Calculate the wavelength of spectral line in Lyman series corresponding to `n_2 = 3`.

Explanation

To calculate the wavelength of the spectral line in the Lyman series corresponding to \( n_2 = 3 \), we can follow these steps:

### Step 1: Identify the Transition
In the Lyman series, the electron transitions from a higher energy level \( n_2 \) to the lowest energy level \( n_1 = 1 \). Here, \( n_2 = 3 \).

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength \( \lambda \) of the spectral lines 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 Values
For the Lyman series:
- \( n_1 = 1 \)
- \( n_2 = 3 \)

Substituting these values into the formula:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \]

Calculating the terms inside the parentheses:

\[ \frac{1}{1^2} = 1 \quad \text{and} \quad \frac{1}{3^2} = \frac{1}{9} \]

So,

\[ 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

### Step 4: Calculate \( \frac{1}{\lambda} \)
Now substituting back into the equation:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{8}{9} \]

Calculating this gives:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 0.8888 \approx 9.75 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Calculate \( \lambda \)
Now, take the reciprocal to find \( \lambda \):

\[ \lambda = \frac{1}{9.75 \times 10^6} \approx 1.0256 \times 10^{-7} \, \text{m} \]

### Step 6: Convert to Nanometers
To convert meters to nanometers, we multiply by \( 10^9 \):

\[ \lambda \approx 1.0256 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \approx 102.56 \, \text{nm} \]

### Final Answer
Thus, the wavelength of the spectral line in the Lyman series corresponding to \( n_2 = 3 \) is approximately:

\[ \lambda \approx 102.56 \, \text{nm} \]

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