Radiation coming from transition `n = 2 to n = 1` of hydrogen atoms falls on helium in `n = 1 and n= 2` state what are the possible transition of helium ions as they absorb energy from the radiation?

Explanation

To solve the problem of possible transitions in helium ions when they absorb energy from radiation emitted during the transition of hydrogen atoms from n = 2 to n = 1, we will follow these steps:

### Step 1: Calculate the energy of the radiation from hydrogen transition
The energy of the radiation emitted during the transition from n = 2 to n = 1 in hydrogen can be calculated using the formula:

\[ E = 13.61 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Here, \( n_1 = 1 \) and \( n_2 = 2 \).

Substituting the values:

\[ E = 13.61 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 13.61 \left( 1 - \frac{1}{4} \right) = 13.61 \left( \frac{3}{4} \right) = 10.206 \text{ eV} \approx 10.2 \text{ eV} \]

### Step 2: Calculate the energy levels for helium
For helium, which has an atomic number \( Z = 2 \), the energy levels can be calculated using the formula:

\[ E = Z^2 \cdot 13.61 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

### Step 3: Calculate energy for transitions in helium
1. **Transition from n = 1 to n = 2:**
\[ E_{1 \to 2} = 2^2 \cdot 13.61 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 4 \cdot 13.61 \left( 1 - \frac{1}{4} \right) = 4 \cdot 13.61 \cdot \frac{3}{4} = 40.83 \text{ eV} \]

2. **Transition from n = 1 to n = 3:**
\[ E_{1 \to 3} = 2^2 \cdot 13.61 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = 4 \cdot 13.61 \left( 1 - \frac{1}{9} \right) = 4 \cdot 13.61 \cdot \frac{8}{9} \approx 48.3 \text{ eV} \]

3. **Transition from n = 2 to n = 3:**
\[ E_{2 \to 3} = 2^2 \cdot 13.61 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = 4 \cdot 13.61 \left( \frac{1}{4} - \frac{1}{9} \right) = 4 \cdot 13.61 \cdot \left( \frac{5}{36} \right) \approx 7.56 \text{ eV} \]

4. **Transition from n = 2 to n = 4:**
\[ E_{2 \to 4} = 2^2 \cdot 13.61 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = 4 \cdot 13.61 \left( \frac{1}{4} - \frac{1}{16} \right) = 4 \cdot 13.61 \cdot \left( \frac{3}{16} \right) \approx 10.2 \text{ eV} \]

### Step 4: Determine possible transitions
Now we compare the energies of the transitions in helium with the energy of the radiation from hydrogen (10.2 eV):

- **Transition from n = 1 to n = 2:** 40.83 eV (not possible)
- **Transition from n = 1 to n = 3:** 48.3 eV (not possible)
- **Transition from n = 2 to n = 3:** 7.56 eV (possible)
- **Transition from n = 2 to n = 4:** 10.2 eV (possible)

### Conclusion
The possible transitions of helium ions when they absorb energy from the radiation emitted during the transition of hydrogen atoms from n = 2 to n = 1 are:
- From n = 2 to n = 3
- From n = 2 to n = 4