Suppose an electron is attracted towards 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),r_(n)propn^(2)`
B. `T_(n)prop(1)/(n^(2)),r_(n)propn^(2)`
C. `T_(n)` is independent then `n,r_(n)propn`
D. `T_(n)prop(1)/(n),r_(n)propn`

Answer

C

Explanation

If the velocity of an electron in `n^(th)` orbit is `v_(n)`, then the centripetal force = the force on electron.
`:.(mv_(n)^(2))/(r_(n))=(K)/(r_(n))`
`:.v_(n)^(2)=(K)/(m)`
`T_(n)=(1)/(2)m((K)/(m))`
`:.T_(n)=(K)/(2)`
Now kinetic energy `=(1)/(2)mv_(n)^(2)`
Here, there is no n term, so the kinetic energy is independent then n.
From Bohr.s hypothesis, the angular momentum
`mv_(n)r_(n)=(nh)/(2pi)`
`:.r_(n)=(nh)/(2pimv_(n))`
`:.r_(n)=(nh)/(2pimxsqrt((k)/(m))):.r_(n)=(nh)/(2pisqrt(mk))`
`:.r_(n)propn`