If the ionization energy for the hydrogen atom is 13.6 eV , the energy required to excite it from the ground state to the next higher state is nearly

Options

A. 3.4 eV
B. 10.2 eV
C. 17.2 eV
D. 13.6 eV

Explanation

To find the energy required to excite a hydrogen atom from the ground state to the next higher state, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ionization Energy**:
The ionization energy for the hydrogen atom is given as 13.6 eV. This is the energy required to remove the electron from the ground state (n=1) to infinity (n=∞).

2. **Determine the Energy of the Ground State (E1)**:
The energy of the ground state (E1) can be expressed as:
\[ E_1 = -13.6 \, \text{eV} \]
This is because the ionization energy is the difference between the energy at infinity (0 eV) and the energy at the ground state.

3. **Use the Energy Formula for Hydrogen Atom**:
The energy of an electron in a hydrogen atom at any energy level n is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
For hydrogen, \( Z = 1 \).

4. **Calculate the Energy of the Second State (E2)**:
For the second energy level (n=2):
\[ E_2 = -\frac{13.6 \, \text{eV} \cdot 1^2}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

5. **Find the Energy Difference (ΔE)**:
The energy required to excite the electron from the ground state (E1) to the second state (E2) is given by:
\[ \Delta E = E_2 - E_1 \]
Substituting the values:
\[ \Delta E = (-3.4 \, \text{eV}) - (-13.6 \, \text{eV}) = -3.4 \, \text{eV} + 13.6 \, \text{eV} = 10.2 \, \text{eV} \]

6. **Conclusion**:
The energy required to excite the hydrogen atom from the ground state to the next higher state is approximately **10.2 eV**.