Total energy of an electron in nth stationary orcit of hydrogen atom can be obtained by

Options

A. `E_n = 13.6/ n^2 eV`
B. `E_n = -13.6/ n^2 eV`
C. `E_n = -13.6/ n^2 eV`
D. `E_n = 13.6 x n^2 eV`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Forces**: The electron in a hydrogen atom moves in a circular orbit around the nucleus (proton). The centripetal force required for this circular motion is provided by the electrostatic force of attraction between the electron and the proton.

2. **Centripetal Force and Electrostatic Force**:
- The centripetal force can be expressed as:
\[ F_c = \frac{mv^2}{r_n} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r_n \) is the radius of the nth orbit.
- The electrostatic force between the electron and the proton is given by Coulomb's law:
\[ F_e = \frac{k \cdot e^2}{r_n^2} \]
where \( k \) is Coulomb's constant, and \( e \) is the charge of the electron.

3. **Setting the Forces Equal**: For the electron to remain in a stable orbit, the centripetal force must equal the electrostatic force:
\[ \frac{mv^2}{r_n} = \frac{k \cdot e^2}{r_n^2} \]
Rearranging gives:
\[ mv^2 = \frac{k \cdot e^2}{r_n} \]

4. **Angular Momentum Quantization**: According to Bohr's model, the angular momentum of the electron is quantized:
\[ mvr_n = n\frac{h}{2\pi} \]
where \( n \) is a positive integer (the principal quantum number) and \( h \) is Planck's constant.

5. **Finding the Radius of the nth Orbit**: By solving the equations for \( r_n \) and substituting the expression for \( v \), we find:
\[ r_n = \frac{n^2 h^2}{4\pi^2 k e^2 m} \]
This can be simplified to:
\[ r_n = 0.529 \frac{n^2}{Z} \text{ Å} \]
for hydrogen, where \( Z = 1 \).

6. **Kinetic Energy (KE)**: The kinetic energy of the electron in the nth orbit can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Using the earlier relationship, we find:
\[ KE = \frac{k \cdot e^2}{2r_n} \]

7. **Potential Energy (PE)**: The potential energy of the electron in the nth orbit is given by:
\[ PE = -\frac{k \cdot e^2}{r_n} \]

8. **Total Energy (E)**: The total energy of the electron is the sum of its kinetic and potential energies:
\[ E_n = KE + PE = \frac{k \cdot e^2}{2r_n} - \frac{k \cdot e^2}{r_n} = -\frac{k \cdot e^2}{2r_n} \]

9. **Substituting for Hydrogen**: For hydrogen (\( Z = 1 \)), we substitute the values to find:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

### Final Result:
Thus, the total energy of an electron in the nth stationary orbit of a hydrogen atom is:
\[ E_n = -\frac{13.6}{n^2} \text{ eV} \]