How do you calculate the ionization energy of a hydrogen atom in its ground state?

By using the Rydberg expression:

The wavelength λ of the emission line in the hydrogen spectrum is given by:

1λ=R[1/n₁²−1/n₂²]

R is the Rydberg Constant and has the value 1.097×107m−1

n1 is the principle quantum number of the lower energy level

n2 is the principle quantum number of the higher energy level.

The energy levels in hydrogen converge and coalesce:

The electron is in the n1=1 ground state we need to consider series 1. These transitions occur in the u.v part of the spectrum and is known as The Lyman Series. The 1/n₂² decreases.

The Rydberg expression then written as

1λ=R[1/n₁²−0]=R/n₁²

Since n1=1 this becomes:

1λ=R

∴1λ=1.097×107

∴λ=9.116×10−8m

c=νλ

∴ν=c/λ=3×10⁸/9.116×10⁻⁸

=3.291×10¹⁵s⁻¹

E=hν

∴E=6.626×10−34×3.291×1015=2.18×10−18J

This is the energy needed to remove 1 electron from 1 hydrogen atom. To find the energy required to ionize 1 mole of H atoms we multiply by the Avogadro Constant:

E=2.18×10−18×6.02×1023=13.123×105J/mol

E=1312kJ/mol