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(A) The total number of molecules $N$ is given by the integral of the number density $n(r)$ over the entire volume.In spherical coordinates,the volume

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$n_0 \alpha^{-3/4}$

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(A) The total number of molecules $N$ is given by the integral of the number density $n(r)$ over the entire volume.In spherical coordinates,the volume element is $dV = 4\pi r^2 dr$.Thus,$N = \int_{0}^{\infty} n(r) dV = \int_{0}^{\infty} n_0 e^{-\alpha r^4} (4\pi r^2) dr$.$N = 4\pi n_0 \int_{0}^{\infty} r^2 e^{-\alpha r^4} dr$.Let $u = \alpha r^4$,then $r = (u/\alpha)^{1/4}$ and $dr = \frac{1}{4} \alpha^{-1/4} u^{-3/4} du$.Substituting these into the integral:$N = 4\pi n_0 \int_{0}^{\infty} (u/\alpha)^{2/4} e^{-u} (\frac{1}{4} \alpha^{-1/4} u^{-3/4}) du$.$N = \pi n_0 \alpha^{-1/2} \alpha^{-1/4} \int_{0}^{\infty} u^{1/2 - 3/4} e^{-u} du = \pi n_0 \alpha^{-3/4} \int_{0}^{\infty} u^{-1/4} e^{-u} du$.The integral $\int_{0}^{\infty} u^{-1/4} e^{-u} du$ is a constant (Gamma function $\Gamma(3/4)$).Therefore,$N \propto n_0 \alpha^{-3/4}$.

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

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