The nucleus and an atom can be assumed to be spherical. The radius of atom is on `Å` if the mass number is 64, the fraction fo the atomic volume that is occupied by the nucleus is:

Options

A. `1.0 xx10^(-3)`
B. `5.0 xx10^(-5)`
C. `2.5 xx10^(-2)`
D. `1.25 xx10^(-7)`

Answer

To solve the problem of finding the fraction of the atomic volume that is occupied by the nucleus, we can follow these steps:

### Step 1: Understand the Given Data
- The mass number (A) is given as 64.
- The radius of the atom (R_a) is given as 1 Å (1 angstrom = \(10^{-8}\) cm).
- The formula for the radius of the nucleus (R_n) is given as:
\[ R_n = 1.25 \times 10^{-13} \times A^{1/3} \]

### Step 2: Calculate the Radius of the Nucleus
Using the mass number A = 64, we can calculate the radius of the nucleus:
\[ R_n = 1.25 \times 10^{-13} \times 64^{1/3} \]
Calculating \(64^{1/3}\):
\[ 64^{1/3} = 4 \]
Now substituting this value back:
\[ R_n = 1.25 \times 10^{-13} \times 4 = 5 \times 10^{-13} \text{ cm} \]

### Step 3: Calculate the Volume of the Nucleus
The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volume of the nucleus (V_n) is:
\[ V_n = \frac{4}{3} \pi R_n^3 = \frac{4}{3} \pi (5 \times 10^{-13})^3 \]

### Step 4: Calculate the Volume of the Atom
Similarly, the volume of the atom (V_a) is:
\[ V_a = \frac{4}{3} \pi R_a^3 = \frac{4}{3} \pi (1 \times 10^{-8})^3 \]

### Step 5: Calculate the Fraction of the Atomic Volume Occupied by the Nucleus
We need to find the fraction of the atomic volume that is occupied by the nucleus:
\[ \text{Fraction} = \frac{V_n}{V_a} = \frac{\frac{4}{3} \pi R_n^3}{\frac{4}{3} \pi R_a^3} = \frac{R_n^3}{R_a^3} \]
Substituting the values:
\[ \text{Fraction} = \frac{(5 \times 10^{-13})^3}{(1 \times 10^{-8})^3} \]
Calculating this gives:
\[ \text{Fraction} = \frac{125 \times 10^{-39}}{1 \times 10^{-24}} = 125 \times 10^{-15} = 1.25 \times 10^{-13} \]

### Final Answer
The fraction of the atomic volume that is occupied by the nucleus is:
\[ 1.25 \times 10^{-13} \]