Finding compressibility from given internal Energy function

[H Psi equal E Psi]

Hi everyone!

1. Homework Statement

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": $$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

The Attempt at a Solution

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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?

Thanks for your help!

 

[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.

 

[Chestermiller]

Hi everyone!

1. Homework Statement

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": $$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

The Attempt at a Solution

[/B]
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?

Thanks for your help!

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

 

[H Psi equal E Psi]

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

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?

 

[Chestermiller]

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?

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

 

[Chestermiller]

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

 

[H Psi equal E Psi]

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

$$dS=\frac{1}{T}\left[(u+P)dV+V\frac{du}{dT}dT\right]$$
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...

 

[Chestermiller]

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?