In hydrogen and hydrogen-like atom , the ratio of `E_(4 n) - E_(2 n) and E_(2 n) - E_(n)` varies with atomic nimber `z` and principal quantum number`n` as

Options

A. `z^(2)//n^(2)`
B. `z^(4)//n^(4)`
C. `z//n`
D. `(n^(2)//z^(2))`

Answer

D

Explanation

`E_(n)= -13.6/n^(2) (Z^(2))`
`E_(2n)= -13.6/((2n)^(2)) Z^(2)=13.6/4 (Z^(2)/n^(2))`
`E_(2n)-E_(n)=[(-13.6/4)Z^(2)/n^(2)]-[-3.6 (Z^(2)/n^(2))]`
`E_(2n)-E_(n)= -(13.6 )/(4) Z^(2)/n^(2)+13.6 Z^(2)/n^(2)`
`E_(2n)-E_(n)=Z^(2)/n^(2) (13.6-13.6/4)=(10.2) (Z^(2)/n^(2))`
`(E_(2n))(E_(n))=((13.6)^(2))/(4) (Z^(4)/n^(4))`
`((E_(2n)-E_(n)))/((E_(2n))(E_(n)))=((10.2)(Z^(2)/n^(2)))/(((13.6)^(2))/4 (Z^(4)/n^(4))) implies ((E_(2n)-E_(n)))/((E_(2n))(E_(n))) prop (n^(2)/Z^(2))`