Potential energy `(PE_(n))` and kinetic energy `(KE_(n))` of electron in nth orbit are related as

Options

A. `PE_(n)=KE_(n)`
B. `PE_(n)=-2KE_(n)`
C. `PE_(n)=2KE_(n)`
D. `PE_(n)=KE_(n)`

Answer

To find the relationship between the potential energy (PE) and kinetic energy (KE) of an electron in the nth orbit, we can follow these steps:

### Step 1: Understand the definitions of KE and PE
- The kinetic energy (KE) of an electron in the nth orbit is given by the formula:
\[ KE_n = \frac{Z e^2}{8 \pi \epsilon_0 r_n} \]
- The potential energy (PE) of the electron in the nth orbit is given by:
\[ PE_n = -\frac{Z e^2}{4 \pi \epsilon_0 r_n} \]

### Step 2: Write the total energy (E) of the electron
- The total energy (E) of the electron in the nth orbit is the sum of its kinetic and potential energy:
\[ E_n = KE_n + PE_n \]
- Substituting the expressions for KE and PE, we get:
\[ E_n = \frac{Z e^2}{8 \pi \epsilon_0 r_n} - \frac{Z e^2}{4 \pi \epsilon_0 r_n} \]

### Step 3: Simplify the total energy expression
- To combine the terms, we need a common denominator:
\[ E_n = \frac{Z e^2}{8 \pi \epsilon_0 r_n} - \frac{2Z e^2}{8 \pi \epsilon_0 r_n} \]
- This simplifies to:
\[ E_n = -\frac{Z e^2}{8 \pi \epsilon_0 r_n} \]

### Step 4: Relate PE and KE
- From the expressions for KE and PE, we can derive the relationship between them:
- We have:
\[ PE_n = -\frac{Z e^2}{4 \pi \epsilon_0 r_n} \]
and
\[ KE_n = \frac{Z e^2}{8 \pi \epsilon_0 r_n} \]
- Now, we can express PE in terms of KE:
\[ PE_n = -2 \times KE_n \]

### Conclusion
Thus, the relationship between the potential energy and kinetic energy of an electron in the nth orbit is:
\[ PE_n = -2 \times KE_n \]