The energy different between the first two levels of hydrogen atom is `10.2eV`. What is the corresponding energy difference for a single ionized helium atom?

Options

A. `10.2 eV`
B. `20.4 eV`
C. `40.8 eV`
D. `81.6eV`

Answer

To find the energy difference between the first two energy levels of a single ionized helium atom, we can use the formula for the energy levels of hydrogen-like atoms:

\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]

where:
- \( E_n \) is the energy of the nth level,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number.

### Step 1: Calculate the energy for the first level (n=1) of Helium (Z=2)

Using the formula for \( n = 1 \):

\[ E_1 = -\frac{13.6 \, \text{eV} \cdot (2)^2}{(1)^2} \]
\[ E_1 = -\frac{13.6 \, \text{eV} \cdot 4}{1} \]
\[ E_1 = -54.4 \, \text{eV} \]

### Step 2: Calculate the energy for the second level (n=2) of Helium (Z=2)

Using the formula for \( n = 2 \):

\[ E_2 = -\frac{13.6 \, \text{eV} \cdot (2)^2}{(2)^2} \]
\[ E_2 = -\frac{13.6 \, \text{eV} \cdot 4}{4} \]
\[ E_2 = -13.6 \, \text{eV} \]

### Step 3: Calculate the energy difference between the first two levels

The energy difference \( \Delta E \) between the first two levels is given by:

\[ \Delta E = E_2 - E_1 \]
\[ \Delta E = (-13.6 \, \text{eV}) - (-54.4 \, \text{eV}) \]
\[ \Delta E = -13.6 + 54.4 \]
\[ \Delta E = 40.8 \, \text{eV} \]

### Conclusion

The corresponding energy difference for a single ionized helium atom is \( 40.8 \, \text{eV} \).