The energy required to ionize a mole of potassium ions is 419 kJ/mol. What is the longest wavelength of light capable of this ionization?

First divide the ionization energy by Avogadro's number to get the energy per atom of potassium;

419 kj/mol / 6.023 x 10²³

= 4.19 x 10⁵ / 6.023 x 10²³ = 6.96 x 10⁻¹⁹

E = hc/λ

where lambda (λ) is the wavelength, h is Planck's constant, c is the speed of light

E = 6.96 x 10⁻¹⁹ j/atom

h = 6.63x10⁻³⁴ Js

c = 3 x 10 ⁸ m/s

λ = ?

λ = hc/E = (6.63x10⁻³⁴ x 3 x 10⁸) / 6.96 x 10⁻¹⁹ = 285.8nm = 286nm

The longest wavelength of light capable of this ionization is 286nm.