The approximate quantum number fo a circular orbit of diamere , ` 20, nm` of the hydrogen atom according to Bohr`s theory is :

Options

A. ` 10`
B. ` 14`
C. `12`
D. `16`

Explanation

To find the approximate quantum number for a circular orbit of diameter 20 nm in a hydrogen atom according to Bohr's theory, we can follow these steps:

### Step 1: Convert the Diameter to Radius
The diameter of the orbit is given as 20 nm. To find the radius (r), we divide the diameter by 2.

\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{20 \, \text{nm}}{2} = 10 \, \text{nm} \]

### Step 2: Convert Nanometers to Meters
We need to convert the radius from nanometers to meters for calculations.

\[ 10 \, \text{nm} = 10 \times 10^{-9} \, \text{m} = 1.0 \times 10^{-8} \, \text{m} \]

### Step 3: Use the Bohr Model Formula
According to Bohr's theory, the radius of the nth orbit (r_n) is given by the formula:

\[ r_n = \frac{n^2}{Z} \cdot r_1 \]

Where:
- \( r_1 \) is the radius of the first orbit (0.529 Å or \( 0.529 \times 10^{-10} \, \text{m} \)),
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number.

### Step 4: Substitute Known Values
We know \( r_n \) and \( r_1 \), and we can substitute these values into the formula.

First, convert \( r_1 \) to meters:

\[ r_1 = 0.529 \, \text{Å} = 0.529 \times 10^{-10} \, \text{m} \]

Now substitute into the formula:

\[ 1.0 \times 10^{-8} = \frac{n^2}{1} \cdot (0.529 \times 10^{-10}) \]

### Step 5: Solve for \( n^2 \)
Rearranging the equation to solve for \( n^2 \):

\[ n^2 = \frac{1.0 \times 10^{-8}}{0.529 \times 10^{-10}} \]

Calculating the right-hand side:

\[ n^2 = \frac{1.0 \times 10^{-8}}{0.529 \times 10^{-10}} \approx 189.4 \]

### Step 6: Calculate \( n \)
Taking the square root to find \( n \):

\[ n \approx \sqrt{189.4} \approx 13.8 \]

Rounding to the nearest whole number, we get:

\[ n \approx 14 \]

### Final Answer
The approximate quantum number for a circular orbit of diameter 20 nm of the hydrogen atom according to Bohr's theory is **14**.

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