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Question: The number density of molecules of a gas depends on their distance \( r \) from the origin as\[n\lef...

The number density of molecules of a gas depends on their distance rr from the origin asn(r)=n0e−αx4n\left( r \right)={{n}_{0}}{{e}^{-\alpha {{x}^{4}}}}. Then the total number of molecules will be proportional to:

& A.{{n}_{o}}{{\alpha }^{\dfrac{1}{4}}} \\\ & B.{{n}_{o}}{{\alpha }^{-3}} \\\ & C.{{n}_{o}}{{\alpha }^{\dfrac{-3}{4}}} \\\ & D.\sqrt{{{n}_{o}}}{{\alpha }^{\dfrac{1}{2}}} \\\ \end{aligned}$$
Explanation

Solution

Here the number density of molecules are given. Find out the total number of molecules by integrating the number density in a volume. From his we will get the proportionality relation we need. This may help us to find out the answer.

Complete step-by-step answer:
Here it is given the number density of molecules of a gas.
n(r)=n0e−αx4n\left( r \right)={{n}_{0}}{{e}^{-\alpha {{x}^{4}}}}
Therefore the total number of molecule is given by integrating this equation with respect to volume,
That is,
N=∫0∞n(r)dVN=\int\limits_{0}^{\infty }{n\left( r \right)dV}
Perform the integration with respect to its volume,
First of all substitute the number density of molecules of a gas in this.
N=∫0∞n0e−αx44πr2drN=\int\limits_{0}^{\infty }{{{n}_{0}}{{e}^{-\alpha {{x}^{4}}}}4\pi {{r}^{2}}dr}
This is because, the volume can be written as,
V=4πr2drV=4\pi {{r}^{2}}dr
Where rrbe the distance from origin.
Now perform the integration over the limits.
Then we will get that,
N∝n0α−34N\propto {{n}_{0}}{{\alpha }^{-\dfrac{3}{4}}}

So, the correct answer is “Option C”.

Note: The number density of molecules is defined as an intensive property which is used to explain the degree of concentration of countable objects such as particles, molecules, phonons, cells, galaxies, and so on. The density of an object is one of its most prime and easily valued physical properties. Densities are generally used to find out the pure substances and to estimate and characterise the composition of so many types of mixtures. Relative density is explained as the measure of whether a material will float in a fluid. An iron cannonball will sink into water due to its density which is greater than that of water. In short each object is having its own density according to their molecular arrangements.