# Specific heat at constant volume

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

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

Homework Helper
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.

Bipolarity
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.

Superb! Thanks!

BiP

Mentor
$$C_{V} = \frac{∂U}{∂T}$$

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

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