Question

An electron in an excited state in a hydrogen atom can return to the ground state in two different ways: (a) via a direct transition in which a photon of wavelength \(\lambda_{1}\) is emitted and (b) via an intermediate excited state reached by the emission of a photon of wavelength \(\lambda_{2}\). This intermediate excited state then decays to the ground state by emitting another photon of wavelength \(\lambda_{3}\). Derive an equation that relates \(\lambda_{1}\) to \(\lambda_{2}\) and \(\lambda_{3}\).

Short answer

\( \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \).

Step-by-step solution

Understanding the Energy Transition

An electron in the hydrogen atom drops directly from an excited state to the ground state emitting a photon of wavelength \( \lambda_1 \). This corresponds to an energy change \( E_1 \). Similarly, for an indirect pathway, the electron first transitions to an intermediate state emitting a photon with wavelength \( \lambda_2 \) and then from the intermediate state to the ground state emitting another photon of wavelength \( \lambda_3 \). The total energy change through this path is the sum of two energy transitions, \( E_2 + E_3 \).

Relate Energy to Wavelength

According to the Planck–Einstein relation, the energy of a photon is given by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength. Therefore, for the direct transition \( E_1 = \frac{hc}{\lambda_1} \), and for the indirect pathway: \( E_2 = \frac{hc}{\lambda_2} \) and \( E_3 = \frac{hc}{\lambda_3} \).

Apply the Conservation of Energy

For the electron's transitions, the energy change through the direct transition must equal the total energy change through the indirect transitions. Thus, we have \( E_1 = E_2 + E_3 \). By substituting the relations from the previous step, we write \( \frac{hc}{\lambda_1} = \frac{hc}{\lambda_2} + \frac{hc}{\lambda_3} \).

Simplify the Equation

The equation \( \frac{hc}{\lambda_1} = \frac{hc}{\lambda_2} + \frac{hc}{\lambda_3} \) can be divided by \( hc \) on both sides to get \( \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \). This relation shows how the wavelength of the direct transition photon relates to the wavelengths of the photons emitted in the indirect transitions.

Key concepts

Photon Emission

When an atom releases energy as light, it emits particles of light called photons. The emission occurs when an electron moves from a higher energy level to a lower one. In the case of a hydrogen atom, electrons in excited states can drop back down to lower energy states, releasing photons in the process. Each emitted photon has an energy equal to the difference in energy levels the electron transitions between. This difference determines the photon's wavelength and color of the light. It's like when you let a ball fall; the higher it drops from, the more energy it has when it lands.

Wavelength Transitions

Wavelength transitions refer to the change in wavelength (or color) of light as electrons move between energy levels in an atom. In hydrogen, if an electron moves directly from an excited state to the ground state, one wavelength is observed. However, if the electron drops to an intermediate state before reaching the ground level, two emissions occur, each with its own wavelength. This creates a sequence of photon emissions, each having different energies and hence different wavelengths. Measuring these wavelengths helps us understand the energy changes in the atom.

Energy Conservation

Energy conservation in physics means that energy can't be created or destroyed, only transformed. During electron transitions within an atom, the energy emitted as photons must equal the energy lost by the electron changing levels. In hydrogen atoms, if an electron falls directly to the ground state, the energy released equals a single photon's energy. Simultaneously, if the same electron first transitions to an intermediate state and then to the ground state, the total energy released remains the same, but it is split across two photons. This ensures that the overall energy balance is maintained, following the law of conservation of energy.

Planck–Einstein Relation

The Planck–Einstein relation provides a way to connect photon energy to its wavelength. It states that the energy (E) of a photon is equal to Planck’s constant (h) times its frequency (\(ν\)): E = hν. Since frequency relates to wavelength by the speed of light (\(c\)), the relation can be rewritten in terms of wavelength (\(λ\)): E = \(\frac{hc}{λ}\). In the context of hydrogen atom transitions, this formula allows us to calculate the energy of the emitted photon based on its wavelength. Understanding this relation helps explain why the wavelength changes in specific steps according to different energy state jumps within the atom.