Thermodynamics Part 3

Given, pT3=constant ...(i)
From ideal gas equation, we have
pV=nRT
⇒p=

nRT

V

From Eq. (i), we get
∴(

nRT

V

)T3=constant
⇒nRT4V−1=constant
Differentiate above expression, we get
nR[4T3V−1‌dT−

T4

V2

‌dV]=0
⇒4‌dT=

T

V

‌dV ...(ii)
We know that, dV=γ‌VdT
Here, γ = coefficient of volume expansion.
⇒γ=

dV

VdT

...(iii)
From Eq. (ii),

4

T

=

dV

VdT

...(iv)
On comparing Eqs. (iii) and (iv), we get
γ=

4

T

Thus, the coefficient of volume expansion is

4

T

.