The Relationship between Entropy and Temperature
Our starting point for this discussion is the definition of a measurable entropy change:
This definition may be used to calculate the entropy of a system at a temperature T_{2} from a knowledge of its entropy at a temperature T_{1} and the heat supplied to change the temperature from T_{1} to T_{2}:
If we consider the situation where the system is subjected to a constant pressure (which is commonly the case in chemistry, many experiments being done under atmospheric pressure) then we can make a substitution in the above expression using the definition of the constant pressure heat capacity, C_{p}:,
so, |
as an enthalpy change can be equated with a heat transfer at constant pressure.
As long as the system does no non-expansion work, then the expression for q_{p} may be substituted into the equation for S(T_{2}) , to give:
Equation A: |
If the constant pressure heat capacity is independent of temperature over the range of interest, we may take it outside the integral and evaluate the integrand directly, to yield:
If the system is considered to be at constant volume rather than constant pressure, exactly analogous equations featuring the constant volume heat capacity, C_{v} , in place of C_{p} are obtained.
The above equation may be used to calculate the entropy of a system at any temperature, T, from a knowledge of its entropy at T = 0. This is done by measuring the system’s heat capacity, C_{p} , and evaluating the integral in Equation A above. Note this integral is equivalent to the area under a graph of C_{p}plotted against ln T . In addition, the entropy of transition (equal to ΔH_{trs} / T ) must be added for any phase change which occurs between T = 0 and the temperature of interest.
This graph illustrates how the entropy of a substance increases with temperature. Note the discontinuities in the graph at phase transitions, corresponding to the entropy change of transition: |
Thus if a substance melts at T_{f} (the subscript f standing for fusion, as the melting process is called) the entropy of the substance at some temperature T at which it is liquid is given by:
All the properties needed to to evaluate this equation can be measured calorimetrically, except S(0) which may be extrapolated from other data. The integrals are normally evaluated by fitting a polynomial to the experimental data and integrating the polynomial analytically.
Heat capacities are difficult to measure close to T = 0. However, there are good theoretical grounds for assuming that the heat capacity is proportional to T^{3} when T is low. This forms the basis of the Debye Extrapolation, where C_{p} is measured to as low a temperature as possible, and a curve of the form cT^{3} (c a constant) is fitted to the data. The fit determines the value of c, and the expression C_{p} = cT^{3} is assumed valid down to T = 0.