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 find the energy of the hydrogen atom in the excited state when the potential energy in the ground state is considered to be zero, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Ground State Energy**:
The energy of a hydrogen atom in the ground state (n=1) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For n=1 (ground state):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

2. **Set Potential Energy to Zero**:
The problem states that the potential energy of the hydrogen atom in the ground state is zero. This means we need to adjust the total energy to account for this:
\[ \text{Total Energy} = \text{Kinetic Energy} + \text{Potential Energy} \]
If the potential energy is set to zero, the total energy becomes:
\[ \text{Total Energy} = 0 - (-13.6 \, \text{eV}) = 13.6 \, \text{eV} \]
However, we need to consider that the total energy in the ground state is actually:
\[ E_{\text{total}} = 13.6 \, \text{eV} + 13.6 \, \text{eV} = 27.2 \, \text{eV} \]

3. **Calculate Energy in the Excited State**:
Now, we need to find the energy of the first excited state (n=2):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

4. **Adjust for the New Reference Point**:
Since we have set the potential energy to zero, we need to adjust the energy in the excited state:
\[ E_{2 \text{ (new)}} = \text{Total Energy} - E_2 \]
Substituting the values:
\[ E_{2 \text{ (new)}} = 27.2 \, \text{eV} - 3.4 \, \text{eV} = 23.8 \, \text{eV} \]

5. **Final Answer**:
Therefore, the energy of the hydrogen atom in the excited state, when the potential energy in the ground state is set to zero, is:
\[ \boxed{23.8 \, \text{eV}} \]