Animated Solution for Physics - Thermodynamics: 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

Imagine a gas cloud spreading outwards from the origin in all directions. The problem states that the number density of the gas molecules, $n(r)$, is not uniform; it decreases as we move further away from the origin according to the relation $n(r)=n_0{e}^{-\alpha r^4}$.

To find the total number of molecules in this entire space, we cannot simply multiply density by total volume. Instead, we must use the power of calculus. We start by considering a very thin spherical shell of radius $r$ and an infinitesimally small thickness $dr$. Because the shell is so thin, the density of the gas within this specific shell is practically constant.

The volume of this thin spherical shell, $dV$, is simply its surface area multiplied by its thickness. Thus, $dV=4\pi r^{2}dr$.

The number of molecules residing just within this shell, let's call it $dN$, is the product of the local density and the shell's volume:

Substituting our given expressions into this relation, we get:

To find the total number of molecules $N$ in the entire universe of this gas cloud, we must sum up all these tiny $dN$ contributions from the origin ($r=0$) all the way to infinity ($r=\infty$). This gives us our master integral:

At first glance, integrating $e^{-\alpha r^4}{r}^2$ looks intimidating. However, a clever algebraic substitution will break it down beautifully. Let's take the troublesome exponent and set it to a new variable $t$:

From this, we can isolate $r$:

Now, we need to find the differential $dr$ in terms of $dt$. Differentiating our substitution equation yields:

We need to replace the $r^{2}dr$ term in our integral. Let's manipulate the differential equation to isolate $r^{2}dr$:

Substitute the expression for $r$ back into the denominator:

Now, we substitute everything back into our master integral. Notice that as $r\to 0$, $t\to 0$, and as $r\to \infty$, $t\to \infty$. So, our limits of integration remain unchanged.

Let's pull all the constants out of the integral. The $4$ in the numerator and denominator cancel out perfectly:

Look closely at the remaining integral: ${\int }_0^{\infty }{e}^{-t}{t}^{-1/4}dt$. This is a definite integral with constant limits. It does not depend on $r$, $t$, $\alpha$, or $n_0$. It evaluates to a pure, constant number (specifically, it is related to the Gamma function, $\Gamma (3/4)$).

Since the integral is just a constant multiplier, we can confidently conclude the proportionality of the total number of molecules:

This elegant result shows how the total particle count scales with the central density $n_0$ and the spatial decay parameter $\alpha$.