# Specific heat at constant volume

## Main Question or Discussion Point

$$C_{V} = \frac{∂U}{∂T}$$

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

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

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.

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

Chestermiller
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