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)`
B. `T_n` is independent of n, `r_n propn`
C. `T_(n)prop(1)/(n)` and `r_n`
D. `T_(n)prop(1)/(n)` and `r_npropn^2`

Explanation

To solve the problem step by step, we will analyze the situation using the Bohr model and the given force acting on the electron.

### Step 1: Understand the Force Acting on the Electron
The electron is attracted toward the origin by a force given by:
\[ F = \frac{k}{r} \]
where \( k \) is a constant and \( r \) is the distance of the electron from the origin.

**Hint**: Identify the nature of the force and how it relates to the motion of the electron.

### Step 2: Relate the Force to Centripetal Force
In circular motion, the centripetal force required to keep the electron in orbit is provided by the attractive force. Therefore, we can equate the two:
\[ \frac{k}{r} = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

**Hint**: Recognize that the centripetal force is necessary for circular motion.

### Step 3: Simplify the Equation
By canceling \( r \) from both sides, we have:
\[ k = mv^2 \]
From this, we can express the kinetic energy \( T \) of the electron:
\[ T = \frac{1}{2}mv^2 = \frac{k}{2} \]

**Hint**: Kinetic energy is related to the velocity of the electron, which is derived from the force acting on it.

### Step 4: Analyze the Kinetic Energy
From the previous step, we see that the kinetic energy \( T \) is a constant value \( \frac{k}{2} \), which does not depend on \( n \) (the quantum number).

**Hint**: Consider how kinetic energy relates to the quantum states of the electron.

### Step 5: Determine the Radius of the nth Orbit
According to the Bohr model, the angular momentum \( L \) of the electron in the nth orbit is quantized:
\[ L = n\frac{h}{2\pi} \]
where \( h \) is Planck's constant. The relationship between the radius \( r_n \) and the angular momentum can be expressed as:
\[ r_n \propto n^2 \]

**Hint**: Recall that in the Bohr model, the radius of the orbit increases with the square of the principal quantum number \( n \).

### Step 6: Conclusion
From the analysis:
- The kinetic energy \( T_n \) is independent of \( n \).
- The radius \( r_n \) is directly proportional to \( n^2 \).

Thus, the correct conclusion is that:
- The kinetic energy \( T_n \) is constant and does not depend on \( n \).
- The radius of the nth orbit \( r_n \) is proportional to \( n^2 \).

**Final Answer**: The kinetic energy \( T_n \) is independent of \( n \), while the radius \( r_n \) is proportional to \( n^2 \).