The electric potential between a proton and as electron is given by `V= V_(0) ln (r )/(r_(0))` , where`r_(0)` is a constant . Assuming Bohr's model to be applicable , write variation of `r_(n)` with `n,n` being the principal quantum number ?

Options

A. `r_(n) prop n`
B. `r_(n) prop 1//n`
C. `r_(n) prop n^(2)`
D. `r_(n) prop 1//n^(2)`

Answer

A

Explanation

Potential enegry `U = eV = eV_(0)In(r )/(r_(0))`
`:.` Force `F =- |(dU)/(dr)| = (eV_(0))/(r )`.
`:.` The force will provide the necessary centripetal
force. Hence `(mv^(2))/(r ) = (eV_(0))/(r ) rArr v = sqrt((eV_(0))/(m))` …..(i)
and `mvr = (nh)/(2pi)` ....(ii)
From equations (i) and (ii), `mr = ((nh)/(2pi)) sqrt((m)/(eV_(0)))`
or `r prop n`