GEORGE J. IAN2 Rensselaer Polytechnic Institute, Troy, New York
INTHERMOCHEMICAL and thermodynamic calculations, slope is constant, the average slope in this temperature a knowledge of the heat capacity of a substance as a interval is function of temperature is frequently required. If the '/~f(dC,/dT)sm~ (dC,/dT)ua~] = b 2c(45Oo) heat capacity can be expressed by a purely empirical power series equation or the slope of the curve a t T = 450°K. Similarly, the C , = a + b T + c T Z + d T a + ... (1) average slope over the temperature range 900-600°K. is 12.2 X and this, as shown above, is also the the constants a, b, c, d must be derived from heat a- slope a t 750°K. pacity data over the range of temperatures of interest The rate of change of slope (d2 C9/dT2) from 450' to with reference to the specific problem under considera- 750°K. is ~ ( 1 . 1 0X 10-s)/300 = ~ 3 . X 6 ~10-O. In tion. This paper describes a method based on dif- this instance, since the slope is decreasing, the rate of ferential calculus for this purpose rather than the al- change of the slope is negative. Using this value in (3) gebraic approach generally used. Thus for a three above, cis found to be - 1 . 8 ~X lo-'. Solving the first parameter power series equation as above, the first dif- differential equation for b, and substituting for c, the ferential of Cp with respect to T defines the slope of the value for b, obtained quite simply, is 14.g6 X 10-8. curve a t any point T, i. e., These values in (1) give 4.231 for the constant a. The dC,/dT = b + 2cT (2) equation for the heat capacity as a function of temperature for methane in the temperature interval 300The second differential 900°K. is given by
+
C, = 4.231
defines the rate of change of slope a t any point T. It is evident from (3) that the latter is constant in this case. With these equations the constmts a, b, and c are readily obtained from a knowledge of the heat capacities a t three temperatures. To reduce the use of letters and symbols to a minimum, let the following data for methane' be used for illustration of this method: T (OK.) C,(cal/deg. mol)
300 8.55
600 12.55
900 16.21
The average slope over the temperature range 600300" K. is simply 4.00/300 = 13.3 X lo-\ This is also the slope a t 450°K., i. e., since the rate of change of
' "Selected Values of Propertiw of Hydrocarbons," Circular C461, National Bureau of Standards, Washington, D. C., 1947.
+
+ 14.9: X 1 0 - T - l.Ss X 10-'TP
Thus, a t 700°K., the heat capacity calculated by this relation is 13.81 cal/deg. mol, and the literature value (1) is 13.88, i. e., the agreement is sufficientfor most problems encountered. The algebraic methods of solution, in this case, of three simultaneous equations rest on the elementary and basic procedure of elimination of the letters one by one from different pairs of equations until only one remains, or the more advanced and neater procedure of determinants, in which four determinants of the third order must be solved. The method described and illustrated above is basically more simple and less laborious than the algebraic methods for obtaining the equations expressing the heat capacity as a power series function of temperature.
