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 = nCVΔT

For an ideal gas, applying the First Law of Thermodynamics tells us that heat is also equal to:

Q = ΔEint + W, although W = 0 at constant volume.
For a monatomic ideal gas we showed that ΔEint =
3
2
nR ΔT

Comparing our two equations
Q = nCV ΔT   and   Q =
3
2
nR ΔT

we see that, 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 = ΔEint when the volume is constant, the change in internal energy can always be written:

ΔEint = n CV ΔT