Gilbert F. Pollnow Wisconsin State University-Oshkosh Oshkosh, 54901
Least Squares Fit for TWO Forms of the Clausius-Clapeyron Equation
M o s t undergraduate physical chemistry laboratory mimuals include in them experiments to measure the vapor pressure of pure substances as a function of temperature which then suffices to determine the heat of vaporizat,ion of the pure substance via the integrated form of the Clausius-Clapeyron equation. Since experimental data are generally taken a t more than t,wo temperatures, statistical methods are usually employed to obtain the best values for the empirical constants of this eqn. (1) logloPi = A
+ B/Ti
(2)
Daniels' suggests that the method of least-squares applied to eqn. (1) to obtain the best values for the experimental constants A and B from which the heat of vaporization can then be determined from eqn. (2). Likewise Schlessinger and de Michielle in a recent call attention to their computer article in THIS JOURNAL program to accomplish this same objective. The purpose of this paper is to describe a method of leastsquares to fit the exponential form of eqn. (1) in such a way as to minimize the deviations in P rather than in the logarithm of P as suggested.= This procedure is more accurate since usually P is measured rather than its logarithm; hence, it is the sum of the squares of the residuals in P which should be minimized. The method of least-squares, however, cannot be applied directly to the exponential form of eqn. (1) since the normal equations are no longer linear in the constants A and B. I n order to get around this difficulty the exponential form of eqn. (1) can be expanded in a Taylor series about an initial approximate set of values of the constants A and B which may be obtained, for example, by arbitrarily substituting the conjugate extremes of T , and P, into eqn. (1) to give eqn. (3)
Equation (3) is linear in the correction terms dA and d B which are small, and the subscripts on the partial derivatives indicate that they are to be evaluated using the initial values of A and B. Application of the Presented at the Great Lakes Regional Meeting of the American Chemical Society in Fargo, N.D., June, 1970. DANIELS, F., et al., "Experimental Physical Chemistry" (7th Ed.), McGraw-HillBook Co., New York, 1970, p. 52. SCHLESBINGP;R, G. F., AND DE MICHIELL, ROBERTL., J. CHEM.EDUC.,47, 119 (1970). Also see reference in footnote 1.
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Journal o f Chemical Education
C (Pi,
- PiJ2 = minimum
(4 )
%,here Pt,is the experimentally measured pressure and
P,, is the computed pressure. In eqn. (4) and those which follow the summations extend over all the experimental data points and the resulting normal eqns. (5) and (6) in dA and dB become
(1)
where P,is the vapor pressure a t the absolute temperature T,,and the heat of vaporieat,ion AH is related to the constant B through eqn. (2) AH = -B(2.303)(1.987)
Principle of Least Squares requires the sum of the squares of the residuals to be a minimum, i.e.
The partial derivatives appearing in eqns. (5) and (6) are given by eqns. (7) and (8)
and dK is defined by eqn. (9) &Ti = (Pi.
- lO(A+B/Td)
(9)
Solution of eqns. (5) and (6) for dA and dB are used to improve the initial estimate f o r A and B and the process repeated until no further significant changes in dA and dB occur which requires four to five iterations when the initial estimate is made as suggested above. The figure shows the output of two remotely accessible computer programs fitting both the logarithmic and exponential forms of eqn. (1) by the method of least squares. In'the figure Tt and P,are the experimental temperatures and pressures. PCAL is the vapor pressure calculated from the least-squared exponential equation and the other column headings are self explanatory. Clearly evident in this figure is the fact that the least-squares method applied to eqn. (1) directly minimizes the deviations in log P(DLGP) but leads to wide deviations in the pressures (DP) themselves, whereas the method suggested in this paper leads to a uniform deviation in the pressures about the fitted line but biased deviations in the logarithm of P as must follow from the foregoing discussion. This is also reflected in the standard deviation in the pressure SIGMA in the two cases which in the latter case is more consistent with the usual accuracy of vapor pressure measurements, i.e., 1.0 mm Hg. Finally, the errors in experimental constants A and B of the Clausius-Clapeyron equation are respectively, 2.029 and -2.883%. The figure also includes the IBM 360/40 RAX terminal commands which the student enters on the IBM 2741 typewriter to call in either of
*
/INPUT /INCLUDE PCEMl /END RUN M.0073 ACTION IN PROGRESS. END OF COMPILATION MAIN 7 0.0 60.0 149.5
SIZE OF COMMON 00000 4.6 85.0
25.0 433.8
LEAST SQUARED FIT TO LOGlO(P(1)) = A f B/T(I)
PlLOGRAhf 01574 23.8 95.0
35.0 634.0
(DATA FOR WATER, DANIELS 6TH El))
PCAL 4.76 23.41 41.20 145.90 433.07 642.08 775.66
A
=
8.93306
B
=
-2255.1
SIGMA = 7.36
/INPUT /INCLUDE PCEM4 /END RUN M ,0073 ACTION IN PROGRESS. SIZE OF COMMON 00000 END OF COMPILATION MAIN 7 0.0 4.6 25.0 60.0 149.5 85.0 433.8 LEAST SQUARED FIT TO P(1) = 10.**(A T(I)
A = 8.75542
p(I)
B
=
-2191.9
760.0
+ B/T(I))
DLGP
PCAL
SIGMA = 1.36
Comparison of the leost-squares flt to the lo~orithmicand exponential forms of the Clauriul-Clapeyron equation.
the two programs, and also contains the actual data as entered via the typewriter into the program in both cases. Students are routinely required to utilize both programs when doing either of the vapor pressure ex-
periments in Daniels and also to graph their results along with the two equations determined by the above programs. Listings of either or both programs may be obtained from the author upon request.
Volume 48, Number 8, August 1971
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