Kinetic energy of electron in nth orbit is given by

Options

A. `(Rhc)/(2n^(2))`
B. `(2Rhc)/(n)`
C. `(Rhc)/(n)`
D. `(Rhc)/(n^(2))`

Answer

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

### Step 1: Write the basic formula for kinetic energy
The kinetic energy (KE) of an electron in the nth orbit is given by the formula:
\[ KE = \frac{2 \pi^2 m e^4}{n^2 h^2} \]
where:
- \( m \) is the mass of the electron,
- \( e \) is the charge of the electron,
- \( n \) is the principal quantum number (orbit number),
- \( h \) is Planck's constant.

### Step 2: Introduce the Rydberg constant
The Rydberg constant \( R \) is defined as:
\[ R = \frac{2 \pi^2 m e^4}{c h^3} \]
where \( c \) is the speed of light.

### Step 3: Rearrange the kinetic energy formula
We want to express the kinetic energy in terms of \( R \), \( h \), and \( c \). To do this, we need to manipulate the original kinetic energy formula.

### Step 4: Modify the kinetic energy formula
We can multiply and divide the original kinetic energy formula by \( h \) and \( c \):
\[ KE = \frac{2 \pi^2 m e^4}{n^2 h^2} \cdot \frac{h}{h} \cdot \frac{c}{c} \]
This gives us:
\[ KE = \frac{2 \pi^2 m e^4 h c}{n^2 h^3} \]

### Step 5: Substitute for \( R \)
Now, we can substitute \( R \) into the equation:
\[ KE = \frac{R h c}{n^2} \]

### Final Result
Thus, the kinetic energy of an electron in the nth orbit can be expressed as:
\[ KE = \frac{R h c}{n^2} \]

### Conclusion
The correct expression for the kinetic energy of an electron in the nth orbit is \( \frac{R h c}{n^2} \).

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