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

1. Nov 6, 2015

### Lagraaaange

1. The problem statement, all variables and given/known data
Show that the internal energy at constant entropy and volume decrease for a spontaneous process

2. Relevant equations
F = U-TS

3. 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?

2. Nov 7, 2015

### Staff: Mentor

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 TB is the temperature at the heat transfer interface between the system and the surroundings, and where we have assumed that TB 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 TB is varying during the process.