Nuclear Energy

Given: The mass numbers of two nuclei are in the ratio 3:2.

Concept: Nuclear density is given by:

Since the mass of a nucleus is approximately proportional to its mass number , and the radius of a nucleus is given by ,

Volume

So, density

Thus, nuclear density is independent of mass number, meaning:

Ratio of nuclear densities = 1:1

Correct Answer: Option A: 1:1

Step 1: An -particle consists of 2 protons and 2 neutrons, meaning its emission reduces the atomic number by 2 and the mass number by 4. Step 2: A -particle is an electron emitted when a neutron decays into a proton, meaning its emission increases the atomic number by 1 but does not change the mass number.
Step 3: Thus, the net effect of emitting one -particle and one -particle is:
- Mass number decreases by 4 (due to the -particle).
- Atomic number decreases by 2 (from ) but increases by 1 (from ), leading to a net decrease of 1.
Step 4: Since the question asks about the mass number, the correct answer is (B).

Step 1: In quantum mechanics, the total energy of a hydrogen atom is given by the negative of its potential energy divided by two, .
Step 2: When an atom is excited, the electron moves to a higher energy level, which means it is further from the nucleus.
Step 3: This increases the potential energy (less negative) because the electron is less tightly bound to the nucleus.
Step 4: Simultaneously, according to the virial theorem for the hydrogen atom, the kinetic energy is , thus decreasing in magnitude as the potential energy becomes less negative.
Step 5: Therefore, upon excitation, the potential energy increases (becomes less negative) and the kinetic energy decreases.

In the given nuclear reaction: The atomic number decreases by 1, implying the emission of a proton, as a proton carries a positive charge and reduces the atomic number by 1.
Hence, the correct answer is (D).