Let `v_n and E_n` be the respective speed and energy of an electron in the n-th orbit radius `r_n` in a hydrogen atom, as predicted by Bohr's model . Then

Options

A. plot of `E_nr_n//E_1r_1` as a function of n is a straigth line of slope 0
B. plot of `r_nv_n//r_1v_1` as a function of n is a straight line of slope 1
C. plot of 1n `(r_n)/(r_1)` as a function of `ln (n)` is a straight line of slope 2
D. plot of 1n `((r_nE_1)/(E_nr_1))` as a function of 1n (n) is a straight line of slope 4

Answer

A, B, C, D

Explanation

According to Bohr's theory , `v_n prop 1/n` ……(1)
`E_n prop 1/(n^2) ……..(2)`
and `r_n prop n^2` …..(3)
From equations (2) and (3) we get ,
`E_nr_n prop n^2 xx1/(n^2) or, E_n r_n prop n^0`
Hence , `E_nr_n` =constant
`therefore (E_nr_n)/(E_1r_1)="constant"[therefore "slope"=0]`
From equations (1) and (3) we get ,`r_nv_n prop n`
`(r_nv_n)/(r_1v_1)=n or, ((r_nv_n)//(r_1v_1))/(n)=1 `
From equation (3) ,`(r_n)/(r_1)=n^2 or, (1n(r_n//r_1))/(1n(n))=2`
From equations (2) and (3) ,`(r_n)/(E_n) prop n^4`
`therefore (r_nE_1)/(E_nr_1)=n^4 or, (1n(r_nE_1//E_nr_1))/(1n(n))=4`