Question

When atoms in excited states collide with unexcited atoms they can transfer their excitation energy to those atoms. The most efficient energy transfer occurs when the excitation energy matches the energy of an excited state in the unexcited atom. Assuming that we have a collection of excited hydrogen atoms in the \(2 s^{1}\) excited state, are there any transitions of \(\mathrm{He}^{+}\) that could be most efficiently excited by the hydrogen atoms?

Short answer

The specific answer will depend on the matching energy levels found in step 2. However, it will be a statement of either yes or no, based on the comparison of energy levels of hydrogen and \( \mathrm{He}^{+} \) atoms.

Step-by-step solution

Determine the Energy of the Excited Hydrogen Atom

The energy of the \(2 s^{1}\) excited state in a hydrogen atom can be calculated using the Rydberg formula for hydrogen, \(E_n = -13.6 eV/n^2\), where \(n\) is the principal quantum number. For the given \(2 s^{1}\) excited state, \(n=2\). Use this in the formula to find the energy.

Find a Matching Energy Level in the \( \mathrm{He}^{+} \) Ion

Next, use the energy level diagram for helium to find any excited states that match the energy of the excited hydrogen state. Check which states of \( \mathrm{He}^{+} \) have the same (or close) energy levels.

Analyze the Energy Match

If there's an energy level of \( \mathrm{He}^{+} \) that matches the energy of the \(2 s^{1}\) excited state of hydrogen, then those transitions of \( \mathrm{He}^{+} \) could most efficiently accept energy from the hydrogen atoms. If not, then energy transfer wouldn't be efficient.

Key concepts

Hydrogen Atom Excited States

Hydrogen atoms can reside in different states based on the energy they possess. These states are referred to as "excited states" when an electron has absorbed energy and moved to a higher energy level. For hydrogen, these energy levels are determined by the principal quantum number, often represented as "n." The lowest state or ground state is when \(n=1\). The energy increases as the electron moves to higher levels, such as \(n=2\), \(n=3\), and so on.

In the given problem, the hydrogen atoms are in the \(2s^1\) excited state, which corresponds to \(n=2\). This indicates that the electron has absorbed energy sufficient to move from its ground state. Understanding these excited states is crucial because they play a role in energy transfer processes where atoms collide, sharing energy with unexcited atoms.

Rydberg Formula

The Rydberg formula is a pivotal equation used to predict the wavelength of light resulting from an electron moving between energy levels in a hydrogen atom. This formula can be adapted to find the energy of these transitions instead, which is handy when dealing with excited states.

The formula is expressed as:

• \(E_n = -13.6 \text{ eV} / n^2\)

Here, \(E_n\) represents the energy of an electron at level \(n\), and \(-13.6 \text{ eV}\) is the ionization energy of hydrogen. For the hydrogen \(2s^1\) state used in the exercise, by plugging \(n=2\) into this formula, you get the energy in electron volts. This equation provides insight into how much energy an electron possesses at any given energy level.

Helium Ion Transitions

Helium ions, specifically \(\mathrm{He}^{+}\), are essentially hydrogen-like in that they are a helium atom that has lost one of its two electrons, creating a single electron in orbit. Just like hydrogen, the energy levels of helium ions are quantized and marked by principal quantum numbers. However, these levels differ due to the different nuclear charge.

When excited states of helium ions are considered, they can undergo transitions, where their single electron moves between these energy levels. These transitions are associated with specific energy differences, which can be compared with those of hydrogen to detect possible energy matches. These matches indicate suitable conditions for energy transfer efficiency between the two atoms when they collide.

Energy Level Matching

Energy level matching is a significant concept in atomic physics, especially relevant to excitation energy transfer. It involves finding a scenario where the energy levels of one atom align closely with the energy levels of another atom or ion.

In the context of the exercise, you're trying to find any helium ion transitions that can be effectively excited by hydrogen atoms in the \(2s^1\) state. A match occurs when the excited state's energy of the hydrogen atom is equal to one of the transition energies of the helium ion. When such a match happens, the energy transfer is most efficient because the quantum condition is satisfied. In practical terms, this means looking at helium's quantized energy levels to identify a possible transition close to the \(n=2\) energy level of hydrogen.