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I know we can write entropy ##S## as ##S(P,T)## and ##S(V,T)## to derive different relations between heat capacities ##C_V## and ##C_P##. I was wondering if it is technically correct to write

$$dS(P,V,T)=\left ( \frac{\partial S}{\partial P} \right )_{V,T}dP+\left ( \frac{\partial S}{\partial V} \right )_{P,T}dV+\left ( \frac{\partial S}{\partial T} \right )_{P,V}dT$$

$$\delta Q_{rev}=TdS(P,V,T)=T\left ( \frac{\partial S}{\partial P} \right )_{V,T}dP+T\left ( \frac{\partial S}{\partial V} \right )_{P,T}dV+T\left ( \frac{\partial S}{\partial T} \right )_{P,V}dT$$

Then how would we define heat capacities ##C,C_V,C_P## in terms of these partial derivatives?

I know we can define them as (at constant volume and constant pressure, respectively)

$$C_V=T\left ( \frac{\partial S}{\partial T} \right )_{V}; C_P=T\left ( \frac{\partial S}{\partial T} \right )_{P}$$

But I can't understand how to reconcile the gap between the second and third equations I typed.