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 CV, the heat capacity at constant volume, and CP, the heat capacity at constant pressure.
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)nRΔT
Comparing our two equations
Q = nCVΔT and Q = (3/2)nRΔT
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
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 = nCPΔT
For an ideal gas, applying the First Law of Thermodynamics tells us that heat is also equal to:
Q = ΔEint + W
At constant pressure W = PΔV = nRΔT
For a monatomic ideal gas, where ΔEint = (3/2)nRΔT, we get:
Q = (3/2)nRΔT + nRΔT = (5/2)nRΔT
So, for a monatomic ideal gas:
CP = (5/2)R
For diatomic and polyatomic ideal gases we get:
diaatomic: CP = (7/2)R
polyatomic: CP = 4R
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 = CP / CV = [5R/2] / [3R/2] = 5/3