OM: Proving the Thermodynamics Relation for Ideal Gases Using Maxwell Relations

[Bipolarity]

Homework Statement

Prove that:
$$(\frac{∂U}{∂V}) = P - T(\frac{∂S}{∂V})$$

Homework Equations

- Maxwell relations
- Total differential
- Euler's test for exact differentials

The Attempt at a Solution

It appears to work for an ideal gas, but I am wondering if it works in the general case.
Thanks!

BiP

 

[clamtrox]

Yes it works with everything.

Here's the rough idea:
Suppose you have the internal energy in terms of two independent variables, V and S: U = U(V,S). Likewise, we can write S in terms of two independent variables, S = S(V,x) where x is some variable we don't need to care about right now, but it's independent of V.
Then
$$\frac{\partial U(V,S(V,x))}{\partial V} = \left( \frac{\partial U(V,S)}{\partial V} \right)_{S = const.} + \left( \frac{\partial U(V,S)}{\partial S}\right)_{V = const.} \frac{\partial S(V,x)}{\partial V}$$

 

[Chestermiller]

Homework Statement

Prove that:
$$(\frac{∂U}{∂V}) = P - T(\frac{∂S}{∂V})$$

Homework Equations

- Maxwell relations
- Total differential
- Euler's test for exact differentials

The Attempt at a Solution

It appears to work for an ideal gas, but I am wondering if it works in the general case.
Thanks!

BiP

We know that, for an ideal gas, the internal energy is a function only of temperature. However, for a real gas, as we increase the pressure and decrease the volume (say at constant temperature), the internal energy begins to change (as we move out of the ideal gas region). This equation is the first step in the derivation of a relationship for calculating the change in the internal energy as we move out of the ideal gas region. However, there is a sign error (there should be a minus sign multiplying the entire right hand side).

For any gas, we have:

dU = TdS - PdV

If we regard U as a function of V and T, and take the partial derivative of U with respect to V at constant T, we obtain:

(∂U/∂V)_T= -P + T (∂S/∂V)_T

The next step in the derivation is to replace (∂S/∂V)_T with an expression involving P, V, and T derived from the Maxwell relation associated with the differential in Helmholtz Free Energy. The resulting equation can be integrated, taking into account the modified gas law involving the compressibility factor z to obtain the change in internal energy of a real gas with volume at constant temperature.