First slide

Work done by gas

Question

A gas expands with temperature according to the relation $V = {KT}^{1 / 3}$ . What is the work done when the temperature changes by 30^oC

Moderate

A

10R
B

20R
C

30R
D

40R

Solution

$\begin{array}{l} V = {KT}^{1 / 3} \\ \Rightarrow dV = \frac{1}{3} {KT}^{- \frac{2}{3}} dT \\ \Rightarrow= \frac{1}{3} \frac{V}{T^{1 / 3}} T^{- \frac{2}{3}} dT \\ \Rightarrow dV = \frac{1}{3} V \cdot T^{- 1} dT \\ \Rightarrow \frac{dV}{V} = \frac{1}{3} \frac{dT}{T} \Rightarrow W = \int_{1 / 3}^{T_{2}} \frac{RT}{T} dT \\ = \frac{1}{3} R (T_{2} - T_{1}) \\ \Rightarrow W = \frac{1}{3} \times R \times 30 = 10 R \end{array}$

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