Total energy of electron in nth stationary orbit of hydrogen atom is

Options

A. `(e^2)/(4piepsilon_0r)`
B. `(-e^2)/(4piepsilon_0r)`
C. `(-e^2)/(8piepsilon_0r)`
D. `(e^2)/(8piepsilon_0r)`

Explanation

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

### Step 1: Understand the Forces Acting on the Electron
In a hydrogen atom, the electron revolves around the proton due to the electrostatic force of attraction between them. This force provides the necessary centripetal force for the electron's circular motion.

### Step 2: Write the Expression for Centripetal Force
The centripetal force required for the electron to move in a circular orbit is given by:
\[ F_{\text{centripetal}} = \frac{mv^2}{r_n} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r_n \) is the radius of the nth orbit.

### Step 3: Write the Expression for Electrostatic Force
The electrostatic force between the electron and the proton can be expressed using Coulomb’s law:
\[ F_{\text{electrostatic}} = \frac{k \cdot e^2}{r_n^2} \]
where \( k \) is Coulomb's constant and \( e \) is the charge of the electron.

### Step 4: Set the Forces Equal
Since the centripetal force is provided by the electrostatic force, we can set these two expressions equal:
\[ \frac{mv^2}{r_n} = \frac{k \cdot e^2}{r_n^2} \]

### Step 5: Rearrange to Find Kinetic Energy
From the above equation, we can express the kinetic energy (KE) of the electron:
\[ mv^2 = \frac{k \cdot e^2}{r_n} \]
The kinetic energy is given by:
\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \cdot \frac{k \cdot e^2}{r_n} \]

### Step 6: Write the Expression for Potential Energy
The potential energy (PE) of the electron in the nth orbit is given by:
\[ PE = -\frac{k \cdot e^2}{r_n} \]
The negative sign indicates that the force is attractive.

### Step 7: Calculate Total Energy
The total energy (E) of the electron in the nth orbit is the sum of its kinetic and potential energies:
\[ E = KE + PE \]
Substituting the expressions for KE and PE:
\[ E = \frac{1}{2} \cdot \frac{k \cdot e^2}{r_n} - \frac{k \cdot e^2}{r_n} \]
\[ E = \frac{1}{2} \cdot \frac{k \cdot e^2}{r_n} - \frac{2}{2} \cdot \frac{k \cdot e^2}{r_n} \]
\[ E = -\frac{1}{2} \cdot \frac{k \cdot e^2}{r_n} \]

### Step 8: Substitute the Value of k
We can substitute \( k \) with \( \frac{1}{4 \pi \epsilon_0} \):
\[ E = -\frac{1}{2} \cdot \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r_n} \]
This simplifies to:
\[ E = -\frac{e^2}{8 \pi \epsilon_0 r_n} \]

### Final Expression
Thus, the total energy of the electron in the nth stationary orbit of a hydrogen atom is:
\[ E = -\frac{e^2}{8 \pi \epsilon_0 r_n} \]

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