What is likely to be orbit number for a circular orbit of diameter `20 nm` of the hydrogen atom:

Options

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

Explanation

To find the orbit number for a circular orbit of diameter 20 nm of the hydrogen atom, we can follow these steps:

### Step 1: Calculate the Radius
The diameter of the orbit is given as 20 nm. To find the radius, we divide the diameter by 2.

\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{20 \text{ nm}}{2} = 10 \text{ nm} \]

### Step 2: Convert Radius to Angstroms
We need to convert the radius from nanometers to angstroms because the formula we will use requires the radius in angstroms.

1 nm = 10 angstroms, so:

\[ \text{Radius in angstroms} = 10 \text{ nm} \times 10 = 100 \text{ angstroms} \]

### Step 3: Use Bohr's Model Formula
According to Bohr's model, the radius of the nth orbit of a hydrogen atom can be calculated using the formula:

\[ R_n = \frac{0.529 \, N^2}{Z} \text{ angstroms} \]

Where:
- \( R_n \) is the radius of the orbit,
- \( N \) is the orbit number,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 4: Rearranging the Formula
We can rearrange the formula to solve for \( N \):

\[ N^2 = \frac{R_n \cdot Z}{0.529} \]

### Step 5: Substitute the Values
Substituting the values we have:

\[ N^2 = \frac{100 \, \text{angstroms} \cdot 1}{0.529} \]

Calculating this gives:

\[ N^2 = \frac{100}{0.529} \approx 189.03 \]

### Step 6: Calculate N
Now, we take the square root to find \( N \):

\[ N = \sqrt{189.03} \approx 13.7 \]

### Step 7: Round to the Nearest Whole Number
Since the orbit number must be a whole number, we round 13.7 to 14.

### Conclusion
The likely orbit number for a circular orbit of diameter 20 nm of the hydrogen atom is approximately **14**.

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