By definition of Specific heat

C_{v} =dQ/(µ+dT)

by first law of thermodynamics

dQ=dU+dW

at constant volume (dW=0) then

dQ=dU

But U=fµRT/2, so that

dU/dT=fµR/2

Then molar specific heat at constant volume,

C_{v}=dU/µdT

Here

C_{v}=fR/2

Here C_{p}-C_{v}=R

then

C_{p}=C_{v}+R

Hence

C_{p}=(f/2+1)R

now, rate of molar specific heat,

C_{p}/C_{v}=Γ (Γ is gamma)

then

Γ=[(f/2+1)R]/(f/2)r

â–ˆ For monoatomic gas (f=3)

C_{v}=3R/2

C_{p}=5R/2

Γ=5/3

â–ˆ For diatomic gas (f=5)

C_{v}=5R/2

C_{p}=7R/2

Γ=7/5

â–ˆ For Triatomic gas

â–„Linear Structural (f=7)

C_{v}=7R/2

C_{p}=9R/2

Γ=9/7

â–„non-linear structural (f=6)

C_{v}=3R

C_{p}=4R

Γ=4/3

Hence Proved.