Heat Capacity of an Ideal Gas

The heat capacity specifies the heat needed to raise a certain amount of a substance by 1 K. For a gas, the molar heat capacity C is the heat required to increase the temperature of 1 mole of gas by 1 K.

Defining statement:  dQ = nC dT

Important: The heat capacity depends on whether the heat is added at constant volume or constant pressure.

Why care about the difference?

Heat Capacity of Ideal Gas at Constant Volume CV

Defining statement:   dQ = nCV dT

1st Law of Thermodynamics gives:   dQ = dE - dW = dE at constant volume.
Thus dQ = dE = nCV dT =
3
2
nR dT ,
Comparing, we get, for a monatomic ideal gas:  CV =
3
2
R     (diatomic: CV =
5
2
R)

Since dQ = dE when the volume is constant, dE = n CV dT.