Thermo: Show that the internal energy at constant entropy and volume decrease

[Lagraaaange]

Homework Statement

Show that the internal energy at constant entropy and volume decrease for a spontaneous process

Homework Equations

F = U-TS

The Attempt at a Solution

Use Clausius: dS-dQ/dT > 0
Assume constant volume: TdS > dU
assume constant entropy
this becomes
0>dU

Since dU is negative, Change in F is negative thus spontaneous?

 

[Chestermiller]

You kind of had the right idea. For a constant volume process that transitions a closed system from thermodynamic equilibrium state A to thermodynamic equilibrium state B, you indicated that:
$$ΔU=Q$$
Also, from the Clausius inequality, for a spontaneous process,

$$ΔS>\frac{Q}{T_B}$$
where T_B is the temperature at the heat transfer interface between the system and the surroundings, and where we have assumed that T_B is held constant during the spontaneous process. Since ΔS is zero in the transition from state A to state B, we have:

$$0>\frac{Q}{T_B}$$

Therefore, Q<0, and ΔU<0.

I don't know how to do this if T_B is varying during the process.