An imaginary particle has a charge equal to that of an electron and mass 100 times the mass of the electron. It moves in a circular orbit around a nucleus of charge + `4 e`. Take the mass of the nucleus to be infinite. Assuming that the Bohr model is applicable to this system. (a)Derive an expression for the radius of `n^(th)` Bohr orbit. (b) Find the wavelength of the radiation emitted when the particle jumps from fourth orbit to the second orbit.

Explanation

We have
`(m_(p) v^(2))/(r_(n)) = (1)/(4 pi epsilon_(0)) = (4 e^(2))/(r_(n)^(2))` (i)
The quatization of angular momentum gives
`m_(p)vr_(n) = (nh)/(2 pi)` (ii)
Solving Eqs. (i) and (ii) we get
`r = (n^(2) h^(2) epsilon_(0))/(2 pi m_(p)) e^(2)`
Substituting `m_(p) = 100 m`, where `m = mass` of electron, we get
`r_(n) = (n^(2) h^(2) epsilon_(0))/(400 pi m e^(2))`
b. As we know , `E_(1) = - 13.60 e V` (for-H-atom)`
and `E_(n) prop ((z^(2))/(n^(2))) m`
For the given partical,
`E_(4) = ((-13.60) (4)^(2))/((4)^(2)) xx 100 = - 13.60 e V`
and `E_(2) = ((-13.60) (4)^(2))/((2)^(2)) xx 100 = - 5440 e V`
`Delta E = E_(4) - E_(2) = 4080 e V`
`lambda ("in" Å) = (12375)/(Delta E ("in" e V)) = (12375)/(4080) xx 3.03 Å`