The total energy of a hydrogen atom in its ground state is `-13.6 eV`. If the potential energy in the first excited state is taken as zero then the total energy in the ground state will be

Options

A. `-3.4 eV`
B. `3.4 eV`
C. `-6.8 eV`
D. `6.8 eV`

Answer

To solve the problem, we need to analyze the total energy of a hydrogen atom in its ground state and how it changes when we consider the potential energy in the first excited state to be zero.

### Step-by-Step Solution:

1. **Understand the Total Energy in Ground State**:
The total energy (E) of a hydrogen atom in its ground state is given as:
\[ E = -13.6 \, \text{eV} \]

2. **Identify the Relationship Between Kinetic and Potential Energy**:
In a hydrogen atom, the total energy is the sum of the kinetic energy (K.E.) and potential energy (P.E.):
\[ E = K.E. + P.E. \]
It is known that the kinetic energy is half the magnitude of the potential energy:
\[ K.E. = -\frac{1}{2} P.E. \]

3. **Express Total Energy in Terms of Potential Energy**:
Since potential energy is negative in bound systems, we can express the total energy as:
\[ E = K.E. + P.E. = -\frac{1}{2} P.E. + P.E. \]
Simplifying this gives:
\[ E = \frac{1}{2} P.E. \]

4. **Relate Total Energy to Potential Energy**:
Rearranging the equation gives:
\[ P.E. = -2E \]
Substituting the total energy of the ground state:
\[ P.E. = -2 \times (-13.6 \, \text{eV}) = 27.2 \, \text{eV} \]

5. **Consider the First Excited State**:
If the potential energy in the first excited state is taken as zero, we need to find the new total energy in the ground state.

6. **Calculate the New Total Energy**:
The total energy in the ground state, when the potential energy in the first excited state is zero, can be calculated as:
\[ E' = -\frac{1}{2} P.E. \]
Since we have found that the potential energy in the ground state is -27.2 eV, we can find the new total energy:
\[ E' = -\frac{1}{2} \times (-27.2 \, \text{eV}) = -13.6 \, \text{eV} \]

7. **Final Answer**:
Thus, the total energy in the ground state, when the potential energy in the first excited state is taken as zero, is:
\[ E' = -3.4 \, \text{eV} \]

### Final Answer:
The total energy in the ground state will be **-3.4 eV**.