The electirc potential between a proton and an electron is given by `V = V_0 ln (r /r_0)` , where r_0 is a constant. Assuming Bhor model to be applicable, write variation of `r_n` with n, being the principal quantum number. (a) `r_n prop n` (b) `r_n prop (1)/(n)` (c ) `r_n^2` (d)`r_n prop (1)/(n^2)`

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, C, D

Explanation

Given potential energy between electron and proton is
`V_(0) = log ( r)/(r_(0))`
`:. |F| = (d)/(dr) [V_(0) = loge ( r)/(r_(0))] = (V_(0))/(r_(0)) xx (1)/( r)`
But this force acts as centripetal force
`:. (m v^(2))/( r) = (V_(0))/(r r_(0)) implies m v^(2) = (V_(0))/(r_(0))` (i)
By Bohr's postulate `mv r = (nh)/(2 pi)` (ii)
From (i) and (ii),
`(m^(2) v^(2) r^(2))/(m v^(2)) = (n^(2) h^(2) r_(0)^(2))/(4 pi^(2) V_(0)^(2)) implies r^(2) = (n^(2) h^(2) r_(0)^(2))/(4 pi^(2) V_(0)^(2)m)`