The energy required for transition in a Hydrogen atom from `1^(st)` energy level to the `2^(nd)` energy level is E. Then , which of the following transitions is possible for the same energy in Helium atom ?

Options

A. `2 rarr3`
B. `2rarr4`
C. `1rarr2`
D. `1rarr3`

Answer

To solve the problem of finding which transition in a Helium atom corresponds to the energy required for the transition from the first energy level to the second energy level in a Hydrogen atom, we can follow these steps:

### Step 1: Calculate the Energy Transition in Hydrogen
The energy of an electron in the nth energy level of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the first energy level (n=1):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
For the second energy level (n=2):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]
The energy required for the transition from the first to the second level in Hydrogen (ΔE) is:
\[ \Delta E = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \]

### Step 2: Determine the Energy Levels for Helium
For a Helium atom, the atomic number \(Z\) is 2. The energy levels are given by:
\[ E_n = -\frac{13.6 \times Z^2 \, \text{eV}}{n^2} = -\frac{13.6 \times 2^2 \, \text{eV}}{n^2} = -\frac{54.4 \, \text{eV}}{n^2} \]
Calculating the energy levels for Helium:
- For \(n=1\):
\[ E_1 = -\frac{54.4 \, \text{eV}}{1^2} = -54.4 \, \text{eV} \]
- For \(n=2\):
\[ E_2 = -\frac{54.4 \, \text{eV}}{2^2} = -\frac{54.4 \, \text{eV}}{4} = -13.6 \, \text{eV} \]
- For \(n=3\):
\[ E_3 = -\frac{54.4 \, \text{eV}}{3^2} = -\frac{54.4 \, \text{eV}}{9} \approx -6.04 \, \text{eV} \]
- For \(n=4\):
\[ E_4 = -\frac{54.4 \, \text{eV}}{4^2} = -\frac{54.4 \, \text{eV}}{16} \approx -3.4 \, \text{eV} \]

### Step 3: Calculate Possible Transitions in Helium
We need to find a transition in Helium that corresponds to the energy \(10.2 \, \text{eV}\) calculated earlier.

1. **Transition from \(n=1\) to \(n=3\)**:
\[ \Delta E = E_3 - E_1 = -6.04 \, \text{eV} - (-54.4 \, \text{eV}) = 48.36 \, \text{eV} \quad (\text{not valid}) \]

2. **Transition from \(n=1\) to \(n=4\)**:
\[ \Delta E = E_4 - E_1 = -3.4 \, \text{eV} - (-54.4 \, \text{eV}) = 51 \, \text{eV} \quad (\text{not valid}) \]

3. **Transition from \(n=2\) to \(n=3\)**:
\[ \Delta E = E_3 - E_2 = -6.04 \, \text{eV} - (-13.6 \, \text{eV}) = 7.56 \, \text{eV} \quad (\text{not valid}) \]

4. **Transition from \(n=2\) to \(n=4\)**:
\[ \Delta E = E_4 - E_2 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \quad (\text{valid}) \]

### Conclusion
The transition in Helium that corresponds to the same energy \(E\) required for the transition from the first energy level to the second energy level in Hydrogen is from \(n=2\) to \(n=4\).