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The image contains a physics problem involving a hydrogen atom and asks for the wavelength of radiation emitted when an electron transitions from the \( n = 5 \) energy level to the \( n = 4 \) energy level. Additionally, it asks in which region of the electromagnetic spectrum this radiation lies.

To solve this, we use the **Rydberg formula** for hydrogen's energy transitions:

\[
\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)
\]

Where:
- \( R_H \) is the **Rydberg constant**, \( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \)
- \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels, respectively
- \( \lambda \) is the wavelength of the emitted radiation

For this problem:
- \( n_2 = 5 \) (higher energy level)
- \( n_1 = 4 \) (lower energy level)

### Step 1: Calculate the inverse wavelength

\[
\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \left( \frac{1}{4^2} - \frac{1}{5^2} \right)
\]

\[
\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \left( \frac{1}{16} - \frac{1}{25} \right)
\]

\[
\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \left( \frac{25 - 16}{400} \right)
\]

\[
\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \times \frac{9}{400}
\]

\[
\frac{1}{\lambda} = 2.46825 \times 10^5 \, \text{m}^{-1}
\]

### Step 2: Calculate the wavelength

\[
\lambda = \frac{1}{2.46825 \times 10^5} \, \text{m}
\]

\[
\lambda = 4.05 \times 10^{-6} \, \text{m} = 4050 \, \text{nm}
\]

### Step 3: Determine the region of the electromagnetic spectrum

A wavelength of 4050 nm lies in the **infrared** region of the electromagnetic spectrum.

### Final Answer:
- **Wavelength**: \( 4050 \, \text{nm} \)
- **Region**: **Infrared**

---

Do you want more details or have any other questions? Here are 5 related questions to expand the discussion:

1. How would the wavelength change if the transition was from \( n = 6 \) to \( n = 4 \)?
2. What is the energy of the photon emitted in this transition?
3. How does the Rydberg formula apply to elements other than hydrogen?
4. What are the different series of spectral lines in the hydrogen atom (Lyman, Balmer, etc.)?
5. Can this radiation be detected using common infrared sensors?

**Tip**: When analyzing spectral transitions, remember that longer wavelengths correspond to lower energy transitions!

What would be the wavelength of radiation emitted from a hydrogen atom when an electron moves from the n = 5 to n = 4 energy level? In what region of the spectrum does this radiation lie?

Calculate the wavelength of radiation emitted when an electron in a hydrogen atom transitions from n = 5 to n = 4. Using the Rydberg formula, determine that the wavelength is 4050 nm and identify this as infrared radiation. This problem involves quantum mechanics and the electromagnetic spectrum.

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