JEE Main · PhysicsMedium

$During an adiabatic process, the pressure $P$ of a gas is found to be proportional to the cube of its absolute temperature $T$, i.e., $P \propto T^3$. What is the adiabatic exponent $\gamma$ of the gas?$

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Correct answer: C — $$1.5$$

Solution

For an adiabatic process,

P^{1 - γ} T^{γ} = constant

. Given

P \propto T^{3}

, we have

P T^{- 3} = constant

. Thus,

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

is incorrect; rather

P \propto T^{\frac{γ}{γ - 1}}

. Setting

\frac{γ}{γ - 1} = 3

, we get

γ = 3 γ - 3

, so

2 γ = 3

γ = 1.5