UTILIZATION OF EQUILIBRIUM VAPOR PRESSURE DATA LEO BREWER and ALAN W. SEARCY University of California, Berkeley, California
IT IS still fairly common practice to determine heats of vaporization from the slope of a plot of R In p versus l/T, and the entropy of vaporization from the intercept of this plotted line with the axis of 1/T = 0%-1. Such a treatment gives a constant value for AHo which is an average of the true AHo's in the temperature range investigated and gives an entropy which may be several cal. mol-I deg.? in error for even the experimental range. Normal boiling points have often been predicted from such equations which assumed that the heat of vaporization determined a t vapor pressures of less than a millimeter was constant up to a vapor pressure of one atmosphere. Boiling points so predicted usually are considerably lower than the actual boiling points because the heat capacities of liquids are almost invariably higher than those of the corresponding gases, so that the plotted slopes are too high a t pressures higher than those experimentally determined. Heats of fusion calculated from the difference between average heats of sublimation and vaporization may also he greatly in error. For example, suppose we have vapor pressure data available for a certain compound for 100° above and 100' below its melting point. If we plot R in p versus l/T and draw the best straight lines through the data above and below the melting point, we find an average value of 23,000 cal. mol-I for the heat of sublimation and of 20,000 cal. mol-I for the heat of vaporization. Assuming that these average heats apply a t the melting point, the heat of fusion would be their difference, 3000 cal. mol-I. If, however, the heat capacity chinges for sublimation of the solid and for vaporization of the liquid are each about -10 cal. mol-' deg.-', the AH0 of sublimation a t the melting point is about 500 cal. mol -' lower and A H o of vaporization 500 cal. mol-' higher than the average values, so that we should subtract 20,500 from 22,500 getting 2000 cal. mol-' as the heat of fusion. The value calculated from the average heats was 50 per cent off. Obviously then, when accurate values are desired the assumption of constant AHo is inadequate if we are to obtain maximum use from our vapor pressure measurements. We must derive equations for AHo and for A S o as functions of temperature which utilize not only our vapor pressure determinations hut also any heat capacity and entropy data available from calorimetric or spectroscopic measurements. With the appropriate data available it is a simple matter to develop equations which will permit very accurate predictions of AHo, AS", and AFo values far outside the experimental
temperature range. Unfortunately, most thermodynamics texts give only sketchy instruction in techniques for applying these equations. In this paper we will consider these techniques in detail, using data concerning the vaporization of Brl to illustrate the methods we discuss. CORRECTION OF DATA FOR DEVIATION FROM THE PERFECT GAS LAW
The first step in utilizing vapor pressure data to develop thermodynamic equations is to determine AFo, that is, the free energy of vaporization of the liquid to the standard thermodynamic gaseous state where the fugacity j equals one atmosphere a t the temperature of measurement. Since A F for vaporization of the gas t o its equilibrium pressure a t the temperature of measurement is zero, to obtain A F o we need only calculate the free energy change when the vapor changes from the equilibrium fugacity to a fugacity of one atmosphere. This change is A F o = -RT In j, , where j, is the fugacity of gas in equilibrium with the liquid a t the given temperature. It is common practice in utilizing vapor data with pressures of an atmosphere or less to assume the vapor a perfect gas, in which case prehure equals fugacity and AFo = -RT In p, , where p, is the equilibrium pressure. Often, however, deviation from the perfect gas law is sufficient a t even an atmosphere of pressure t o warrant correction of pressures to fugacity. Whether or not the pressure and fugacity are essentially equal under given conditions can be easily determined from the following empirical equation:
where h is the molal volume of the liquid in ~ m . p~ , is the pressure in atmospheres, and Tb is the normal boiling temperature in degrees Kelvin. This equation may be found by solving the van der Waals equation for v, obtaining v = RT/p b - a/pv ab/pv2, where the letters have their usual significance, neglecting the second order correction term ab/pv2, and making the substitution a/pu = a/RT. For normal liquids a is approximately equal to the product of the energy of vaporization and the liquid molal volume 0 expressed in appropriate units, and from Trouton's rule, the energy of vaporization is approximately 20T, calories. For most gases b averages 1.351 We now know v as a function of temperature, pressure, and two readily available constants, the boiling temperature and the liquid
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OCTOBER, 1949
549
molal volume. By well-known thermodynamic relations' we can now get equation (1). Equation (1) is much more generally applicable than either van der Waals' equation or Berthelot's equation, since these latter equations require vapor density or critical point data. For a comparison of fugacity to pressure a t the boiling point equation (1) reduces to simply: . f /.p = .
1
- 0.1072' 1b
(2)
which tells us immediately that among liquids of similar boiling points the liquid with highest molal volume will show the greatest deviation of vapor pressure from the fugacity, while among liquids with similar molal volumes compared near their respective boiling points the liquid of lowest boiling point will show the greatest deviation of vapor pressure from fugacity. It should be noted that the equation is strictly valid only for liquids consisting of gaseous molecules held together mainly by van der Waals and similar forces. It would not be expected to hold well for an ionic substance such as NaCI. Let us apply equation (1) to the data of Scheffer and Voogd2 a t 2.5 atmospheres to determine if corrections of vapor pressures to fugacities are necessary for their low pressure results. With p = 2.5 atmospheres, T = 362.51°K., ir, = 55.2 ~- m ~-,-=- . TB 331.g°K., we find that f/p
=
2.5
x 55.2
1 -0.0158 X -362.5
7'78
331'9 -1 or 0.963. So a correction .is 362.5 desirable for accurate work. Whenever the critical constants are available it is preferable to use Berthelot's equation for correcting pressures to fugacities. Berthelot's equation written in terms of the critical constants gives
(
where T, and p, are the critical temperature and pressure of the substance considered. Subaituting into this equation the critical constants for Br2, 584'K. and 102 atmospheres, we find that a t the same temperature and pressure considered in using (1) the ratio f/p is 0.958. For Brs a t 2.5 atmospheres, then, equation (1) gives the same ratio of fugacity to pressure as the Berthelot equation to within the probable error in the Berthelot correction.
HEAT CAPACITIES Heat capacities for many common substances are now known, and whenever they are available should be used in developing thermodynamic equations from vapor pressure data. Ideally, these.C/s are listed as functions of temperature in forms such as a bT cT2
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'LEWIS,G. N., AND M. RANDALL, "Thermodynamics," 1st ed., McGraw-Hill, Inc., New York, 1923, Chap. XVII. F. E. C., AND M. VOOGD, Rec. trav. ehim., 45, SCHEFFER, 214 (1926).
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. . . or A BT - CT-', where the coe5cients in the various terms of the heat capacity expression are empirical constants; however, even a constant value for AC, greatly increases the usefulness of the free energy equation, and when C, values are not known for the molecules being studied they should be estimated from data for similar molecules. It is almost always possible to estimate ACis within 1 or 2 cal. mol-I deg.-1. Sunnose that we have no emerimental value for the ' AC9 of vaporization of bromine, but that we do know the C,'s for chlorine and iodine. Proceeding by a useful general method, we will estimate separately the heat capacities of liquid and of gaseous bromine, because the heat capacity for a single phase is usually more easily predicted than the heat capacity difference between two phases. Obviously, in the present case with only two related substances from which to estimate, we would find the same answer by estimating directly from the ACis of vaporization as by considering each separately. For chlorine the Cn's are 15.9 cal. mol-I deg.-I for the liquids and 8.1 for the gas4; for iodine the values are 19.55and 8.S4. Since bromine falls between chlorine and iodine in the periodic table, we know that the C,'s for bromine must be approximately the average of the corresponding values for chlorine and iodine; therefore. we estimate 17.7 to be the heat cavacitv of liquid bromine and 8.5 to be that for the gas; making AC, = -9.2 cal. mol-' deg.? for vaporization. The actual values are about 17.3 for liquid Br8' and 8.6 for gaseous bromine, making AC, -8.7 cal. mol-I deg.?, so that our estimated value is in error by 0.5 cal. mol-I deg.?, or less than 6 per cent of the error introduced by assuming that ACI, = 0. Kelley,8J Brewer, Bromley, Gilles, and Lofgren,Io and the National Bureau of Standards4give convenient summaries of heat capacity values for many substances. The references listed in the Bureau of Mines Bulletin, "Entropies of Inorganic Substanced,!' contain much useful heat capacity data. When desired C,'s are not available one can almost always find C,'s of related compounds in these works from which to estimate the desired C.'s.
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~GIA~QU W.E F., , iurn T. M. POWELL, J. Am. Chem: Soc., 61, 1970 (1939).
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National Bureau of Standards, "Selected Values of Chemical Thermodynamic Properties," Series I, 1948. FREDER~CK, K. J., AND J. H. H I L D E B ~ NJ.D ,Am. Chem. Soe., 60, 1436 (1938). ANDUEWS, T., Quart. .lour. Chem. Soe., London, 1,18 (1849). RE~NAULT, M. V., Ann. Chim. Phys., 111, 26, 268 (1849). KELLEY.K. K.. "The Free Enereies of Va~orimtionand Vapor ~ros&s of 'inorganic ~ u h s t a k . , "~ u r e a uof Mines Bulletin No. 383 (1935). KELLEY, K. K., "High Temperature Speeifio-Heat Equations for Inorganic Substances," Bureau of Mines Bulletin No. 371 11496) ,-"v*,. 'OBREWER,L., L. A. BROMLEY, P. W. GILLES,AND N. L. LOFGREN, "Declassified Atomic Energy Commission Paper," MDDC-438-F (1945). KELLEY, K. K., "The Entropies of Inorganic Substances," Bureau of Mines Bulletin No. 434 (1941).
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JOURNAL OF CHEMICAL EDUCATION
ENTROPIES
I n estimating the entropy of vaporization of a substance such as bromine we can use the well-known genWith the heat capacity known, we can evaluate the eralization, Trouton's rule. This rule tells us that for available entropy data. The relations between the B T most chemical substances the entropy of vaporization thermodynamic functions when AC9 = A -C/T-2 are given by Kelley. For our study with is 20 to 25 cal. mol-I deg.-I when the partial pressure AC, expressed as a constant, the equations of interest of the substance considered is one atmosphere. However, we can often estimate the value more accurately reduce to: by considering the entropies of vaporization of moleAC; = a (4) cules having a family resemblance to the substance of AH" = AH; aT (5) interest than by employing this rule, for example, A F 0 = AH: - a T l n T + I T (6) we find that the entropies of vaporization of fluorine, a S 0 = a + a l n T - I (7) chlorine, and iodine are 19.3, 20.4, and 22.8 cal. mol-' deg.-I a t the normal boiling points. These values If no entropy is available from calorimetric or spectro- show a regular trend and we would estimate the corscopic investigations we must find the entropy change responding entropy for bromine to lie in a regular place for our process by determining I,the constant appearing in this series a t about 21.6 cal.mo1-I deg.-I. Our in equation (6). equation gives a value of 21.8 cal. mol-I deg.-', a t the Equation ( G ) can be rewritten in the form boiling uoint,. in eood aereement. I n
