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 measurements at constant volume are dangerous because the container can explode! It's safer to measure heat capacity C_P.
• However it is easier to calculate heat capacity at constant volume C_V.

Heat Capacity of Ideal Gas at Constant Volume C_V

Defining statement:   dQ = nC_V 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 C_V dT.