For balmer series in the spectrum of atomic hydrogen the wave number of each line is given by
`bar(V)=R_(H)(1)/(n_(1)^(2))-(1)/(n_(2)^(2))` where `R_(H)` is a constant and `n_(1)` and `n_(2)` are integers which of the following statement (S) is / are correct
1 as wavelength decreases the lines in the series converge
2 The integer `n_(1)` is equal to 2
3 The ionization energy of hydrogen can be calculated from the wave number of these lines
4 The line of longest wavelength corresponds to `n_(2)=3`

Options

A. 1,2 and 3
B. 2,3 and 4
C. 1,2 and 4
D. 2 and 4 only

Answer

To solve the question regarding the Balmer series in the spectrum of atomic hydrogen, we will analyze each statement provided in the question step by step.

### Given:
The wave number of each line in the Balmer series is given by:
\[ \bar{V} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R_H \) is a constant, and \( n_1 \) and \( n_2 \) are integers.

### Analyzing Each Statement:

1. **Statement 1**: As wavelength decreases, the lines in the series converge.
- **Explanation**: In the context of wave numbers, as the wavelength decreases, the wave number \( \bar{V} \) increases. For the Balmer series, \( n_1 \) is fixed at 2, while \( n_2 \) can take values 3, 4, 5, etc. As \( n_2 \) increases, the difference \( \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \) becomes smaller, leading to convergence of the lines. Thus, this statement is **True**.

2. **Statement 2**: The integer \( n_1 \) is equal to 2.
- **Explanation**: In the Balmer series, transitions occur from higher energy levels \( n_2 \) (3, 4, 5, ...) to \( n_1 = 2 \). Therefore, this statement is **True**.

3. **Statement 3**: The ionization energy of hydrogen can be calculated from the wave number of these lines.
- **Explanation**: The ionization energy refers to the energy required to remove an electron completely from the atom. The Balmer series only involves transitions where \( n_1 = 2 \) and \( n_2 \) is greater than 2. To calculate ionization energy, we would need to consider transitions from \( n = 1 \) to \( n = \infty \). Therefore, this statement is **False**.

4. **Statement 4**: The line of longest wavelength corresponds to \( n_2 = 3 \).
- **Explanation**: The longest wavelength corresponds to the smallest energy transition, which occurs when \( n_2 \) is at its minimum value. For the Balmer series, the transition from \( n_2 = 3 \) to \( n_1 = 2 \) gives the longest wavelength. Therefore, this statement is **True**.

### Conclusion:
Based on the analysis:
- Statement 1: True
- Statement 2: True
- Statement 3: False
- Statement 4: True

Thus, the correct statements are 1, 2, and 4.

### Final Answer:
The correct option is **1, 2, and 4**.

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