To solve the problem regarding the Balmer series for the hydrogen atom, we will analyze each statement provided in the question step by step.
### Step 1: Analyze the first statement
**Statement 1:** `n_1 = 2`
The Balmer series corresponds to transitions where the electron falls to the second energy level (n=2) from higher energy levels (n=3, 4, 5, ...). Thus, for the Balmer series, the value of `n_1` is indeed 2.
**Conclusion:** This statement is **True**.
### Step 2: Analyze the second statement
**Statement 2:** Ionization energy of H atom can be calculated from the above formula.
The formula given is for calculating the wave number (ν̄) of the emitted light during electronic transitions in the hydrogen atom. While we can calculate the energy of the emitted photon, the ionization energy specifically refers to the energy required to remove an electron completely from the atom, which cannot be directly calculated using this formula.
**Conclusion:** This statement is **False**.
### Step 3: Analyze the third statement
**Statement 3:** `λ_max` is for `n_2 = 3`.
To find the longest wavelength (λ_max) in the Balmer series, we need to consider the transition from n=3 to n=2. The longest wavelength corresponds to the smallest energy difference, which occurs when the electron transitions from the closest higher level (n=3) to n=2.
Using the formula:
\[
\bar{\nu} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)
\]
For `n_1 = 2` and `n_2 = 3`, we can calculate:
\[
\bar{\nu} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right)
\]
This indeed gives us the wave number for the transition from n=3 to n=2, confirming that `λ_max` corresponds to `n_2 = 3`.
**Conclusion:** This statement is **True**.
### Step 4: Analyze the fourth statement
**Statement 4:** If `λ` decreases, then spectrum lines will converge.
As the wavelength (λ) decreases, it implies that the energy of the emitted photons is increasing, which corresponds to transitions from higher energy levels to lower energy levels. As we consider transitions from higher levels (n=3, 4, 5, ...) to n=2, the energy differences become smaller as n increases, leading to the lines converging in the spectrum.
**Conclusion:** This statement is **True**.
### Final Summary of Statements:
1. **True**: `n_1 = 2`
2. **False**: Ionization energy cannot be calculated from this formula.
3. **True**: `λ_max` is for `n_2 = 3`.
4. **True**: If `λ` decreases, spectrum lines will converge.
### Correct Options:
The correct options regarding the statements are **1, 3, and 4**.
---
