Find the amount of work done to increase the temperature of 1 mol of an ideal gas by 30ºC if it is expanding under the condition V ∝ T2/3

Engineering

Physics

Thermodynamics

Question

Find the amount of work done to increase the temperature of 1 mol of an ideal gas by 30ºC if it is expanding under the condition V ∝ T^2/3

166.2 J

136.2 J

126.2 J

None of these

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

Given: no. of moles = 1 mol.
Increase in temperature = 30ºC.

Condition = V ∝ T^2/3

To find the amount of work done to increase the given temperature.

We know PV = nRT (n-moles)
for 1 mole PV = RT.

$W = \int PdV = \int \frac{RT}{V} dV$

V = CT^2/3 (∵ V ∝ P^2/3)

$∴ dv = \frac{2}{3} {CT}^{- 1 / 3} dT$

$\frac{dV}{V} = \frac{2}{3} \frac{C}{V} T^{- 1 / 3} dT \Rightarrow \frac{dV}{V} = \frac{2}{3} \frac{dT}{T}$

Now T₂ the work done is given as:

$W = \int_{T_{1}}^{T_{2}} RT \frac{2}{3} \frac{dT}{T} = \frac{2}{3} R (T_{2} - T_{1}) = 166.2 J$

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

∴ (A)166.2 J is the correct option