Final answer:
To determine the final orbital after an electron emits a photon, the Rydberg formula is applied. Given the initial orbital (n=5) and the emitted photon's wavelength (1284 nm), the calculations show that the final orbital is nf = 3.
Step-by-step explanation:
To determine the final orbital (nf) after an electron emits a photon from the n = 5 orbital, we need to use the Rydberg formula for hydrogen:
Rydberg equation: \( \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \)
Here, \( \lambda \) is the wavelength of the emitted photon, R is the Rydberg constant (1.097 \times 10^7 m^-1), \( n_i \) is the initial energy level, and \( n_f \) is the final energy level.
Given data: \( n_i = 5 \) and \( \lambda = 1284 \mathrm{nm} \) or \( 1284 \times 10^{-9} m \).
Step 1: Convert the given wavelength to meters.
\( \lambda = 1284 \times 10^{-9} m \)
Step 2: Insert known values into the Rydberg formula and solve for \( n_f \).
\( \frac{1}{1284 \times 10^{-9} m} = 1.097 \times 10^7 m^-1 \left( \frac{1}{n_f^2} - \frac{1}{5^2} \right) \)
Step 3: Rearrange to solve for \( n_f^2 \) and then find \( n_f \).
After calculations, we find that the final orbital is n_f = 3.
This means the electron emitted a photon and transitioned from the \( n = 5 \) orbital to the \( n = 3 \) orbital.