The nucleus and an atom can be assumed to be spherical. The relation between radius of the nucleus and mass number A is given by `1.25 times 10^(-13)times A^(1/3)`.The radius of atom is one `Å` if the mass number is 64, the fraction fo the atomic volume that is occupied by the nucleus is:

Options

A. `1.0xx10^(-13)`
B. `5.0xx10^(-5)`
C. `2.0xx10^(-2)`
D. `1.25xx10^(-13)`

Explanation

To solve the problem, we need to find the fraction of the atomic volume that is occupied by the nucleus. We will follow these steps:

### Step 1: Understand the relationship between the radius of the nucleus and mass number (A)
The radius of the nucleus (R_n) is given by the formula:
\[ R_n = 1.25 \times 10^{-13} \times A^{1/3} \]

### Step 2: Calculate the radius of the nucleus for A = 64
Substituting A = 64 into the formula:
\[ R_n = 1.25 \times 10^{-13} \times 64^{1/3} \]

First, we need to calculate \( 64^{1/3} \):
\[ 64^{1/3} = 4 \]
Now substituting this value back:
\[ R_n = 1.25 \times 10^{-13} \times 4 = 5.0 \times 10^{-13} \, \text{cm} \]

### Step 3: Calculate the volume of the nucleus
The volume (V_n) of a sphere is given by the formula:
\[ V_n = \frac{4}{3} \pi R_n^3 \]
Substituting \( R_n = 5.0 \times 10^{-13} \, \text{cm} \):
\[ V_n = \frac{4}{3} \pi (5.0 \times 10^{-13})^3 \]
Calculating \( (5.0 \times 10^{-13})^3 \):
\[ (5.0 \times 10^{-13})^3 = 1.25 \times 10^{-38} \, \text{cm}^3 \]
Now substituting this into the volume formula:
\[ V_n = \frac{4}{3} \pi (1.25 \times 10^{-38}) \]
\[ V_n \approx \frac{4}{3} \times 3.14 \times 1.25 \times 10^{-38} \]
\[ V_n \approx 5.24 \times 10^{-38} \, \text{cm}^3 \]

### Step 4: Calculate the volume of the atom
The radius of the atom is given as 1 Å (1 Angstrom = \( 1 \times 10^{-8} \, \text{cm} \)). Therefore, the volume (V_a) of the atom is:
\[ V_a = \frac{4}{3} \pi R_a^3 \]
Substituting \( R_a = 1 \times 10^{-8} \, \text{cm} \):
\[ V_a = \frac{4}{3} \pi (1 \times 10^{-8})^3 \]
Calculating \( (1 \times 10^{-8})^3 \):
\[ (1 \times 10^{-8})^3 = 1 \times 10^{-24} \, \text{cm}^3 \]
Now substituting this into the volume formula:
\[ V_a = \frac{4}{3} \pi (1 \times 10^{-24}) \]
\[ V_a \approx \frac{4}{3} \times 3.14 \times 1 \times 10^{-24} \]
\[ V_a \approx 4.19 \times 10^{-24} \, \text{cm}^3 \]

### Step 5: Calculate the fraction of the atomic volume occupied by the nucleus
The fraction (F) of the atomic volume occupied by the nucleus is given by:
\[ F = \frac{V_n}{V_a} \]
Substituting the volumes we calculated:
\[ F = \frac{5.24 \times 10^{-38}}{4.19 \times 10^{-24}} \]
Calculating this fraction:
\[ F \approx 1.25 \times 10^{-14} \]

### Final Answer
The fraction of the atomic volume that is occupied by the nucleus is approximately \( 1.25 \times 10^{-14} \).

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