If `R` is the Rydberg's constant for hydrogen the wave number of the first line in the Lyman series will be

Options

A. `( R)/(2)`
B. `2 R`
C. `( R)/(4)`
D. `(3 R)/(4)`

Explanation

To find the wave number of the first line in the Lyman series, we can use the Rydberg formula for hydrogen. The wave number (ν̅) is given by the formula:

\[ \nu̅ = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where:
- \( R \) is the Rydberg constant,
- \( n_1 \) is the principal quantum number of the lower energy level,
- \( n_2 \) is the principal quantum number of the higher energy level.

### Step 1: Identify the values of \( n_1 \) and \( n_2 \) for the Lyman series.
In the Lyman series, the transitions occur to the \( n_1 = 1 \) level. The first line in the Lyman series corresponds to the transition from \( n_2 = 2 \) to \( n_1 = 1 \).

### Step 2: Substitute the values into the Rydberg formula.
Now, we can substitute \( n_1 = 1 \) and \( n_2 = 2 \) into the formula:

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

### Step 3: Calculate the individual terms.
Calculating the individual terms gives:

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

### Step 4: Simplify the expression.
Now simplify the expression:

\[ \nu̅ = R \left( 1 - 0.25 \right) = R \left( \frac{3}{4} \right) \]

### Step 5: Write the final answer.
Thus, the wave number of the first line in the Lyman series is:

\[ \nu̅ = \frac{3R}{4} \]

### Summary of the Solution:
The wave number of the first line in the Lyman series is \( \frac{3R}{4} \).

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