Transition in He ion with same wave number as first line in Balmer series of H atom

Introduction
The Balmer series in hydrogen atom refers to the emission lines of visible light that are produced when an excited electron in the hydrogen atom falls back to the second energy level. Similarly, He+ ion also emits radiation when an electron jumps from a higher energy level to a lower energy level. In this answer, we will explore which transition in He+ ion has the same wave number as the first line in Balmer series of H atom.

Balmer Series in Hydrogen Atom
The Balmer series in hydrogen atom consists of the emission lines that are produced when an excited electron in the hydrogen atom falls back to the second energy level (n=2) from higher energy levels. The first line in the Balmer series (H-alpha line) corresponds to the transition from n=3 to n=2 energy level and has a wavelength of 656.3 nm.

Transitions in He+ Ion
He+ ion is a hydrogen-like ion with a single electron orbiting around a nucleus with two protons. Just like hydrogen atom, He+ ion also emits radiation when an electron jumps from a higher energy level to a lower energy level. The energy levels in He+ ion are given by:

E = -13.6 * Z^2 / n^2

where Z=2 is the atomic number of He+ ion and n is the principal quantum number.

Comparison of Energy Levels in Hydrogen Atom and He+ Ion
In hydrogen atom, the energy levels are given by:

E = -13.6 / n^2

Comparing the two equations, we can see that the energy levels in He+ ion are four times smaller than the energy levels in hydrogen atom. This means that the transitions in He+ ion will have four times higher wave numbers than the transitions in hydrogen atom.

Answer
The first line in Balmer series of H atom has a wavelength of 656.3 nm. The corresponding wave number is given by:

wave number = 1 / wavelength = 1.524 x 10^3 cm^-1

Since the transitions in He+ ion have four times higher wave numbers than the transitions in hydrogen atom, the transition in He+ ion that has the same wave number as the first line in Balmer series of H atom will be:

wave number = 4 * 1.524 x 10^3 cm^-1 = 6.096 x 10^3 cm^-1

Using the equation for energy levels in He+ ion, we can solve for the principal quantum number (n) that corresponds to this wave number:

E = -13.6 * 2^2 / n^2 = -54.4 / n^2
n^2 = 54.4 / 6.096 x 10^3 = 8.907 x 10^-3
n = 0.094

This is a non-physical result since the principal quantum number can only take positive integer values. Therefore, there is no transition in He+ ion that has the same wave number as the first line in Balmer series of H atom.

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