RALPH SPITZER Oregon State College, Corvallis, Oregon
A
FAMILIAR experiment in the elementary physical chemistry laboratory is the determination of C,/C, by the method of Clement. and DQ~ormes.~ The equation usually given for determining
Because of the rapidity of the expansion and the large size of the carboy, it may be assumed, with some justification, that the expansion is adiabatic and, hence, that q = 0. Application of the first. law of thermodynamics leads to AE = C,(Tf - T t ) = 0 - w = -P/(V - %) (2)
from this experiment is not correct because it is based on the expression pv' = constant (1) which is correct only for a reversible, adiabatic expansion of a perfect gas. The expansion in the method under discussion is very definitely irreversible and, hence, a different treatment must be given. Because the only readily available treatment of this experiment2is based on this error, which is a rather widespread one in thermodynamics, we shall discuss the derivation in some detail. Fortunately, because of the small excess pressures which are used in the laboratory, the difference between the equation which will be derived here and that which has been in common use is small. The present derivation still makes use of the simplifying assnmptions that the process is perfectly adiabatic, that the gas is ideal, and that the gas has constant heat capacity. In the method of Clement and DBsormes, a large (40 to 50 liter) carboy of volnme v, is filled with air or some other gas a t temperature T , to a pressure P , = P! AP, where p, is atmospheric pressure, The stopper is removed from the carboy and the gas expands to p ~ its , temperature, meanwhile, falling to T'., The stopper is immediately replaced in the bottle and, after the temperature of the gas returns to T , (the ambient or thermostat temperature), the pressure is found to be
This expression may be contrasted with the first law equation from which (1) is derived, namely:
+
Equation (3) is the correct equation to use only if the confining pressure is kept always infinitesimally less than the internal pressure of the gas. Such a process would be the infinitely slow expansion of a gas in a cylinder, when the pressure on the confining piston is constantly decreased by infinitesimal amounts to maintain a balance between internal and external pressure. Such a balance is not achieved in the Clement and DBsormes experiment, in which the gas expands rapidly and irreversibly through a finite pressure drop. Such irreversible expansions, which are quite common in practice, are often neglected in elementary thermodynamics courses (except for the-example of expansion into a vacuum). One of the purposes of this paper is to point out that an irreversible expansion against constant atmospheric pressure can be treated by thermodynamic methods quite as simply as the usual reversible expansion. The application of the perfect gas law to (2) yields:
Dividing both numerator and denominator by pp,, p, is necessarily less than p, because some gas has been lost to the atmosphere in expansion. Inasmuch as the expansion is performed a t constant atmospheric pressure, the work done is
where v, is the volume to which the gas expands-that is, the volume of the gas remaining in the carboy plus that of the escaped gas a t either T, and pf or T , and po. ~
Published with the approval oi the Oregon State Monographs Committee. Oreeon State Colleee. research DaDer number 108. School of Science, Department 07 chemistry. MACK,E., AND W. G. FRANCE, "A Lahoratory Manual of Physical Chemistry," 2nd Ed., D. Van Nostrand Company, Inc., New York, 1934, pp. 55-8. Tho symbols used in Mack and France are adopted in this paper. 25
..
I n order to be able to calculate v in terms of the experimental variables, we require the ratio v,/v,. Taking account of the gas that is lost in the expansion and the fact that p, is measured a t the original temperature, it is easy to show2that
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The final result is
Inasmuch asp, is usually within one to two per cent of p,equation (6) may be simplified by cancelling p, against p, to give Y =
Api Api
-
AP~
(7)
the approximate equation found in most elementary textbooks. The use of equation (1) in place of (2) results in log Pi y e -
Pi
log p
Pa
Expansion of the logarithms in eqnation (8) with neglect of higher powers leads again to equation (7). Although it might appear that equation (6) mould give better results than equation (8) when Ap, is fairly large, i t is probable that under such conditions both equations would be far from the truth because of the large temperature change and resultant heat leak. Under practical conditions of a small pressure drop, therefore, equations (6) through (8) would give results which are identical within experimental error. From the pedagogic standpoint, however, the derivation of equation (6) presents a good opportunity to stress the neglected fact that equation (1)applies only to reversible adiabatic processes. Equation (6) has the additional advantage of being easier to use in numerical calculations than is (8).
