Question

Show that for an ideal gas Cp- Cv = R.

Answer 1

According to the first law of thermodynamics:

ΔQ = ΔU + ΔW

⇒ ΔW = PΔV

ΔQP​ = ΔU + PΔV  

Since ΔQ_p​ = nC_p​ΔT 

and ΔU = nC_v​ΔT , therefore,

nC_p​ΔT = nC_v​ΔT + PΔV    .....(1)

We know that PV = nRT

At T₁​Kelvin: PV₁​ = nRT₁​   .....(a)

At T₂​Kelvin: PV₂​ = nRT₂​   .....(b)

Subtracting (a) from (b)

PV₂ ​− PV_1​ = nRT₂​ − nRT₁​

P(V₂​ − V₁​) = nR(T₂​ − T₁​)        

PΔV = nRΔT

Putting the value of PΔV in equation (1)

nC_p​ΔT = nC_v​ΔT + nRΔT

nC_p​ΔT = nΔT(C_v​ + R)

Cp = (Cv + R)

Cp − Cv = R

Answer 2

When a gas is heated under constant pressure, the heat is required for raising the temperature of the gas and also for doing mechanical work against the external pressure during expansion.

At constant volume, the heat capacity, C is written as Cv and at constant pressure this is denoted by Cp.

we write heat q

at constant volume as qv = Cv ΔT = DU

at constant pressure as qp = CpΔT = D+H

The difference between Cp and Cv can be derived for an ideal gas as :

For a mole of an ideal gas,

On putting the values of ΔH and ΔU, we have;