Question

Given that the hydrogen atom has an infinite number of energy levels, why can't a hydrogen atom in the ground state absorb all possible wavelengths of light?

Short answer

A hydrogen atom in the ground state cannot absorb all possible wavelengths of light because the energy difference between the ground state and other energy levels must exactly match the energy of the absorbed photon. Although there are infinite energy levels, the energy differences between them decrease as n increases, limiting absorption to specific wavelengths that correspond to these energy differences.

Step-by-step solution

Understanding Hydrogen Atom Energy Levels

A hydrogen atom consists of a single proton in the nucleus and a single electron orbiting it. These electrons can occupy different energy levels, and the energy of these levels is given by the formula: E_n = -\frac{13.6 eV}{n^2} where E_n is the energy of the nth energy level and n is the principal quantum number (n = 1, 2, 3, ...). The ground state is the lowest energy level, with n = 1.

Photon Absorption and Emission

When a hydrogen atom absorbs a photon of light, it can undergo a transition from a lower energy level to a higher energy level. The absorbed photon's energy must exactly match the energy difference between the two levels, given by the formula: \Delta E = E_{final} - E_{initial} As a result, only certain specific wavelengths of light can be absorbed, corresponding to the energy differences between energy levels.

Ground State Absorption Limitations

When a hydrogen atom is in the ground state, it can only absorb photons that have enough energy to transition to another energy level. However, even though there are an infinite number of energy levels, the energy difference between them decreases as n increases: \Delta E = -\frac{13.6 eV}{n_f^2} - \left(-\frac{13.6 eV}{n_i^2}\right) In the ground state, n_i = 1, and n_f can take any integer value greater than 1. This means that as n_f increases, the absorbed photon's energy decreases, and the corresponding wavelength increases. When n_f approaches infinity, the energy difference approaches zero, and the wavelength approaches infinity. Therefore, even though there are an infinite number of energy levels in hydrogen, there are specific wavelengths (and corresponding energies) that can be absorbed by the atom when it is in the ground state. This is why a hydrogen atom in the ground state cannot absorb all possible wavelengths of light.