Question

In the Bohr model, which of the following electron transitions in a hydrogen atom results in the emission of the highest-energy photon? $$ \begin{aligned} &n=3 \text { to } n=2 \\ &n=4 \text { to } n=3 \end{aligned} $$

Short answer

The electron transition from \( n=3 \) to \( n=2 \) results in the emission of the highest-energy photon.

Step-by-step solution

Define the Energy Difference Formula

The change in energy is given by the Rydberg formula: \( \Delta E = 13.6 (1/n_{f}^{2} - 1/n_{i}^{2}) \) eV, where \( n_{f} \) is the final energy level, \( n_{i} \) is the initial energy level, and the energy is measured in electronvolts (eV).

Calculate the Energy Difference for First Transition

Substitute \( n_{i} = 3 \) and \( n_{f} = 2 \) into the formula: \( \Delta E = 13.6 (1/2^{2} - 1/3^{2}) \approx 1.89 \) eV. This is the energy emitted during the transition from n=3 to n=2.

Calculate the Energy Difference for Second Transition

Similarly, substitute \( n_{i} = 4 \) and \( n_{f} = 3 \) into the formula: \( \Delta E = 13.6 (1/3^{2} - 1/4^{2}) \approx 0.66 \) eV. This is the energy emitted during the transition from n=4 to n=3.

Compare the Energy Differences

Comparing the energy emitted during the two transitions, it is clear that the transition from \( n=3 \) to \( n=2 \) emits the highest energy photon.

Key concepts

Electron Transitions

Understanding electron transitions in the hydrogen atom provides insight into how photons are emitted or absorbed within this simplest of atomic systems. When an electron changes its position from a higher energy level (one with a larger principal quantum number) to a lower energy level, it emits a photon—a particle of light. The energy of the emitted photon corresponds directly to the difference in energy between these two states.

The Bohr model simplifies these electron transitions to a process we can easily analyze: by using specific energy levels, we can predict the energy of photons emitted as an electron falls from a higher orbital to a lower one. The higher the initial energy level or quantum number, and the greater the difference between the initial and final levels, the more energy the photon carries. For instance, a transition from an energy level of 3 to 2 releases more energy—and hence a higher-energy photon—than a transition from level 4 to 3, as indicated by the exercise problem.

Rydberg Formula

The Rydberg formula is a key piece of the puzzle in understanding atomic spectra and electron transitions in the hydrogen atom. It's an equation that tells us the wavelength—or equivalently, the energy—of the photons emitted or absorbed when an electron jumps between energy levels. Mathematically, it's expressed as: \[\begin{equation} \frac{1}{\lambda} = R_H \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \end{equation}\]where \( \lambda \) is the wavelength of the emitted or absorbed light, \( R_H \) is the Rydberg constant for hydrogen, \( n_{i} \) is the initial energy level, and \( n_{f} \) is the final energy level. When applied to exercise problems, as in the given example, we usually use a shorthand version of this formula that gives us the energy directly in electronvolts (eV), allowing us to bypass wavelength calculations and focus on the energy aspect of the electron transitions.

Energy Levels in Hydrogen Atom

Dwelling deeper into the Bohr model, the energy levels of the hydrogen atom are quantized, meaning that the electron can only exist in certain discrete energy levels, defined by the principal quantum number ($n$). These are often depicted as orbits or shells around the atomic nucleus; the larger the value of \(n\), the further away the orbit is from the nucleus, and the higher the energy of the electron in that orbit.

Each energy level is associated with a specific amount of energy and is given by the formula: \[\begin{equation} E_n = - \frac{13.6}{n^2} \text{ eV} \end{equation}\]where \(E_n\) is the energy in electronvolts of the level defined by the quantum number \(n\), and where \(n = 1, 2, 3, ...\). The negative sign indicates that the electron in orbit is considered bound to the atom, and the energy is taken to be negative relative to a free electron at rest, which is assigned zero energy. As \(n\) increases, the electron has higher energy and is less tightly bound to the nucleus. Understanding these energy levels is helpful for analyzing the behavior of electrons in atoms and predicting the results of specific transitions, as was seen in the exercise solution.