Kinetic Theory

JEE Main level, single correct, medium

154 of 168

Question 154 of 168

The number density of molecules of a gas depends on their distance r from the origin as, n(r)=n0e−αr4.n\left(r\right)={n}_{0}{e}^{-\alpha {r}^{4}}. Then the numer of molecules is proportional to:

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Answer: (C) n0α−34{n}_{0}{\alpha }^{\dfrac{-3}{4}}

Density of molecules of gas as a of r is, n(r)=n0e−αr4n(r)={n}_{0}{e}^{-\alpha {r}^{4}}
Total no. of molecules =∫0∞n(r)dv=\int _{0}^{\infty }n(r)dv
∫0∞n0e−αr44πr2dr\int _{0}^{\infty }{n}_{0} {e}^{-\alpha {r}^{4}}4\pi {r}^{2} dr
To integrate this, substitute u=α3/4r3u={\alpha }^{3/4}{r}^{3} this will give us dr=13α3/4r2dudr=\dfrac{1}{3{\alpha }^{3/4}{r}^{2}}du
Now, the integration will become, 4πn03α3/4∫0∞e−u4/3du\dfrac{4\pi {n}_{0}}{3{\alpha }^{3/4}}\int _{0}^{\infty } {e}^{-{u}^{4/3}}du
This shows that the numer of molecules is proportional to n0α−3/4{n}_{0}{\alpha }^{-3/4}

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