The kinetic energy of the electron in the nth orbit of an atom is given by the relation

Options

A. `-(4pi^(2) me^(4))/(n^(2)h^(2))`
B. `-(2pi^(2) me^(4))/(n^(2)h^(2))`
C. `-(2pi^(2) me^(4))/(n^(2)h^(2))`
D. None of these

Explanation

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

### Step 1: Understand the relationship between kinetic energy (KE) and potential energy (PE)
The kinetic energy of an electron in an orbit is related to its potential energy by the equation:
\[ KE = -2 \times PE \]

### Step 2: Express the potential energy in terms of kinetic energy
From the relationship established in Step 1, we can express potential energy as:
\[ PE = -\frac{KE}{2} \]

### Step 3: Substitute the expression for potential energy
If we know the expression for kinetic energy, we can substitute it into the potential energy equation. The given expression for kinetic energy is:
\[ KE = -\frac{2\pi^2 m e^4}{n^2 h^2} \]
Thus, substituting this into the potential energy equation:
\[ PE = -\frac{1}{2} \left(-\frac{2\pi^2 m e^4}{n^2 h^2}\right) = \frac{\pi^2 m e^4}{n^2 h^2} \]

### Step 4: Calculate the total energy (TE)
The total energy of the electron in the nth orbit is the sum of kinetic and potential energy:
\[ TE = KE + PE \]
Substituting the expressions we derived:
\[ TE = -\frac{2\pi^2 m e^4}{n^2 h^2} + \frac{\pi^2 m e^4}{n^2 h^2} \]
\[ TE = -\frac{2\pi^2 m e^4}{n^2 h^2} + \frac{1\pi^2 m e^4}{n^2 h^2} = -\frac{\pi^2 m e^4}{n^2 h^2} \]

### Step 5: Finalize the kinetic energy expression
Thus, the kinetic energy of the electron in the nth orbit is given by:
\[ KE = -\frac{2\pi^2 m e^4}{n^2 h^2} \]

### Summary of Results
- Kinetic Energy (KE) of the electron in the nth orbit:
\[ KE = -\frac{2\pi^2 m e^4}{n^2 h^2} \]
- Potential Energy (PE) of the electron in the nth orbit:
\[ PE = \frac{\pi^2 m e^4}{n^2 h^2} \]
- Total Energy (TE) of the electron in the nth orbit:
\[ TE = -\frac{\pi^2 m e^4}{n^2 h^2} \]