Constant Volume

A constant volume process is also known as an isochoric process. An example is when heat is added to a gas in a container with fixed walls.

Because the walls can't move, the gas can not do work:

W = 0

In that case the First Law states:

Q = ΔU

The P-V diagram for this process is simple - it's a vertical line going up if heat is added, and going down if heat is removed.

In the case of a monatomic ideal gas:
U =
3

2
NkT =
3

2
nRT

Therefore Q = ΔU =
3

2
nRΔT

Heat Capacity of a Gas

The heat capacity of a substance tells us how much heat is required to raise a certain amount of the substance by one degree. For a gas we can define a molar heat capacity C - the heat required to increase the temperature of 1 mole of the gas by 1 K.

Q = nCΔT

The value of the heat capacity depends on whether the heat is added at constant volume, constant pressure, etc.

Heat Capacity at Constant Volume

Q = nC_VΔT

For an ideal gas, applying the First Law of Thermodynamics told us that:
Q =
3

2
nR ΔT

Comparing our two equations we get, for a monatomic ideal gas:
C_V =
3

2
R

For diatomic and polyatomic ideal gases we get:
diatomic: C_V =
5

2
R

polyatomic: C_V = 3R

This is from the extra 2 or 3 contributions to the internal energy from rotations.

Because Q = ΔU when the volume is constant, the change in internal energy can always be written:

ΔU = n C_V ΔT