JEE Main · 2015 · 04 Apr 2015 · Offline · Heat & Thermodynamics

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Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion the average time of collision between molecules increases as

V^{q}

, where V is the volume of the gas. The value of q is :

(γ = \frac{C _{p}}{C _{v}})

(A) $\frac{3 γ - 5}{6}$
(B) $\frac{γ + 1}{2}$ ✓ Correct
(C) $\frac{3 γ + 5}{6}$
(D) $\frac{γ - 1}{2}$

Answer: (B)

Solution

The mean time between collisions is the mean free path divided by the typical speed,

τ \sim \frac{λ}{v _{r m s}}

. The mean free path grows with the room available,

λ \propto V

, while

v_{r m s} \propto T

, so

τ \propto \frac{V}{T}

. For an adiabatic process

T V^{γ - 1} = constant

, i.e.

T \propto V^{- (γ - 1)}

and

T \propto V^{- (γ - 1) /2}

. Putting this in,

τ \propto V \cdot V^{(γ - 1) /2} = V^{(γ + 1) /2}

, so the exponent is

q = \frac{γ + 1}{2}

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option

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JEE Main · 2015 · 04 Apr 2015 · Offline · Heat & Thermodynamics

Solution

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