OLIVER I. I. BROWN Syracuse University, Syracuse, New York
ALMOST without
exception textbooks of physical chemistry follow the original derivation of Clausius in arriving a t the Clausius-Clapeyron equation for vapor pressures. The assumptions are made that (1) the volume occupied by one mol of liquid under its saturation pressure is negligible compared to the volume occupied by one mol of vapor, (2)the vapor behaves as an ideal gas, and (3) AH of vaporization does not change with temperature. The exact form of the Clapeyron equation
is then readily converted to the Clausius-Clapeyron equation
$($) + constant This equation predicts that In P will be a straight line when plotted against 1/T and that the slope will be - AH/R. It is usually pointed out that the assumptions limit the use of this equation to low pressures. I t then becomes impossible to explain the experimental fact that the plot of In P versus 1/T is frequently very nearly linear up to the critical point, even though a11 three of the assumptions made are evidently no longer
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It is the purpose of this paper to show a simple alternative derivation which avoids these difficulties and which shows that linearity of In P versus 1/T does not require that AH be independent of temperature except in the limit of zero pressure. This derivation also shows that even at one atmosphere pressure a simple correction should be applied in calculating AH from the slope of the straight line. The derivation is as follows:
where z, the compressibility coefficient, is a function of temperature and pressure, and varies from unity at zero pressure to about 0.2 or 0.3 at the critical point. Substituting the right-hand member of equation (3) into equation (1)we obtain
Then
Equation (6) is exact, but contains three terms which vary with temperature. Now if In P is linear with 1/T with slope B, that is, if
+ constant
(7)
AH = -zBR(I - (VL/V.)
(8)
In P = B(1/1')
it follows that Since z varies from unity to 0.2 or 0.3 as the pressure increases along the saturation curve from zero to the critical pressure, and since (1 - (VJV,)) varies from unity at zero pressure to zero at the critical point, it follows that linearity of In P versus 1/T demands that AH vary from -BR at very low pressures to zero at the critical point. At pressures near one atmosphere AH values calculated as -BR are usually several per cent too large. Multiplication by the 2 factor will restore them to agreement with the calorimetric values of AH if the equation in P = B/T A really fits the data, since the term (1 - (VL/V0)3will still be essentially unity at one atmosphere pressure, except in the case of extremely low boiling liquids. The recent measurements of Friedman and White1 on the vapor pressure of liquid nitrogen serve as an excellent example of a case in which the In P versus 1/T plot is practically linear up to the critical point. The AH of vavorization calculated from the vauor nressure curve-is tabulated by these authors at even temperatures and found to vary from 1320 calories per mol a t the boiling point to 280 calories a t lo below the critical temperature. It should be noted that equation (6) is perfectly general and without approximations. I t shows that the slope of the tangent to the plot of In P versus 1/T at any point when multiplied by -zR(l - V,/Vo)) will give the AH of vaporization at that temperature. Here again the z term is increasingly important with increasing pressure. The term involving volumes makes AH zero at the critical temperature since then V , will be equal to V,. The z factor is, to a good approximation, a general function of reduced temperature and reduced pressure so that where critical data are available it is readily possible to calculate this quantity even if P-V-T data are not available for the region of interest. The Berthelot equation of state is also readily solved for P V J R T , which is the z factor. The alternative derivation of the Clausius-Clapeyron equation suggested here is certainly not too difficult for the beginning student in physical chemistry. B y postponing the three approximations to the last step,
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'FRIEDMAN, A. S., 3931-2 (1960).
AND
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D. WHITE,J. Am. Chem. Soc., 72,
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AUGUST, 1951
the student can obtain a clearer idea of how these approximations enter into the final result. For instance, it will be found that a t low pressures the assumption of ideal gas behavior introduces more error than the as-
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sumption of negligible liquid volume, but that a t higher pressures this situation is reversed. Only in the limit of extremely low pressures does theassumption of the constancy of AH with temperature enter at all.
