JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Heat & Thermodynamics

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During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of

\frac{C _{p}}{C _{v}}

for the gas is :

(A) $\frac{5}{3}$
(B) $\frac{3}{2}$ ✓ Correct
(C) $\frac{7}{5}$
(D) $\frac{9}{7}$

Answer: (B)

Solution

For an adiabatic process,

P V^{γ} = const

. Substituting

V = \frac{n R T}{P}

from the ideal gas law,

P (\frac{n R T}{P})^{γ} = const ⟹ P^{1 - γ} T^{γ} = const ⟹ P T^{γ / (1 - γ)} = const .

Given

P \propto T^{3}

, i.e.

P T^{- 3} = const

, matching exponents gives

\frac{γ}{1 - γ} = - 3 ⟹ γ = - 3 + 3 γ ⟹ γ = \frac{3}{2} .

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Single correct option

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