To determine the wavelength of a photon emitted by a hydrogen atom when its electron transitions from the [tex]\( n=6 \)[/tex] energy level to the [tex]\( n=2 \)[/tex] energy level, we can use the Rydberg formula for hydrogen:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength of the emitted photon,
- [tex]\(R\)[/tex] is the Rydberg constant ([tex]\(1.097373 \times 10^7 \text{ m}^{-1}\)[/tex]),
- [tex]\(n_1\)[/tex] is the principal quantum number of the final state ([tex]\( n_1 = 2 \)[/tex]),
- [tex]\(n_2\)[/tex] is the principal quantum number of the initial state ([tex]\( n_2 = 6 \)[/tex]).
### Step-by-step solution:
1. Identify the values for [tex]\(n_1\)[/tex], [tex]\(n_2\)[/tex], and [tex]\(R\)[/tex]:
- Final state, [tex]\(n_1 = 2\)[/tex]
- Initial state, [tex]\(n_2 = 6\)[/tex]
- Rydberg constant, [tex]\(R = 1.097373 \times 10^7 \; \text{m}^{-1}\)[/tex]
2. Substitute these values into the Rydberg formula:
[tex]\[
\frac{1}{\lambda} = 1.097373 \times 10^7 \left( \frac{1}{2^2} - \frac{1}{6^2} \right)
\][/tex]
3. Calculate the inverse of the wavelength:
- First, compute [tex]\(\frac{1}{2^2}\)[/tex]:
[tex]\[
\frac{1}{2^2} = \frac{1}{4} = 0.25
\][/tex]
- Next, compute [tex]\(\frac{1}{6^2}):
\[
\frac{1}{6^2} = \frac{1}{36} \approx 0.02778
\]
- Subtract these values:
\[
0.25 - 0.02778 \approx 0.22222
\]
- Now multiply by the Rydberg constant:
\[
1.097373 \times 10^7 \times 0.22222 = 2438606.6666666665 \; \text{m}^{-1}
\]
4. Find the wavelength \(\lambda\)[/tex] (in meters):
[tex]\[
\lambda = \frac{1}{2438606.6666666665} \approx 4.1007023136162464 \times 10^{-7} \; \text{meters}
\][/tex]
5. Convert the wavelength from meters to nanometers:
- Since [tex]\(1 \; \text{meter} = 10^9 \; \text{nanometers}\)[/tex]:
[tex]\[
\lambda_{\text{nm}} = 4.1007023136162464 \times 10^{-7} \times 10^9 \approx 410.07023136162456 \; \text{nanometers}
\][/tex]
### Conclusion:
The wavelength of the photon emitted when an electron in a hydrogen atom transitions from the [tex]\(n=6\)[/tex] state to the [tex]\(n=2\)[/tex] state is approximately 410.07 nanometers.
