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 10⁸ 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.