Nuclear Physics

To determine how the mass density of a nucleus varies with the mass number , consider the definition of mass density in terms of mass and volume. The mass density is given by:

Where is the mass of the nucleus and is its volume. For most nuclei, the mass can be approximated as proportional to the mass number , assuming that each nucleon (proton or neutron) has roughly the same mass. Mathematically, this can be expressed as:

The volume of a nucleus is related to its radius , which in turn is proportional to the cube root of the mass number , i.e.,

The volume of a sphere (which is a simplification for the shape of a nucleus) is given by:

Substituting the proportionality for , the volume becomes:

Therefore, the mass density becomes:

This shows that the mass density of a nucleus is independent of the mass number and is constant.

Hence, the correct option is:

When an alpha particle collides head-on with a gold nucleus, the potential energy at the closest approach can be calculated using Coulomb’s law. The total energy at the closest approach is purely electrostatic potential energy because the kinetic energy of the alpha particle becomes zero at the point of closest approach.
The potential energy at the closest approach, , is given by the formula: Where: - (Coulomb constant), - (charge of the alpha particle), - (charge of the gold nucleus), - (elementary charge), - (closest approach). Now, substitute the values into the equation: Simplifying: Thus, the kinetic energy of the alpha particle is .