#### Heat Capacity of a Gas

The heat capacity of anything tells us how much heat is required to raise a certain amount of it 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. Instead of defining a whole set of molar heat capacities, let's focus on C_{V}, the heat capacity at constant volume, and C_{P}, the heat capacity at constant pressure.

#### Heat Capacity at Constant Volume

Q = nC_{V}ΔT

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

Q = ΔE_{int} + W, although W = 0 at constant volume.

For a monatomic ideal gas we showed that ΔE_{int} = (3/2)nRΔT

Comparing our two equations

Q = nC_{V}ΔT and Q = (3/2)nRΔT

we see that, 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 = ΔE_{int}
when the volume is constant, the change in internal energy can always be written:

ΔE_{int} = n C_{V} ΔT

#### Heat Capacity at Constant Pressure

For an ideal gas at constant pressure, it takes more heat to achieve the same temperature change than it does at constant volume. At constant volume all the heat added goes into raising the temperature. At constant pressure some of the heat goes to doing work.

Q = nC_{P}ΔT

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

Q = ΔE_{int} + W

At constant pressure W = PΔV = nRΔT

For a monatomic ideal gas, where ΔE_{int} = (3/2)nRΔT, we get:

Q = (3/2)nRΔT + nRΔT = (5/2)nRΔT

So, for a monatomic ideal gas:

C_{P} = (5/2)R

For diatomic and polyatomic ideal gases we get:

diaatomic: C_{P} = (7/2)R

polyatomic: C_{P} = 4R

#### The ratio C_{P} / C_{V}

It turns out that the ratio of the specific heats is an important number. The symbol we use for the ratio is γ. For a monatomic ideal gas we have:

γ = C_{P} / C_{V} = [5R/2] / [3R/2] = 5/3