Question

What is the longest wavelength of photon that can be ionized a hydrogen atom in its ground state?

Answer 1

Assuming that the hydrogen atom is in the ground state ie principal quantum number n = 1, the wavelength of a photon required to excite an electron in that state up to an energy level with quantum number m is given by the Rydberg equation

1/λ = R.(1/n² - 1/m²)

where R is the Rydberg constant, 1.0974 x 10^7 m^(-1). As m increases, the electron orbits at a greater distance from the nucleus until it effectively becomes detached and the hydrogen atom is ionised into a free electron and the hydrogen nucleus, a proton. At this point the term 1/m² is negligible, so that the corresponding wavelength λe is

1/λe = R or λe = 9.113 x 10^(-8) m = 91.1 nm