The second member of Lyman's series is hydrogen spectrum has a wavelength of 5400 Å. Calculate the wavelength of the first member.

Answer

To solve the problem of finding the wavelength of the first member of the Lyman series in the hydrogen spectrum, we can follow these steps:

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

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength of emitted light during electronic transitions is given by:

\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant (\( R \approx 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( n_f \) is the final energy level,
- \( n_i \) is the initial energy level.

### Step 3: Identify the Values
For the first member of the Lyman series:
- \( n_f = 1 \) (final level),
- \( n_i = 2 \) (initial level).

### Step 4: Substitute the Values into the Rydberg Formula
Substituting the values into the Rydberg formula:

\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

Calculating the right-hand side:

\[ \frac{1}{\lambda_1} = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]

### Step 5: Calculate the Wavelength
Now substituting the value of \( R \):

\[ \frac{1}{\lambda_1} = 1.097 \times 10^7 \times \frac{3}{4} \]

Calculating this gives:

\[ \frac{1}{\lambda_1} = \frac{3.291 \times 10^7}{4} = 8.2275 \times 10^6 \, \text{m}^{-1} \]

Now, taking the reciprocal to find \( \lambda_1 \):

\[ \lambda_1 = \frac{1}{8.2275 \times 10^6} \approx 1.215 \times 10^{-7} \, \text{m} \]

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

\[ \lambda_1 \approx 1.215 \times 10^{-7} \, \text{m} = 1215 \, \text{Å} \]

### Final Answer
The wavelength of the first member of the Lyman series is approximately **1215 Å**.

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