Question

A collection of hydrogen atoms have all been placed into the $n=4$ excited state. What wavelengths of photons will be emitted by the hydrogen atoms as they transition back to the ground state?

Short answer

Answer: The wavelength of the photon emitted when a hydrogen atom transitions from the n=4 excited state to the n=1 ground state is approximately $9.723\times {10}^{-8}\text{m}$.

Step-by-step solution

Understanding the Rydberg formula for hydrogen

The Rydberg formula for hydrogen is given by: ${\displaystyle \frac{1}{\lambda }}=R_H({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}})$ where $\lambda$ is the wavelength of the emitted photon, $R_H$ is the Rydberg constant for hydrogen (approximately $1.097\times {10}^7{\text{m}}^{-1}$), $n_1$ is the principal quantum number of the lower energy level, and $n_2$ is the principal quantum number of the higher energy level. Since the hydrogen atoms are initially in the $n=4$ excited state, we know that $n_2=4$. The ground state of an atom corresponds to $n_1=1$.

Calculate the wavelength for the transition from $n=4$ to $n=1$

Now we can use the Rydberg formula to calculate the wavelength of the emitted photon when the hydrogen atom transitions from the $n=4$ state to the $n=1$ ground state: ${\displaystyle \frac{1}{{\lambda }_{4\to 1}}}=R_H({\displaystyle \frac{1}{1^2}}-{\displaystyle \frac{1}{4^2}})$ ${\displaystyle \frac{1}{{\lambda }_{4\to 1}}}=R_H({\displaystyle \frac{1}{1}}-{\displaystyle \frac{1}{16}})$ ${\displaystyle \frac{1}{{\lambda }_{4\to 1}}}=R_H({\displaystyle \frac{15}{16}})$ Now, we'll find the value of ${\lambda }_{4\to 1}$: ${\lambda }_{4\to 1}={\displaystyle \frac{1}{R_H({\displaystyle \frac{15}{16}})}}$ ${\lambda }_{4\to 1}={\displaystyle \frac{1}{(1.097\times {10}^7{\text{m}}^{-1})({\displaystyle \frac{15}{16}})}}$ ${\lambda }_{4\to 1}\approx 9.723\times {10}^{-8}\text{m}$ So, the wavelength of the photon emitted when the hydrogen atom transitions from the $n=4$ state to the $n=1$ ground state is approximately $9.723\times {10}^{-8}\text{m}$.