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