The energy of a hydrogen atom in its ground state is `-13.6 eV`. The energy of the level corresponding to the quantum number n=5 is

Options

A. `-0.54 eV`
B. `-5.40 eV`
C. `-0.85 eV`
D. `-2.72 eV`

Explanation

To find the energy of the hydrogen atom at the quantum level corresponding to \( n = 5 \), we can use the formula for the energy levels of a hydrogen atom:

\[ E_n = \frac{E_1}{n^2} \]

where:
- \( E_n \) is the energy at level \( n \),
- \( E_1 \) is the energy of the ground state (which is given as \(-13.6 \, \text{eV}\)),
- \( n \) is the principal quantum number.

### Step-by-step Solution:

1. **Identify the ground state energy**:
The energy of the hydrogen atom in its ground state is given as:
\[ E_1 = -13.6 \, \text{eV} \]

2. **Substitute the value of \( n \)**:
We need to find the energy for \( n = 5 \). Substitute \( n \) into the formula:
\[ E_5 = \frac{E_1}{5^2} \]

3. **Calculate \( 5^2 \)**:
Calculate \( 5^2 \):
\[ 5^2 = 25 \]

4. **Substitute \( E_1 \) and \( 5^2 \) into the formula**:
Now substitute the values into the energy formula:
\[ E_5 = \frac{-13.6 \, \text{eV}}{25} \]

5. **Perform the division**:
Calculate the value:
\[ E_5 = -0.544 \, \text{eV} \]

6. **Round the answer**:
Rounding to two decimal places gives:
\[ E_5 \approx -0.54 \, \text{eV} \]

Thus, the energy of the hydrogen atom at the quantum level corresponding to \( n = 5 \) is approximately \(-0.54 \, \text{eV}\).

### Final Answer:
\[ E_5 \approx -0.54 \, \text{eV} \]