According to Bohr's theory of hydrogen atom , for the electron in the nth premissible orbit

Options

A. Linear momentum `prop (1)/(n)`
B. radius of orbit`prop n`
C. Kinetic energy `prop (1)/(n^(2))`
D. angular momentum `prop n`

Explanation

To solve the question regarding Bohr's theory of the hydrogen atom and the properties of the electron in the nth permissible orbit, we will analyze each property step by step.

### Step 1: Linear Momentum
The linear momentum \( p \) of the electron can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity. According to Bohr's theory, the angular momentum \( L \) of the electron in the nth orbit is quantized and given by:
\[ L = mvr = n\frac{h}{2\pi} \]
From the above equation, we can express the velocity \( v \) as:
\[ v = \frac{nh}{2\pi mr} \]
Substituting this back into the expression for linear momentum:
\[ p = mv = m\left(\frac{nh}{2\pi mr}\right) = \frac{nh}{2\pi r} \]
Since the radius \( r \) of the orbit is proportional to \( n^2 \) (i.e., \( r \propto n^2 \)), we can express \( p \) as:
\[ p \propto \frac{n}{n^2} = \frac{1}{n} \]
Thus, the linear momentum is inversely proportional to \( n \).

### Step 2: Radius of Orbit
The radius of the nth orbit is given by:
\[ r \propto n^2 \]
This means that the radius increases with the square of the principal quantum number \( n \).

### Step 3: Kinetic Energy
The kinetic energy \( K \) of the electron in the nth orbit can be derived from the total energy, which is given by:
\[ E = -\frac{Z^2 e^4 m}{2\hbar^2 n^2} \]
For hydrogen, \( Z = 1 \), thus:
\[ E \propto -\frac{1}{n^2} \]
This indicates that the kinetic energy is inversely proportional to \( n^2 \).

### Step 4: Angular Momentum
The angular momentum \( L \) of the electron is given by:
\[ L = mvr = n\frac{h}{2\pi} \]
From this, we can see that angular momentum is directly proportional to \( n \):
\[ L \propto n \]

### Summary of Results
1. **Linear Momentum**: \( p \propto \frac{1}{n} \) (Correct)
2. **Radius of Orbit**: \( r \propto n^2 \) (Incorrect)
3. **Kinetic Energy**: \( K \propto \frac{1}{n^2} \) (Correct)
4. **Angular Momentum**: \( L \propto n \) (Correct)

### Final Conclusion
Thus, the correct options according to Bohr's theory of the hydrogen atom are:
- Linear momentum is inversely proportional to \( n \) (Correct).
- Radius of orbit is proportional to \( n^2 \) (Incorrect).
- Kinetic energy is inversely proportional to \( n^2 \) (Correct).
- Angular momentum is directly proportional to \( n \) (Correct).