A gas has molar heat capacity C = 4.5 R in the process PT = constant. The number of degrees of freedom of molecules in the gas is

Let us take one mole  of an ideal gas, the equation of state becomes, PV = RT------------------------(i)

We are given, PT = constant = K (say) 

or P = $\frac{\mathrm{K}}{\mathrm{T}}------------\left(\mathrm{ii}\right)$

From (i) and (ii), $\left(\frac{\mathrm{K}}{\mathrm{T}}\right)\mathrm{V} = \mathrm{RT}$

Work done, when temperature changes from T to (T+dT),

or V = $\frac{\mathrm{R}}{\mathrm{K}}{\mathrm{T}}^2$-----------------(iii)

Differentiating (iii), dW = P dv = $\left(\frac{\mathrm{K}}{\mathrm{T}}\right)\frac{2\mathrm{RT}}{\mathrm{K}}\mathrm{dT}$

or dW = 2R dT   --------------(iv)

Now using first law of thermodynamics

dQ = dU + dW

CdT = ${\mathrm{C}}_{\mathrm{v}}\mathrm{dT} + 2\mathrm{R} \mathrm{dT}$

${\mathrm{C}}_{\mathrm{v}} = \mathrm{C} - 2\mathrm{R} = 4.5 \mathrm{R} - 2\mathrm{R} = \frac{5}{2}\mathrm{R}$

But we define ${\mathrm{C}}_{\mathrm{v}} = \frac{\mathrm{f}}{2}\mathrm{R}$

Hence, degree of freedom n = 5