The number density of molecules of a gas depends on their distance $r$ from the origin as, $n(r)=n_{0} e^{-\alpha r^{4}}$. Then the total number of molecules is proportional to :

(D) $\sqrt{\mathrm{n}_{0}} \alpha^{1 / 2}$

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Heat & Thermodynamics

The number density of molecules of a gas depends on their distance

r

from the origin as,

n (r) = n_{0} e^{- α r^{4}}

. Then the total number of molecules is proportional to :

(A) $n_{0} α^{- 3/4}$ ✓ Correct
(B) $n_{0} α^{- 3}$
(C) $n_{0} α^{1/4}$
(D) $n_{0} α^{1/2}$

Answer: (A)

Solution

The total number is

\int_{0}^{\infty} n (r) 4 π r^{2} d r = 4 π n_{0} \int_{0}^{\infty} r^{2} e^{- α r^{4}} d r

. Substituting

u = α r^{4}

pulls out a factor

α^{- 3/4}

and leaves a pure number (a Gamma-function constant), so the total number of molecules is proportional to

n_{0} α^{- 3/4}

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option

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