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 propn`
C. `T_(n)prop(1)/(n)` and `r_n`
D. `T_(n)prop(1)/(n)` and `r_npropn^2`

Answer

B

Explanation

Applying bohr model to the given system, `(mv^2)/(r_n)=(k)/(r_n)`........(i)
and `mvr_n=(nh)/(2pi)` or `v=(nh)/(2pimr_n)`
put in (i),`(m)/(r_n)xx(n^2h^2)/(4pi^2m^2r_n^2)=(k)/(r_n)rArrr_n^2=(n^2h^2)/(4pi^2mk)`......(ii)
`therefore r_n^2propn^2` or `r_npropn`
K.E. of the electron,
`T_n=(1)/(2)mv^2=(1)/(2)m(n^2h^2)/(4pi^2m^2r_n^2)=(n^2h^2)/(8pi^2mr_n^2)`
Using (ii), we get , `T_n=(n^2h^24pi^2mk)/(8pi^2mn^2h^2)=(k)/(2)`
`thereforeT_n` is independent of n.