The volume V of a given mass of monoatomic gas changes with temperature T according to the relation $\mathrm{V}=\mathrm{kT}^{\frac{2}{3}}$. The work done when temperature changes by 90 K will be xR . The value of x is. ____ [R = universal gas constant]

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Heat & Thermodynamics

The volume V of a given mass of monoatomic gas changes with temperature T according to the relation

V = kT^{\frac{2}{3}}

. The work done when temperature changes by 90 K will be xR . The value of x is. ____
[R = universal gas constant]

Answer: 60

Solution

For one mole,

P V = R T

, so with

V = k T^{2/3}

the pressure is

P = \frac{R T}{k T ^{2/3}} = \frac{R}{k} T^{1/3}

Differentiating the volume relation gives

d V = \frac{2 k}{3} T^{- 1/3} d T

, so the element of work is

P d V = \frac{R}{k} T^{1/3} \cdot \frac{2 k}{3} T^{- 1/3} d T = \frac{2 R}{3} d T

Integrating over a rise of

Δ T = 90

K gives

W = \frac{2 R}{3} \times 90 = 60 R

, so

x = 60

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Integer answer

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