if the potential energy of a hydrogen atom in the ground state is assumed to be zero, then total energy of `n=oo` is equal to

Options

A. `13.6 eV`
B. `27.2 eV`
C. zero
D. None of these

Explanation

To solve the problem, we need to determine the total energy of a hydrogen atom when it is in the state where \( n = \infty \), given that the potential energy in the ground state is assumed to be zero.

### Step-by-Step Solution:

1. **Understanding Energy Levels in Hydrogen Atom**:
The total energy \( E_n \) of a hydrogen atom at a principal quantum number \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( 13.6 \, \text{eV} \) is the energy of the ground state (when \( n = 1 \)).

2. **Calculating Energy at \( n = \infty \)**:
To find the energy at \( n = \infty \):
\[ E_{\infty} = -\frac{13.6 \, \text{eV}}{\infty^2} = -\frac{13.6 \, \text{eV}}{\infty} = 0 \, \text{eV} \]
Thus, the energy at \( n = \infty \) is \( 0 \, \text{eV} \).

3. **Considering the Given Condition**:
The problem states that the potential energy of the hydrogen atom in the ground state is assumed to be zero. In this case, the total energy at the ground state (when \( n = 1 \)) is:
\[ E_1 = -13.6 \, \text{eV} \]
If we assume this potential energy is zero, we need to adjust the total energy accordingly.

4. **Adjusting for the New Reference Point**:
If we assume the potential energy is zero, we need to add \( 13.6 \, \text{eV} \) to the energy at all states to account for this change in reference. Therefore, the energy at \( n = \infty \) becomes:
\[ E_{\infty} = 0 \, \text{eV} + 13.6 \, \text{eV} = 13.6 \, \text{eV} \]

5. **Conclusion**:
Therefore, the total energy of the hydrogen atom when \( n = \infty \) is:
\[ E_{\infty} = 13.6 \, \text{eV} \]

### Final Answer:
The total energy of the hydrogen atom at \( n = \infty \) is \( 13.6 \, \text{eV} \).

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