The shortest and longest wave number is H spectrum of Lyman series is (R = Rydherg constant)

Options

A. `(3)/(4)R,R`
B. `(1)/(R ),(4)/(3)R`
C. `R,(4)/(3) R`
D. `R (3)/(4) R`

Explanation

To find the shortest and longest wave numbers in the Lyman series of the hydrogen spectrum, we will follow these steps:

### Step 1: Understand the Lyman Series
The Lyman series corresponds to electronic transitions in hydrogen where the electron falls to the first energy level (n=1) from higher energy levels (n=2, 3, 4, ...). The wave number (μ̅) is inversely proportional to the wavelength (λ) and is given by the formula:

\[ \mu̅ = \frac{1}{\lambda} \]

### Step 2: Determine the Shortest Wave Number
The shortest wave number corresponds to the transition with the least energy difference, which occurs when the electron transitions from n=2 to n=1.

Using the Rydberg formula for wave numbers:

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

For the transition from n=2 to n=1:
- \( n_1 = 1 \)
- \( n_2 = 2 \)
- \( Z = 1 \) (for hydrogen)

Substituting these values into the formula:

\[ \mu̅_{\text{shortest}} = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

Calculating this gives:

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

### Step 3: Determine the Longest Wave Number
The longest wave number corresponds to the transition where the electron falls from an infinite energy level to the first energy level (n=1).

For this transition:
- \( n_1 = 1 \)
- \( n_2 = \infty \)

Using the Rydberg formula again:

\[ \mu̅_{\text{longest}} = RZ^2 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) \]

This simplifies to:

\[ \mu̅_{\text{longest}} = R \cdot 1^2 \left( 1 - 0 \right) = R \]

### Final Answers
- The shortest wave number in the Lyman series is \(\frac{3R}{4}\).
- The longest wave number in the Lyman series is \(R\).