In a certain thermodynamical process, the pressure of a gas depends on its volume as $\mathrm{kV}^{3}$. The work done when the temperature changes from $100^{\circ} \mathrm{C}$ to $300^{\circ} \mathrm{C}$ will be \_\_\_\_ nR , where n denotes numbers of moles of a gas.

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Heat & Thermodynamics

In a certain thermodynamical process, the pressure of a gas depends on its volume as

kV^{3}

. The work done when the temperature changes from

10 0^{\circ} C

30 0^{\circ} C

will be ____ nR , where n denotes numbers of moles of a gas.

Answer: 50

Solution

For an ideal gas

P V = n R T

, and the process obeys

P = k V^{3}

, so

k V^{3} \cdot V = n R T

, that is

k V^{4} = n R T

.

The work done is the area under the

P

V

curve:

W = \int_{V_{1}}^{V_{2}} P d V = \int_{V_{1}}^{V_{2}} k V^{3} d V = \frac{k ( V _{2}^{4} - V _{1}^{4} )}{4} = \frac{n R ( T _{2} - T _{1} )}{4}

With

T_{1} = 373

K and

T_{2} = 573

K the temperature rise is

200

K, so

W = \frac{n R ( 200 )}{4} = 50 n R

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Integer answer

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