Angular velocity of an electron in nth Bohr orbit is proportional to `(1)/(n^((p)/(2)))`.Find the value of p

Explanation

To find the value of \( p \) in the expression for the angular velocity of an electron in the nth Bohr orbit, we start with the relationship given in the question:

\[ \omega_n \propto \frac{1}{n^{\frac{p}{2}}} \]

### Step 1: Understand the angular velocity in Bohr's model
In Bohr's model of the atom, the angular velocity \( \omega \) of an electron is related to its linear velocity \( v \) and the radius \( r \) of the orbit. The relationship can be expressed as:

\[ \omega = \frac{v}{r} \]

### Step 2: Determine the linear velocity \( v_n \)
The linear velocity \( v_n \) of an electron in the nth orbit is given by:

\[ v_n \propto \frac{Z}{n} \]

where \( Z \) is the atomic number.

### Step 3: Determine the radius \( r_n \)
The radius \( r_n \) of the nth orbit is given by:

\[ r_n \propto \frac{n^2}{Z} \]

### Step 4: Substitute \( v_n \) and \( r_n \) into the angular velocity formula
Now, substituting the expressions for \( v_n \) and \( r_n \) into the formula for \( \omega_n \):

\[ \omega_n = \frac{v_n}{r_n} = \frac{\frac{Z}{n}}{\frac{n^2}{Z}} = \frac{Z^2}{n^3} \]

### Step 5: Express \( \omega_n \) in terms of \( n \)
From the above expression, we can see that:

\[ \omega_n \propto \frac{Z^2}{n^3} \]

### Step 6: Relate this to the given proportionality
We can rewrite this as:

\[ \omega_n \propto \frac{1}{n^{3}} \]

### Step 7: Compare with the original expression
Now, comparing this with the original expression \( \omega_n \propto \frac{1}{n^{\frac{p}{2}}} \):

\[ \frac{p}{2} = 3 \]

### Step 8: Solve for \( p \)
Multiplying both sides by 2 gives:

\[ p = 6 \]

### Final Answer
Thus, the value of \( p \) is:

\[ \boxed{6} \]