# 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 T2 from a knowledge of its entropy at a temperature T1 and the heat supplied to change the temperature from T1 to T2:

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, Cp:,

 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 qp may be substituted into the equation for S(T2) , 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, Cv , in place of Cp 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, Cp , and evaluating the integral in Equation A above. Note this integral is equivalent to the area under a graph of  Cpplotted against ln T . In addition, the entropy of transition  (equal to ΔHtrs / 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 Tf (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 T3 when T is low. This forms the basis of the Debye Extrapolation, where Cp is measured to as low a temperature as possible, and a curve of the form cT3 (c a constant) is fitted to the data. The fit determines the value of c, and the expression Cp = cT3 is assumed valid down to T = 0.