The approximate radius of a H-atom is `0.05` nm, and that of proton is `1.5 xx 10^-15` m. Assuming both hydrogen atom and the proton to be spherical, calculate fraction of the space in an atom of hydrogen that is occupied by the nucleus.

Options

A. `2.3 xx 10^(-14)`
B. `2.3 xx 10^(-7)`
C. `2.3 xx 10^(-8)`
D. `2.3 xx 10^(-18)`

Answer

To solve the problem of calculating the fraction of space in a hydrogen atom occupied by its nucleus (proton), we will follow these steps:

### Step 1: Identify the given values
- Radius of hydrogen atom (r_H) = 0.05 nm = \(0.05 \times 10^{-9}\) m
- Radius of proton (r_p) = \(1.5 \times 10^{-15}\) m

### Step 2: Convert the radius of the hydrogen atom to meters
- \( r_H = 0.05 \, \text{nm} = 0.05 \times 10^{-9} \, \text{m} = 5.0 \times 10^{-11} \, \text{m} \)

### Step 3: Calculate the volume of the hydrogen atom
The volume \( V_H \) of a sphere is given by the formula:
\[ V_H = \frac{4}{3} \pi r_H^3 \]
Substituting the radius of the hydrogen atom:
\[ V_H = \frac{4}{3} \pi (5.0 \times 10^{-11})^3 \]

### Step 4: Calculate the volume of the proton
Similarly, the volume \( V_p \) of the proton is:
\[ V_p = \frac{4}{3} \pi r_p^3 \]
Substituting the radius of the proton:
\[ V_p = \frac{4}{3} \pi (1.5 \times 10^{-15})^3 \]

### Step 5: Calculate the fraction of the volume occupied by the nucleus
The fraction of the volume occupied by the nucleus is given by:
\[ \text{Fraction} = \frac{V_p}{V_H} \]

### Step 6: Substitute the volumes into the fraction formula
Since both volumes have the factor \( \frac{4}{3} \pi \), it cancels out:
\[ \text{Fraction} = \frac{(1.5 \times 10^{-15})^3}{(5.0 \times 10^{-11})^3} \]

### Step 7: Calculate the numerical values
Calculating the volumes:
- \( (1.5 \times 10^{-15})^3 = 3.375 \times 10^{-45} \)
- \( (5.0 \times 10^{-11})^3 = 1.25 \times 10^{-31} \)

Now substituting these values:
\[ \text{Fraction} = \frac{3.375 \times 10^{-45}}{1.25 \times 10^{-31}} = 2.7 \times 10^{-14} \]

### Final Result
Thus, the fraction of the space in a hydrogen atom that is occupied by the nucleus is approximately \( 2.7 \times 10^{-14} \).

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