The energy of an electron in the nth Bohr orbit of hydrogen atom is

Options

A. `-(13.6)/(n^(4))`eV
B. `-(13.6)/(n^(3))`eV
C. `-(13.6)/(n^(2)) ` eV
D. `-(13.6)/(n) ` eV

Explanation

To find the energy of an electron in the nth Bohr orbit of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formula**:
The energy of an electron in the nth Bohr orbit is given by the formula:
\[ E_n = -\frac{2 \pi^2 m e^4 z^2}{n^2 h^2} \]
where:
- \( E_n \) = energy of the electron in the nth orbit
- \( m \) = mass of the electron
- \( e \) = charge of the electron
- \( z \) = atomic number (for hydrogen, \( z = 1 \))
- \( n \) = principal quantum number (orbit number)
- \( h \) = Planck's constant

2. **Substituting Values**:
For hydrogen (\( z = 1 \)):
\[ E_n = -\frac{2 \pi^2 m e^4}{n^2 h^2} \]

3. **Using Known Constants**:
We can substitute the known values for the constants:
- \( m \) (mass of electron) ≈ \( 9.11 \times 10^{-31} \) kg
- \( e \) (charge of electron) ≈ \( 1.6 \times 10^{-19} \) C
- \( h \) (Planck's constant) ≈ \( 6.63 \times 10^{-34} \) Js

4. **Calculating the Energy**:
After substituting the constants and simplifying, we find:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
This means the energy of the electron in the nth orbit is inversely proportional to the square of the principal quantum number \( n \).

5. **Final Result**:
Therefore, the energy of an electron in the nth Bohr orbit of a hydrogen atom is:
\[ E_n = -13.6 \, \text{eV} \cdot \frac{1}{n^2} \]