# Calculating specific heat capacity from entropy

approx12
Homework Statement:
Given the entropy S of system, calculate the specific heat capacity C_V and C_p
Relevant Equations:
$$C_P=T \left(\frac{\partial S}{\partial T}\right)_{N,P}$$ and
$$C_V=T \left(\frac{\partial S}{\partial T}\right)_{N,V}$$
Hey guys! I'm currently struggling with a specific thermodynamics problem.
I'm given the entropy of a system (where ##A## is a constant with fitting physical units): $$S(U,V,N)=A(UVN)^{1/3}$$I'm asked to calculate the specific heat capacity at constant pressure ##C_p## and at constant volume ##C_V##.
I know that the two are given by the following equation:
$$C_P=T \left(\frac{\partial S}{\partial T}\right)_{N,P}$$$$C_V=T \left(\frac{\partial S}{\partial T}\right)_{N,V}$$I've tried to eleminate ##U## from the equation by calculating: $$\left(\frac{\partial S}{\partial V}\right)=\frac{P}{T}=\frac{1}{3}(NU)^{1/3}V^{-2/3}$$ Solving for ##U## and plugging it back into the original equation gives me: $$S(P,V,T)=\frac{PV}{T}$$
I don't know if my steps were correct so far but what I'm now struggling with is calculating ##\left(\frac{\partial S}{\partial T}\right)_{P}## and ##\left(\frac{\partial S}{\partial T}\right)_{V}##. For me they would be both equal to $$\left(\frac{\partial S}{\partial T}\right)=-\frac{PV}{T^2}$$ But I don't think that is correct because the relationship $$C_P-C_V=\frac{TV\alpha_P^2}{\kappa_T}$$ needs to bet true.

It would be awesome if anyone could help me out with this one and point me in the right direction. Thank you!

Homework Helper
Gold Member
Looks like you dropped a factor of 3 in getting to the equation ##S = \frac{PV}{T}##.

I don't know if my steps were correct so far but what I'm now struggling with is calculating ##\left(\frac{\partial S}{\partial T}\right)_{P}## and ##\left(\frac{\partial S}{\partial T}\right)_{V}##. For me they would be both equal to $$\left(\frac{\partial S}{\partial T}\right)=-\frac{PV}{T^2}$$

When calculating ##\left(\frac{\partial S}{\partial T}\right)_{P}## you can't treat ##V## as a constant. Likewise when calculating ##\left(\frac{\partial S}{\partial T}\right)_{V}## you can't treat ##P## as a constant.

One way to proceed is to find the equation of state that relates the 3 variables ##P##, ##V##, and ##T##. You can then use this to write ##S## as a function of just ##T## and ##P## which then makes it straightforward to evaluate ##\left(\frac{\partial S}{\partial T}\right)_{P}##. Or, you can write ##S## as a function of just ##T## and ##V## so that you can evaluate ##\left(\frac{\partial S}{\partial T}\right)_{V}##. I don't know if this is the best way, but it's one way.

Last edited: