Let the potential energy of the hydrogen atom in the ground state be zero . Then its energy in the excited state will be

Options

A. `10.2 eV`
B. `13.6 eV`
C. `23.8 eV`
D. `27.2 eV`

Explanation

To solve the problem, we need to find the energy of the hydrogen atom in the excited state given that the potential energy of the hydrogen atom in the ground state is set to zero.

### Step-by-Step Solution:

1. **Understand the Energy Levels of Hydrogen Atom:**
The energy levels of a hydrogen atom can be described by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number. For the ground state, \( n = 1 \).

2. **Calculate the Ground State Energy (E1):**
For \( n = 1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
However, we are told to set the potential energy of the ground state to zero. Therefore, we need to adjust the energy levels accordingly.

3. **Adjust the Ground State Energy:**
If we set the potential energy of the ground state to zero, we need to add \( +13.6 \, \text{eV} \) to the ground state energy:
\[ E_1' = E_1 + 13.6 \, \text{eV} = -13.6 \, \text{eV} + 13.6 \, \text{eV} = 0 \, \text{eV} \]

4. **Calculate the Excited State Energy (E2):**
For the first excited state, \( n = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

5. **Adjust the Excited State Energy:**
Now, we also need to adjust the excited state energy by adding \( +13.6 \, \text{eV} \):
\[ E_2' = E_2 + 13.6 \, \text{eV} = -3.4 \, \text{eV} + 13.6 \, \text{eV} = 10.2 \, \text{eV} \]

6. **Final Answer:**
Therefore, the energy of the hydrogen atom in the excited state, when the potential energy of the ground state is set to zero, is:
\[ E_2' = 10.2 \, \text{eV} \]