If an electron is revolving around the hydrogen nucleus at a distance of 0.1 nm,what should be its speed?

Options

A. `2.188xx10^(6) ms^(-1)`
B. `1.094xx10^(6) ms^(-1)`
C. `4.376xx10^(6) ms^(-1)`
D. `1.60xx10^(6) ms^(-1)`

Explanation

To find the speed of an electron revolving around a hydrogen nucleus at a distance of 0.1 nm, we can follow these steps:

### Step 1: Understand the relationship between radius and principal quantum number (n)
The radius of the orbit of an electron in a hydrogen atom is given by the formula:
\[ r = a_0 n^2 \]
where:
- \( r \) is the radius of the orbit,
- \( a_0 \) is the Bohr radius (approximately \( 0.53 \) Å or \( 0.53 \times 10^{-10} \) m),
- \( n \) is the principal quantum number.

### Step 2: Convert the given radius to meters
The given distance is \( 0.1 \) nm. We need to convert this to meters:
\[ 0.1 \text{ nm} = 0.1 \times 10^{-9} \text{ m} = 10^{-10} \text{ m} \]

### Step 3: Set up the equation to find n
Using the formula for the radius:
\[ 10^{-10} = 0.53 \times 10^{-10} \times n^2 \]
Now, rearranging for \( n^2 \):
\[ n^2 = \frac{10^{-10}}{0.53 \times 10^{-10}} \]
\[ n^2 = \frac{1}{0.53} \approx 1.8868 \]

### Step 4: Calculate n
Taking the square root:
\[ n \approx \sqrt{1.8868} \approx 1.375 \]
Since \( n \) must be a whole number, we round \( n \) to the nearest whole number, which is \( n = 1 \).

### Step 5: Use the speed formula
The speed of the electron in the nth orbit is given by:
\[ v = \frac{2.188 \times 10^6}{n} \text{ m/s} \]

### Step 6: Substitute n into the speed formula
Substituting \( n = 1 \):
\[ v = \frac{2.188 \times 10^6}{1} = 2.188 \times 10^6 \text{ m/s} \]

### Step 7: Conclusion
Thus, the speed of the electron revolving around the hydrogen nucleus at a distance of 0.1 nm is:
\[ v \approx 2.188 \times 10^6 \text{ m/s} \]

### Final Answer
The correct option is \( 2.188 \times 10^6 \text{ m/s} \).

---