How is the variation of entropy with temperature mathematically represented?

The variation of entropy with temperature is mathematically represented by the integral formulation, which connects entropy changes to heat capacity at constant pressure, ( C_p ).

In thermodynamics, the change in entropy, ( \Delta S ), when a system is heated or cooled from an initial temperature ( T_i ) to a final temperature ( T_f ) at constant pressure can be expressed using the integral of the heat capacity divided by temperature. This relationship is derived from the definition of entropy and provides a way to calculate the change in entropy for a process where the heat added or removed is dependent on the temperature of the system.

The equation ( S(T_f) = S(T_i) + \int_{T_i}^{T_f} \frac{C_p}{T} dT ) reflects this relationship, where ( S(T_f) ) is the entropy at the final temperature, ( S(T_i) ) is the initial entropy, and the integral accounts for the temperature dependence of ( C_p ). This means that as the temperature changes, so does the amount of heat absorbed per unit temperature, affecting the overall change in entropy.

Utilizing this equation allows one to calculate the entropy change for processes involving various materials and conditions