Q: Find the amount of work done to increase the temperature of 1 mol of an ideal gas by 30°C if it is expanding under the condition V ∝ T^{2/3}.

(a) 166.2J

(b) 136.2J

(c) 126.2J

(d) none of these

Ans: (a)

Sol: As V = K T^{2/3}

dV = K(2/3)K^{2/3 -1}dT

dV = (2K/3)T^{1/3}dT …(i)

On dividing

dV/V = (2/3)dT/T

As, dW = PdV

dW = (RT/V) dV

$ \displaystyle W = \int_{T_1}^{T_2} \frac{RT}{V}dV $

$ \displaystyle W = \int_{T_1}{T_2}\frac{2}{3}\frac{dT}{T} $

$ \displaystyle W = \frac{2R}{3}(T_2 – T_1) $

$ \displaystyle W = \frac{2R}{3}(30) $

W = 20R = 20 x 8.314

W = 166.28 J