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

Answer

A

Explanation

Given : `V=V_(0)ln((r)/(r_(0)))`
`therefore" Potential energy, "U=eV`
or `U=eV_(0) ln ((r)/(r_(0))) therefor e(dU)/(dr)=-(eV_(0))/(r ) or |F|=(eV_(0))/(r )`
This force provides the necessary centripetal force.
`therefore (mv^(2))/(r )=(eV_(0))/(r ) or v=sqrt((eV_(0))/(m))" ...(i)"`
By Bohr.s postulate, `mvr=(nh)/(2pi)`
or `v=(nh)/(2pimr)`
From equation (i) and (ii), we get
`(nh)/(2pimr)=sqrt((eV_(0))/(m)) or r=(nh)/(2pim)xxsqrt((m)/(eV_(0)))`
`"or "r=[(h)/(2pi)sqrt((l)/(meV_(0)))]xxn therefore r_(n) prop n`