# Specific heat at constant volume

1. Sep 18, 2012

### Bipolarity

$$C_{V} = \frac{∂U}{∂T}$$

This is the specific heat at constant volume so I assume it can only be used at constant volume. However, my textbook uses this to derive the following equation for reversible adiabatic expansion:

$$P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}$$

Why are we allowed to use $C_{V}$ when it only works in isovolumetric processes?

BiP

2. Sep 18, 2012

### nasu

How is Cv used to derive the equation for adiabatic transformation?
Can you show it here?

3. Sep 18, 2012

### Bipolarity

4. Sep 19, 2012

### nasu

The change in internal energy has the same expression for any process between two states. For ideal gas is
$$\Delta U = nC_v\Delta T$$
The amount of heat is dependent on the type of process. It is $$Q = nC_v\Delta T$$
only for constant volume process.

5. Sep 19, 2012

### Bipolarity

Superb! Thanks!

BiP

6. Sep 19, 2012

### Staff: Mentor

For an ideal gas, the internal energy is a function only of temperature, such that dU = CvdT always. For an adiabatic expansion, dQ = 0, so that

dU = CvdT = -pdV