Suppose an electron is attracted toward the origin by a force`(k)/(r )` where `k` is a constant and `r` is the distance of the electron from the origin .By applying Bohr model to this system the radius of the `n^(th)` orbital of the electron is found to be `r_(n)` and the kinetic energy of the electron to be `T_(n)` , Then which of the following is true ?

Options

A. `T_(n) prop 1//n^(2), r_(n) prop n^(2)`
B. `T_(n) " independent of " n, r_(n) prop n`
C. `T_(n) prop 1/n, r_(n) prop n`
D. `T_(n) prop 1//n, r_(n) prop n^(2)`

Answer

To solve the problem, we will apply the principles of the Bohr model to determine the relationships between the radius of the nth orbital \( r_n \) and the kinetic energy \( T_n \) of the electron.

### Step-by-Step Solution:

1. **Understanding the Force**:
The electron is attracted towards the origin by a force given by:
\[ F = \frac{k}{r} \]
where \( k \) is a constant and \( r \) is the distance from the origin.

2. **Centripetal Force**:
For an electron moving in a circular orbit, the centripetal force required to keep the electron in that orbit is provided by the attractive force. Thus, we can write:
\[ \frac{mv^2}{r} = \frac{k}{r} \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

3. **Solving for Velocity**:
Rearranging the equation gives:
\[ mv^2 = k \quad \Rightarrow \quad v^2 = \frac{k}{m} \]

4. **Kinetic Energy**:
The kinetic energy \( T_n \) of the electron is given by:
\[ T_n = \frac{1}{2} mv^2 \]
Substituting \( v^2 \) from the previous step:
\[ T_n = \frac{1}{2} m \left(\frac{k}{m}\right) = \frac{k}{2} \]
This shows that \( T_n \) is a constant and does not depend on \( n \).

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

6. **Finding the Radius**:
From the angular momentum equation, we can express \( r \):
\[ r_n = \frac{n h}{2\pi mv} \]
Substituting \( v \) from our earlier expression:
\[ r_n = \frac{n h}{2\pi m \sqrt{\frac{k}{m}}} = \frac{n h}{2\pi} \cdot \frac{1}{\sqrt{k/m}} = \frac{n h \sqrt{m}}{2\pi \sqrt{k}} \]
This shows that \( r_n \) is directly proportional to \( n \).

### Conclusion:
From the analysis, we have:
- The kinetic energy \( T_n \) is independent of \( n \) (constant).
- The radius \( r_n \) is directly proportional to \( n \).

Thus, the correct statement is:
- \( T_n \) is independent of \( n \) and \( r_n \) is directly proportional to \( n \).