B

Step 1: Calculate the volume of the nucleus. The radius of the nucleus is given by the formula:
R = 1.3 A^{1/3} \times 10^{-15} \text{ m}
The volume (V) of the nucleus can be expressed using the formula for the volume of a sphere:
V = \frac{4}{3} \pi R^3
Substituting for R:
V = \frac{4}{3} \pi (1.3 A^{1/3} \times 10^{-15})^3
Step 2: Simplifying the volume computation:
V = \frac{4}{3} \pi (1.3^3 A) (10^{-15})^3
Now calculate 1.3^3 = 2.197:
V = \frac{4}{3} \pi \times 2.197 A \times 10^{-45}
Step 3: Now, compute the mass of the nucleus:
Since each nucleon has a mass of 1.67 \times 10^{-27} \text{ kg}, the total mass of the nucleus (M) is:
M = A \times 1.67 \times 10^{-27} kg
Step 4: Now, the density () of the nucleus can be defined as mass per unit volume:
\rho = \frac{M}{V}
Substituting the expressions for M and V:
\rho = \frac{A \times 1.67 \times 10^{-27}}{\frac{4}{3} \pi \times 2.197 A \times 10^{-45}}
This simplifies down to:
\rho = \frac{1.67 \times 10^{-27}}{\frac{4}{3} \pi \times 2.197 \times 10^{-45}}
Step 5: Calculate this value:
\rho \approx \frac{1.67 \times 10^{-27}}{8.1 \times 10^{-45}} \approx 2.06 \times 10^{17} \text{ kg/m}^3
Thus, the density of the nucleus is approximately 2.06 \times 10^{17} \text{ kg/m}^3.
Step 6: To find the number of nucleons per unit volume, we take the reciprocal of the volume per nucleon:
The volume per nucleon is:
V/A = \frac{4}{3} \pi R^3 / A = K A^{-2/3} where K = \frac{4}{3} \pi (1.3^3 \times 10^{-45})
This gives the number of nucleons per unit volume as approximately 3.33 \times 10^{38} \text{ nucleons/m}^3.
Conclusion: The density of the nucleus can be evaluated and corresponds closely to a density of \approx 2 \text{ g/cm}^3, and the number of nucleons per unit volume is correctly calculated as indicated.