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The number density of molecules of a gas...

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 :

A

`n_0 alpha^(-3//4)`

B

`sqrt(n_0) alpha^(1//2)`

C

`n_0 alpha^(1//4)`

D

`n_0 alpha^(-3)`

Text Solution

Verified by Experts

The correct Answer is:
A
`n(r )=n_0e^(-alpha r^4)`
Molecules in shell element of radii r and `r+dr." "dN=n(r )dV= n(r )(4pir^2dr)`
Total molecules `=N=int dN implies N=4pi n_0 int r^2e^(-alpha r^4)dr`
Put `alphar^4=t implies 4alphar^3dr =dt`
`implies r^2dr= (dt)/(alpha r)= alpha^(-3/4)t^(-1/4)dt implies N=4pi n_0 alpha^(-3/4) int t^(-1/4) e^(-t) dt implies n propto n_0 alpha^(-3/4)`.
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