Suppose an electron is attracted toward the origin by a force`(k)/(r )` where `k` is a constant and `r` is the distance of the electron from the origin .By applying Bohr model to this system the radius of the `n^(th)` orbital of the electron is found to be `r_(n)` and the kinetic energy of the electron to be `T_(n)` , Then which of the following is true ?

Options

A. `T_(n) prop 1/(n^(2))`
B. `T_(n)` is independent of` n,r_(n), prop n`
C. `T_(n) prop 1/(n) , r_(n) prop n`
D. `T_(n) prop 1/(n) and r_(n) prop n^(2)`

Answer

B

Explanation

Applying Bohr model to the given system,
`(mv^(2))/(r_(n))=k/(r_(n)).........(i)`
and `mvr_(n)=(nh)/(2pi), v=(nh)/(2pimr_(n))`
Put in (i), `m/(r_(n))xx(n^(2)h^(2))/(4pi^(2)m^(2)r_(n)^(2))=k/(r_(n))`
`r_(n)^(2)=(n^(2)h^(2))/(4pi^(2)mk)......(ii)`
`:. r_(n)^(2) prop n^(2) or r_(n) propn`
K.E. of electron, `T_(n)=1/2mv^(2)`
`=1/2m (n^(2)h^(2))/(4pi^(2)m^(2)r_(n)^(2)) = (n^(2)h^(2))/(8pi^(2)m^(2)r_(n)^(2))`
Using (ii), we get
`T_(n)=(n^(2)h^(2).4pi^(2)mk) /(8pi^(2)mn^(2)h^(2))=k/2`
`:. T_(n)` is independent of n.