According to Bohr's theory, the radius of the `n^(th)` orbit of an atom of atomic number `Z` is proportional

Options

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

Explanation

To solve the question regarding the radius of the nth orbit of an atom according to Bohr's theory, we can follow these steps:

### Step 1: Understand the Coulomb Force
According to Bohr's theory, the centripetal force acting on the electron in its orbit is provided by the Coulomb force between the positively charged nucleus and the negatively charged electron. The expression for the Coulomb force is given by:

\[ F = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2} \]

where:
- \( Z \) is the atomic number,
- \( e \) is the charge of the electron,
- \( r \) is the radius of the orbit,
- \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Set Up the Centripetal Force Equation
The centripetal force required to keep the electron in circular motion is given by:

\[ F = \frac{mv^2}{r} \]

where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit.

### Step 3: Equate the Forces
Setting the Coulomb force equal to the centripetal force, we have:

\[ \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2} = \frac{mv^2}{r} \]

### Step 4: Use Bohr's Quantization Condition
Bohr's postulate states that the angular momentum of the electron is quantized and given by:

\[ mvr = n \frac{h}{2\pi} \]

where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 5: Solve for Velocity
From the angular momentum equation, we can express \( v \) as:

\[ v = \frac{n h}{2 \pi m r} \]

### Step 6: Substitute Velocity into the Force Equation
Substituting this expression for \( v \) back into the centripetal force equation gives:

\[ \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2} = \frac{m \left(\frac{n h}{2 \pi m r}\right)^2}{r} \]

### Step 7: Simplify the Equation
After substituting and simplifying, we will find that:

\[ \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2} = \frac{n^2 h^2}{4 \pi^2 m r^3} \]

### Step 8: Rearranging to Find the Radius
Rearranging this equation leads to:

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

This shows that the radius \( r \) of the nth orbit is proportional to \( n^2 \) and inversely proportional to \( Z \).

### Conclusion
Thus, according to Bohr's theory, the radius of the nth orbit of an atom of atomic number \( Z \) is proportional to \( \frac{n^2}{Z} \).