# Finding compressibility from given internal Energy function

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1. Jun 28, 2017

### H Psi equal E Psi

Hi everyone!

1. The problem statement, all variables and given/known data

Given is a function for the internal energy: $U(T,V)=Vu(T)$
Asked is to derive the entropy balance equation. In order to do so i need to find the "isothermal and adiabatic compressibility": $$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

3. The attempt at a solution

In order to calculate $\kappa_{T}$ I need to find a function $V(P,T)$ right? But how do I get this function from the given internal energy function?

2. Jun 28, 2017

### Useful nucleus

The isothermal and adiabatic compressibilities are second derivatives for U. So you may try taking derivatives of the internal energy equation using the technique of implicit differentiation.

3. Jun 28, 2017

### Staff: Mentor

What is the general equation (not for this specific material) for dU in terms of dT and dV?

4. Jun 28, 2017

### H Psi equal E Psi

I guess its $$dU(T,V)=\left(\frac{\partial U}{\partial T}\right)_{V} dT+\left(\frac{\partial U}{\partial V}\right)_{T} dV$$ Or one can write: $$dU(T,V)=C_{V}dT-PdV$$
right?

5. Jun 28, 2017

### Staff: Mentor

Wrong. $$dU(T,V)=C_{V}dT-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]dV$$

6. Jun 28, 2017

### Staff: Mentor

Please state the entropy balance equation that you are supposed to derive?

7. Jun 28, 2017

### H Psi equal E Psi

$$dS=\frac{1}{T}\left[(u+P)dV+V\frac{du}{dT}dT\right]$$
I managed to derive the second term by simply using the definition of $C_{V}$ and taking the first derivative of $U(T,V)$ with respect to $T$ since: $$\left(\frac{\partial U}{\partial T}\right)_{V}=T \left(\frac{\partial S}{\partial T}\right)_{V}$$ This obviously leads to $V\frac{du}{dT}$
But I'm still struggling with the first term...

8. Jun 28, 2017

### Staff: Mentor

This is much simpler than you think. Start out with:$$dS=\frac{dU}{T}+\frac{P}{T}dV$$From your equation for U(V,T), what is dU in terms of dV and dT?