# 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: Option 1) $n_{0}\alpha ^{-3/4}$   Option 2) $\sqrt{n_{0}}\alpha ^{1/2}$ Option 3) $n_{0}\alpha ^{1/4}$ Option 4) $n_{0}\alpha ^{-3}$

Maxwell's distribution of molecular speed -

$n_{v}dv= 4\pi N\cdot \left ( \frac{m}{2\pi KT} \right )^{3/2}\cdot v^{2}\cdot e\frac{-mv^{2}}{2KT}\cdot dv$

- wherein

N = total no. of molecules.

m = mass of molecule.

$\pi$,K,T all have usual meanings.

u = velocity of one particle.

Total number of molecule = $\int ^{\infty }_{o}n(r)dv$

$=\int ^{\infty }_{o}n_{0}e^{-\alpha r^{4}}4\pi r^{2}dr$

Number of molecules is proportional to $n_{0}\alpha ^{-3/4}$

Option 1)

$n_{0}\alpha ^{-3/4}$

Option 2)

$\sqrt{n_{0}}\alpha ^{1/2}$

Option 3)

$n_{0}\alpha ^{1/4}$

Option 4)

$n_{0}\alpha ^{-3}$

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