One kg of a diatomic gas is at a pressure of $8 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}$. The density of the gas is $4 \mathrm{~kg} / \mathrm{m}^{-3}$. What is the energy of the gas due to its thermal motion?

(B) $5 \times 10^{4} \mathrm{~J}$

(A) $3 \times 10^{4} \mathrm{~J}$

(C) $6 \times 10^{4} \mathrm{~J}$

(D) $7 \times 10^{4} \mathrm{~J}$

JEE Main · 2009 · 12 Apr 2009 · Offline · Heat & Thermodynamics

One kg of a diatomic gas is at a pressure of

8 \times 1 0^{4} N / m^{2}

. The density of the gas is

4 kg / m^{- 3}

. What is the energy of the gas due to its thermal motion?

(A) $3 \times 1 0^{4} J$
(B) $5 \times 1 0^{4} J$ ✓ Correct
(C) $6 \times 1 0^{4} J$
(D) $7 \times 1 0^{4} J$

Answer: (B)

Solution

The thermal energy of the gas is its internal energy, which for a given pressure and volume is

U = \frac{f}{2} P V

. The volume follows from mass and density as

V = \frac{m}{ρ} = \frac{1}{4} m^{3}

. A diatomic gas has five degrees of freedom, so

U = \frac{5}{2} P V = \frac{5}{2} \times 8 \times 1 0^{4} \times \frac{1}{4}

, which equals

5 \times 1 0^{4}

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option

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