In a hydrogen atom, an electron undergoes transition from an energy level with n= 4 to an energy level with n= 2. Calculate the wavelength of the spectral line thus obtained. In which region of electromagnetic spectrum this line would be observed and what would be its colour?

Answer

To solve the problem of calculating the wavelength of the spectral line obtained when an electron in a hydrogen atom transitions from an energy level of n=4 to n=2, we can follow these steps:

### Step-by-Step Solution:

**Step 1: Identify the Energy Levels**
- The initial energy level (n2) is 4, and the final energy level (n1) is 2.

**Step 2: Use Rydberg's Formula**
- The Rydberg formula for the wavelength of the emitted light during a transition is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \).

**Step 3: Substitute the Values into the Formula**
- Substitute \( n1 = 2 \) and \( n2 = 4 \) into the formula:
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) \]
- Calculate \( \frac{1}{2^2} = \frac{1}{4} \) and \( \frac{1}{4^2} = \frac{1}{16} \):
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{4} - \frac{1}{16} \right) \]

**Step 4: Simplify the Expression**
- Find a common denominator to simplify:
\[ \frac{1}{4} - \frac{1}{16} = \frac{4}{16} - \frac{1}{16} = \frac{3}{16} \]
- Substitute back into the equation:
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{3}{16} \]

**Step 5: Calculate \( \lambda \)**
- Now calculate \( \frac{1}{\lambda} \):
\[ \frac{1}{\lambda} = \frac{1.097 \times 10^7 \times 3}{16} \]
- Calculate the right side:
\[ \frac{1.097 \times 3}{16} \approx 0.2053125 \times 10^7 \, \text{m}^{-1} \]
- Therefore, \( \lambda \) is:
\[ \lambda = \frac{1}{0.2053125 \times 10^7} \approx 4.86 \times 10^{-7} \, \text{m} \]

**Step 6: Convert to Nanometers**
- Convert meters to nanometers:
\[ \lambda \approx 4.86 \times 10^{-7} \, \text{m} = 486 \, \text{nm} \]

**Step 7: Determine the Region of the Electromagnetic Spectrum**
- The wavelength of 486 nm falls within the visible region of the electromagnetic spectrum, which ranges from approximately 400 nm to 700 nm.

**Step 8: Identify the Color**
- The color corresponding to a wavelength of 486 nm is blue, as blue light typically ranges from about 450 nm to 495 nm.

### Final Answer:
- The wavelength of the spectral line is **486 nm**.
- It is observed in the **visible region** of the electromagnetic spectrum.
- The color of this spectral line is **blue**.