Question

An excited hydrogen atom emits a photon with an energy of \(1.133 \mathrm{eV}\). What were the initial and final states of the hydrogen atom before and after emitting the photon?

Short answer

Answer: The initial state is 3, and the final state is 2.

Step-by-step solution

Recall the formula for energy levels in the hydrogen atom

The energy levels of the hydrogen atom can be determined using the following formula: \(E_n = -\dfrac{13.6 \mathrm{eV}}{n^2}\), where \(n\) is the principal quantum number (state) and \(E_n\) denotes the energy of the nth state.

Calculate the energy difference

The energy of the emitted photon is the difference in energy between the initial and final states of the atom. Thus, we can write the energy difference as: \(\Delta E = E_{initial} - E_{final} = 1.133 \mathrm{eV}\)

Use the formula for energy levels to find the possible state transitions

Since the energy difference is caused by a transition between two energy states, we can set up an equation to find the possible initial and final states: \(E_{initial} - E_{final} = \dfrac{13.6}{n^2_{final}} - \dfrac{13.6}{n^2_{initial}} = 1.133 \mathrm{eV}\)

Find the ratio of initial and final states

To simplify our calculation, let's call this ratio \(k\): \(k = \dfrac{1}{n^2_{initial}} - \dfrac{1}{n^2_{final}}\). Now, we can rewrite our equation as: \(13.6 \times k = 1.133 \mathrm{eV}\) Solving for \(k\), we get: \(k =\dfrac{1.133}{13.6} = 0.0833\)

Find the initial and final states that satisfy this ratio

Since \(n_{initial}\) and \(n_{final}\) are integers and \(n_{initial}>n_{final}\), we can find pairs of values \((n_{initial}, n_{final})\) that satisfy the condition \(k = \dfrac{1}{n^2_{initial}} - \dfrac{1}{n^2_{final}}\). After checking possible energy state transitions, we find that only the following pair of states satisfies the obtained ratio: \((n_{initial}, n_{final}) = (3, 2)\)

Conclude the initial and final states

Based on our calculations, the initial and final states of the hydrogen atom before and after emitting the photon are: Initial state: \(n_{initial} = 3\) Final state: \(n_{final} = 2\)