Specific heat or Heat Capacity of a gas:

The amount of heat energy required to raise the temperature of one kilogram (1 kg) of gas through 1K or 1°C is called the specific heat or heat capacity of a gas.

Molar Specific heat of a gas:

The amount of heat energy required to raise the temperature of one mole of the gas through 1K or 1°C  is called molar specific heat or molar heat capacity of a gas. If n moles of an ideal gas are heated volume so that its temperature rises by ΔT then the heat transferred is given by

        ΔQ ∝ n 
and
        ΔQ ∝ ΔT 

Combing the above equations

        ΔQ ∝ nΔT
or
        ΔQ = n Cₘ ΔT 

Where Cₘ is molar heat capacity and its SI unit is J mol⁻¹ K⁻¹.

Molar Specific heat of a gas at constant pressure:

The amount of heat energy required to raise the temperature of one mole of the gas through 1K or 1°C, while keeping the pressure constant is called molar specific heat or molar heat capacity of a gas at constant pressure.

Mathematically
 
        ΔQₚ = n Cₚ ΔT 

Where Cₚ is the molar heat capacity at constant pressure and its SI unit is J mol⁻¹ K⁻¹.

Pressure can be kept constant by having gas enclosed in a cylinder of conducting base, non-conducting walls, and a frictionless movable piston.

Molar Specific heat of a gas at constant volume:

The amount of heat energy required to raise the temperature of one mole of the gas through 1K or 1°C, while keeping the volume constant is called molar specific heat or molar heat capacity of a gas at constant volume.

Mathematically
 
        ΔQᵥ = n Cᵥ ΔT 

Where Cᵥ is the molar heat capacity at constant pressure and its SI unit is J mol⁻¹ K⁻¹.

Volume can be kept constant by keeping the cylinder containing gas of conducting base, non-conducting walls, and fixed piston.

Derivation of Cₚ - Cᵥ = R

At constant Volume:

We know that if n moles of an ideal gas is heated at a constant volume so that its temperature rises by ΔT then the heat transferred is given by
        
        ΔQᵥ = n Cᵥ ΔT  ------------- (1)

Since volume remains constant (i.e. ΔV = 0), so work done ΔW by the system is zero. (i.e. ΔW = 0)

Applying the first law of thermodynamics at constant volume,

ΔQᵥ = ΔU + ΔW

Putting the value of from equation (1) and ΔW = 0

n Cᵥ ΔT = ΔU + 0

or

ΔU = n Cᵥ ΔT   ---------------(2)


At constant Pressure:

If n moles of an ideal gas are heated at constant pressure so that its temperature rises by then the heat
transferred is given by

        ΔQₚ = n Cₚ ΔT  ------------- (3)

Since the gas expands to keep the pressure constant, so the work done by the gas is ΔW = PΔV

Whereas

PΔV = n R ΔT  (Ideal gas equation)

So

ΔW = n R ΔT

Now applying the first law of thermodynamics at constant pressure,

n Cₚ ΔT = ΔU + ΔW


Putting the value of PΔW = n R ΔT  and equation (3)

n Cₚ ΔT = ΔU + n R ΔT  ------------- (4)

Since the change of internal energy ΔU is the same in both cases (i.e.) (at constant pressure and constant volume)

Thus by putting equation (3) in (4) we have

n Cₚ ΔT = n Cᵥ ΔT + n R ΔT

or

n Cₚ ΔT = n ΔT (Cᵥ + R)

or

Cₚ = Cᵥ + R

or

Cₚ - Cᵥ = R ------------(5)

It is clear from equation (5) that by the amount equal to the universal gas constant R ( R = 8.315  J mol⁻¹ K⁻¹ ) and also shows that specific heat at constant pressure is greater than specific heat at constant volume  i.e. Cₚ ＞ Cᵥ

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