The wavelength of the first spectral line of the Lyman series of hydrogen spectrum is

Options

A. 912 Å
B. 1215 Å
C. 1512 Å
D. 6563 Å

Explanation

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

### Step 1: Understand the Lyman Series
The Lyman series corresponds to electronic transitions in hydrogen where the electron falls to the n=1 energy level from higher energy levels (n=2, 3, 4,...). The first spectral line corresponds to the transition from n=2 to n=1.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength 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 (\( R = 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( n_1 \) is the lower energy level (for Lyman series, \( n_1 = 1 \)),
- \( n_2 \) is the higher energy level (for the first line, \( n_2 = 2 \)).

### Step 3: Substitute the Values
Substituting \( n_1 = 1 \) and \( n_2 = 2 \) into the formula:

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

Calculating the terms inside the parentheses:

\[ \frac{1}{1^2} - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} \]

Now substituting this back into the equation:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{3}{4} \]

### Step 4: Calculate \( \frac{1}{\lambda} \)
Calculating \( \frac{1}{\lambda} \):

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 0.75 = 8.2275 \times 10^6 \, \text{m}^{-1} \]

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

\[ \lambda = \frac{1}{8.2275 \times 10^6} \approx 1.215 \times 10^{-7} \, \text{m} \]

### Step 6: Convert to Angstroms
To convert meters to angstroms (1 angstrom = \( 10^{-10} \) m):

\[ \lambda \approx 1.215 \times 10^{-7} \, \text{m} = 1.215 \times 10^{3} \, \text{angstroms} = 121.5 \, \text{angstroms} \]

### Conclusion
The wavelength of the first spectral line of the Lyman series of the hydrogen spectrum is approximately **121.5 angstroms**.

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