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 = DU

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 = DU =
3
2
nRDT

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 = nCDT

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 = nCVDT

For an ideal gas, applying the First Law of Thermodynamics told us that:
Q =
3
2
nR DT

Comparing our two equations we get, for a monatomic ideal gas:
CV =
3
2
R

For diatomic and polyatomic ideal gases we get:
diatomic:   CV =
5
2
R

polyatomic: CV = 3R

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

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

DU = n CV DT