To find the initial energy level ($n_{\text{initial}}$) of the electron in the hydrogen atom that emitted a photon of wavelength 102.6 nm, we can use the Rydberg formula for hydrogen emission lines:

$\frac{1}{\lambda} = R_H \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$

Where:

• $\lambda$ is the wavelength of the emitted photon (in meters),
• $R_H$ is the Rydberg constant for hydrogen, approximately $1.097 \times 10^7 \, \text{m}^{-1}$,
• $n_{\text{final}}$ is the final energy level (which is 1 for ground state),
• $n_{\text{initial}}$ is the initial energy level.

Step 1: Convert the wavelength to meters:

$\lambda = 102.6 \, \text{nm} = 102.6 \times 10^{-9} \, \text{m}$

Step 2: Rearrange the Rydberg equation to solve for $n_{\text{initial}}$:

$\frac{1}{\lambda} = R_H \left( 1 - \frac{1}{n_{\text{initial}}^2} \right)$ $\frac{1}{n_{\text{initial}}^2} = 1 - \frac{1}{R_H \lambda}$

Step 3: Calculate the right-hand side:

$R_H \lambda = (1.097 \times 10^7) \times (102.6 \times 10^{-9}) = 1.125 \times 10^0 = 1.125$ $\frac{1}{n_{\text{initial}}^2} = 1 - \frac{1}{1.125} \approx 1 - 0.889 = 0.111$

Step 4: Solve for $n_{\text{initial}}$:

$n_{\text{initial}}^2 = \frac{1}{0.111} \approx 9.00$ $n_{\text{initial}} = 3$

Thus, the initial energy level ($n_{\text{initial}}$) is 3.

Would you like further details on the steps, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How does the Rydberg constant relate to energy transitions in hydrogen atoms?
2. What is the significance of the Balmer and Lyman series in hydrogen spectroscopy?
3. How would the energy of the emitted photon change for transitions between different $n$ levels?
4. How can we use the energy of a photon to calculate the transition energy of an electron in any atom?
5. How does the wavelength of emitted photons vary with increasing energy level transitions?

Tip: When dealing with atomic transitions, the higher the energy level difference between the initial and final states, the shorter the wavelength of the emitted photon.