Given number density

The number density varies with distance as n(r)=n_0 e^{-lpha r^4}.

The total number of molecules is obtained by integrating the number density over the whole volume: N=int n(r)\,dV.

Use spherical symmetry

Since $n(r)$ depends only on $r$, in spherical coordinates: $dV=4\pi r^2 dr.$

Therefore, N=4\pi n_0 \int_0^\infty r^2 e^{-lpha r^4}dr.

Extract the dependence on $\alpha$

We only need proportionality. Let $x=\alpha^{1/4}r \quad \Rightarrow \quad r=\frac{x}{\alpha^{1/4}}, \quad dr=\frac{dx}{\alpha^{1/4}}.$

Then, $r^2dr=\left(\frac{x^2}{\alpha^{1/2}}\right)\left(\frac{dx}{\alpha^{1/4}}\right)=\frac{x^2}{\alpha^{3/4}}dx.$

Also, e^{-lpha r^4}=e^{-x^4}.

So the integral becomes $N=4\pi n_0 \int_0^\infty \frac{x^2}{\alpha^{3/4}} e^{-x^4}dx.$

Hence, $N=4\pi n_0 \alpha^{-3/4}\int_0^\infty x^2 e^{-x^4}dx.$

Identify the proportionality

The integral $\int_0^\infty x^2 e^{-x^4}dx$ is just a constant, independent of $n_0$ and $\alpha$.

Therefore, $N \propto n_0\alpha^{-3/4}.$

Match with the options

The correct option is: $\boxed{A:\ n_0\alpha^{-3/4}}$

Comparison with stored answer

Stored correct answer: A

Our derived answer: A

So they agree.