what are heat capacities at constant …

Ans. 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.

$Q = nC∆T$

The value of the heat capacity depends on whether the heat is added at constant volume, constant pressure, etc.

C_V, the heat capacity at constant volume, and C_P, the heat capacity at constant pressure.

Heat Capacity at Constant Volume C_V: 

Q = nC_VΔT

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.

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_PΔT

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