Question

With the increase in quantum number the energy difference between consecutive energy levels
A.remains constant
B.decreases
C.increases
D.sometimes increases sometimes decreases

Answer 1

Hint:Since, we know that the energy levels of an electron around a nucleus is represented by,
\[{E_n} = \dfrac{{m{e^4}{Z^2}}}{{8{n^2}{h^2}\varepsilon _0^2}}\]
And the energy of a given atomic orbital is therefore proportional to the inverse square of the principal quantum number i.e., ${E_n}\alpha \dfrac{1}{{{n^2}}}$

Complete step by step answer:
The energy levels of an electron around a nucleus:
\[{E_n} = \dfrac{{m{e^4}{Z^2}}}{{8{n^2}{h^2}\varepsilon _0^2}}\]
Where,
m - the rest mass of the electron;
e - the elementary charge;
Z - the atomic number;
${\varepsilon _0}$- the permittivity of free space;
h - the Planck constant;
n - the principal quantum number.
Where variables have their usual meanings
$ \Rightarrow {E_n}\alpha \dfrac{1}{{{n^2}}}$
The energy difference between adjacent levels with quantum numbers n and (n-1):\[\Delta En,n - 1 = En - En - 1 = - \dfrac{{m{e^4}{Z^2}}}{{8{h^2}\varepsilon _0^2}}[\dfrac{1}{{{{(n - 1)}^2}}} + \dfrac{1}{{{n^2}}} = \dfrac{{2n - 1}}{{{n^2} - 1}}\]
This can be approximated to \[\dfrac{2}{n}\] ​ when n is very large.
Thus, energy difference between consecutive levels decreases as n increases.
Energy of levels in hydrogen atom is $\dfrac{{ - 13.6}}{{{n^2}}}$
So, as the n increases, energy between the consecutive levels will decrease.
Because energy decreases as $\dfrac{1}{{{n^2}}}$
Hence, with increasing quantum numbers the energy difference between adjacent levels in atoms decreases.

Therefore, the correct answer is option (B).

Note:
The energy which is represented by ${E_n} = - \:\dfrac{{m{e^4}{Z^2}}}{{8{n^2}{h^2}\varepsilon _0^2}}$ is negative and it approaches zero as the quantum number n approaches infinity. Because the hydrogen atom is used as a foundation for multi-electron systems, it is useful to remember the total energy (binding energy) of the ground state hydrogen atom, \[{E_H} = - 13.6eV\]. The spacing between electronic energy levels for small values of n is very large while the spacing between higher energy levels gets smaller very rapidly.