For a gas sample with `N_(0)` number of molecules, function `N(V)` is given by: `N(V) = (dN)/(dV) = ((3N_(0))/(V_(0)^(3)))V^(2)` for `0 lt V lt V_(0)` and `N(V) = 0` for `V gt V_(0)`. Where `dN` is number of molecules in speed range `V` to `V+ dV`. The rms speed of the molecules is:

Options

A. `sqrt((2)/(5))V_(0)`
B. `sqrt((3)/(5))V_(0)`
C. `sqrt(2)V_(0)`
D. `sqrt(3)V_(0)`

Answer

B

Explanation

`V_(rms)^(2) = lt V^(2)gt`
`= (V_(1)^(2)+V_(2)^(2)+V_(3)^(2)+….)/(N_(0)) = (intV^(2)dN)/(intdN)`
here `(dN)/(dV) = N(V)`
`V_(rms)^(2) = (1)/(N_(0)) overset(V_(0))underset(0)int N(V)V^(2) dV = (1)/(N_(0)) overset(V_(0))underset(0)int ((3N_(0))/(V_(0)^(3))V^(2))V^(2)dV`
`= (3)/(5) V_(0)^(2) rArr V_(rms) = sqrt((3)/(5)) V_(0)`