The wavelength of Lymen series for first number is

Options

A. ` ( 4 xx 1.097 xx 10 ^( 7 )) /( 3 ) ` m
B. ` ( 3) /( 4 xx 1.097 xx 10^7 ) `m
C. ` ( 4 ) /(3xx 1.097 xx 10^ 7 ) ` m
D. ` ( 3 ) / ( 4 ) xx 1.097 xx 10 ^ 7 ` m

Answer

To find the wavelength of the first line of the Lyman series in the hydrogen spectrum, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Series and Transition Levels**:
- The Lyman series corresponds to transitions where the electron falls to the first energy level (n1 = 1).
- For the first line of the Lyman series, the electron transitions from n2 = 2 to n1 = 1.

2. **Use the Rydberg Formula**:
- The wavelength (λ) can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
- Here, \( R \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \).

3. **Substitute the Values**:
- Substitute \( n_1 = 1 \) and \( n_2 = 2 \) into the formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]
- This simplifies to:
\[ \frac{1}{\lambda} = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]

4. **Calculate the Wavelength**:
- Now substituting the value of \( R \):
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{3}{4} \]
- Calculate \( \frac{3}{4} \):
\[ \frac{3}{4} = 0.75 \]
- Therefore:
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 0.75 = 0.823 \times 10^7 \]
- To find \( \lambda \), take the reciprocal:
\[ \lambda = \frac{1}{0.823 \times 10^7} \approx 1.215 \times 10^{-8} \, \text{m} = 121.5 \, \text{nm} \]

5. **Final Result**:
- The wavelength of the first line of the Lyman series is approximately \( 121.5 \, \text{nm} \).