# Definition of heat capacity in thermodynamics

The following wikipedia page derives the expression of heat capacity of a system when holding different state variables constant: https://en.wikipedia.org/wiki/Relations_between_heat_capacities#Ideal_gas

It is shown that the relationship is given by ("blah" is the state variable being held constant): $$C_{blah} = T \left(\frac{\partial S}{\partial T}\right)_{blah}$$

In the derivation, it is explicitly assumed that, when the gas (or whatever thermodynamic system interested) is being heated, it undergoes a reversible change. May I ask why is this assumption valid? Is it just from the definition of heat capacity?

• Why is any assumption valid? Because it's pretty good? What would you expect to be different - should $C_{\text{blah}}$ be variable? Should the change be irreversible? Reversibility is a really powerful assumption but it requires that there aren't any chemical changes in the mixture (among other things). – user121330 Dec 12 '17 at 16:52

The definition of heat capacity at constant volume $C_v$ is $$C_v=\left(\frac{\partial U}{\partial T}\right)_V$$ and the definition of heat capacity at constant pressure $C_p$ is $$C_p=\left(\frac{\partial H}{\partial T}\right)_P$$where U is the internal energy and H is the enthalpy. If we make use of the property relationship $$dU=TdS-PdV$$ and the equivalent property relationship $$dH=TdS+PdV$$ we obtain: $$C_v=T\left(\frac{\partial S}{\partial T}\right)_V$$ and $$C_p=T\left(\frac{\partial S}{\partial T}\right)_P$$ Since, at equilibrium, all the thermodynamic functions can be regarded as depending on T and any one other thermodynamic parameter, the above relationships provide the basis for extending the definition to $C_{blah}$ to the equation you provided in the original post.