaporization and vapor pressures as well as in distinguivhing "normal" from associated liquids. Six empirical equations have been proposed expressing the heat of vaporization as a function of the boiling point only. These equations usually take their simplest form when expressed as entropy of vaporization. I n this form they may b t written as follows:
B2 P2t
5 B s 21 h
gt o
t
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FIQ.I Trouton,' A S = 21 (1) Nernst,2 A s = 9.5 log Tb - 0.007 Tb (2) Bingham,s A S = 17 0.011 Tb (3 ) F o r ~ r a n d , ~ A s = IO. I log Tb - 1.5 - 0.009 Tt, +0.0000026 Tb2(4) Kistiakowsky,s A S = 4.578 log (82.07 Tbj (5) Mortimer,6 A s = 4.23 (-68/Tb 4.877 0.0005 T b ) (6)
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Curves representing these equations in the temperature range 300' to I. The points, representing observed entropies of vaporization of unassociated liquids, are taken chiefly from measurements by Mathews;' a few are quoted from Mortimer. It will be observed that any of theequationsmaybe usedin this range without serious error but that equations I , 5 and 6 appear to fit more of the data than do the others. For a more complete test of the relative accuracy of the equations i t is necessary to comjoo°K are plotted in Fig.
Trouton: Phil. Mag., (5)18, 54 (1884). Nernst: Gott. Nachr., 1906. 3Bingham: J. Am. Chem. SOC., 28, 723 (1906). 4 de Forcrand: Compt. rend., 156, 1493,1648,1809 (1913). 6Kistiakowsky: J. Russ. Phys. Chem. SOC., 53, zj6 (1921). 6 Mortimer: J. Ab.Chem. SOC.,44, 1429 (1922). 7 Mathews: J. Am. Chem. SOC.,48, 562 (1926).
ENTROPY O F VAPORIZATION O F UNASSOCIATED LIQUIDS
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pare them a t lower and a t higher temperatures. This comparison is made in Figures z and 3. The points shown in these figures are taken from the following sources : oxygen and nitrogen, Dana;8 other low-boiling liquids, International Critical table^;^ cadmium and zinc, calculated from free energy equations of Maier ;IO lead, tin and gold, International Critical table^;^ other high-boiling liquids, Jones, Langmuir and IvlacKay.ll I n the low-temperature range Equation 5 agrees with the observed values, except helium and hydrogen, with surprising accuracy; in the high temperature range it is the only one of the six that approaches the estimates of Jones, Langmuir and NacKay. For these reasons and because of its simplicity it is to be regarded as far superior to the other equations.’*
Kistiakowsky’sEquation 20 Kistiakowsky’s derivation of Equation 5 is open t o the criticism that it involves the cancellation of two quantities which 1 are not apparently cancellable. The equa- 5 1 5 tion itself however contains no empiri- $ cal constants other than the gas constant, R, and may be regarded as belonging to the class of semi-empirical equations which have been so useful in 5 approximate calculations of entropy. I n 50‘ its simplest form t h e equation is written:
iro E
e5v
50.
r4.
,oo.
FIG.2
A S = RlnV (7) where In V is the natural logarithm of the molal volume of the vapor in equilibrium with the liquid (expressed in c.c.). R is of course expressed in the same units as A S. Assuming that the vapor is an ideal gas we may substitute R T for V at the boiling point (since P = I ) ; then: A s = RlnRTb (8) or A H = RTb In RTb (9) The second R must be expressed in c.c.-atmospheres and when this is done Equation 5 results. Hildebrand’s Rule HildebrandI3 has shown that the entropy of vaporization for normal liquids is the same when evaporated to the same vapor concentration. Choosing arbitrarily temperatures at which the vapor concentration is 0.00507 mols *Dana: Proc. Am. Acad., 60, 241 (192j). “International Critical Tables”, 1, 102. 10 Maier: J. Am. Chem. SOC.,48, 356 (t926). l1 Jones, Langmuir and MacKay: Phys. Rev., (2),30, 201 (1927). l2 Although Kistiakowsky’s equation antedates that of Mortimer, the latter had no knowledge of It. Mortimer’s equation fits the data from argon to copper fully as well as does Kistiakowsky’s. Hildebrand: J. rim. Chem. Soc., 37,970 (1915).
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JOHN CHIPMAN
per liter he found an average value of AS of about 27.4 calories per degree. Metals gave a somewhat lower value, about 26.3. Equation 7 is evidently another statement of this principle. According to this equation the entropy of vaporization to a concentration of 0.00507 mols per liter is 24.3 calories per degree. The discrepancy of two to three entropy units makes it seem worth while to recalculate some of Hildebrand's values on the basis of more accurate vapor pressure data than were available at the time of his pub-
y $15
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ABSOLUTE BO/L/NG
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FIG.3 Most investigators express the vapor pressure of a liquid by an equation of the form, log P,,,, = A/T B CT. If the vapor concentration is 0.00507 mol9 per liter, then by the gas law, log P,, = -0.5 log T. By combining these equations we may solve for the temperature T1 a t which the concentration of the vapor is 0.00507. The entropy of vaporization, ASl at this temperature is then obtained by differentiating the vapor pressure equation with respect to temperature and employing the Clausius-Clapeyron relationship. This is the analytical equivalent of the graphical method used
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TABLE I Entropy of Vaporization to a Concentration of 0.00507 Mols per Liter Substance
T1"K
Nitrogen 55.I 55.1 Nitrogen Oxygen 65.8 143.5 Ethane Naphthalene 427. 544. Anthracene Cadmium 962 1107 Zinc
AS1
Vapor pressure data
26.6 Cath14 (extrapolated) 2 6 . 8 Henningls " 27.2 Cath14 26. g Loomis and Waltersl6 26.4 Mortimer and Murphy'' 26. o Mortimer and Murphy 25. 2 Maierlo 25.3 Maier
"Cath: Proc. Acad. Sci. Amsterdam, 21, 656 (1919). l5 Henning: Z.Physik, 40, 775 (1927). 16 Loomis and Waltem: J. Am. Chem. SOC., 48, 2051 (1926). 17Mortimer and Murphy: Ind. Eng. Chem., 15, 1140 (1923).
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by Hildebrand. The results of the calculations for several liquids are given in Table I . The values are slightly lower than those obtained by Hildebrand but higher than the value predicted by Equation 7 . Kistiakowsky’s equation is therefore less reliable a t lower temperatures than a t the boiling point. Entropy of a Monatomic Liquid The entropy of a monatomic gas is given by the Sackur equation:18 S = R In (T3’*w 3’2 V) C
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(10)
By subtracting Equation 7 an expression is obtained for the entropy of a monotornic liquid : S = 3/2 R In (Tw) C (11)
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This equation holds fairly well for the liquefied rare gases (except helium) a t their boiling points; for mercury however it is seriously in error. It may be considered a t best a rough approximation to be used only in the neighborhood of the boiling point. summary
Equations expressing the entropy of vaporization of a normal liquid as a function of its boiling point have been compared graphically with recent experimental data. The equation of Kistiakowsky is found valid over a wide range of temperature. This equation is in accord with Hildebrand’s rule. Recent data lead to values for the entropy of vaporization to a concentration of 0.00507 mols per liter somewhat higher than that predicted by the Kistiakowsky equation. An expression is obtained by which the entropy of a monatomic liquid in in the neighborhood of its boiling point may be roughly estimated. Department of Chemistry, Geol-gza School of Technology, AUanta, Ga. See Lewis and Randall: “Thermodynamics”, p. 455 (1923).
