The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be 𝐸� � and the binding energy of a neutron be 𝐸� � in the nucleus. Which of the following statement(s) is(are) correct? (A) 𝐸� � − 𝐸� � is proportional to 𝑍(𝑍 − 1) where Z is the atomic number of the nucleus. (B) 𝐸� � − 𝐸� � is proportional to 𝐴� � � where A is the mass number of the nucleus. (C) 𝐸� � − 𝐸� � is positive. (D) 𝐸� � increases if the nucleus undergoes a beta decay emitting a positron.

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be 𝐸� � and the binding energy of a neutron be 𝐸� � in the nucleus. Which of the following statement(s) is(are) correct? (A) 𝐸� � − 𝐸� � is proportional to 𝑍(𝑍 − 1) where Z is the atomic number of the nucleus. (B) 𝐸� � − 𝐸� � is proportional to 𝐴� � � where A is the mass number of the nucleus. (C) 𝐸� � − 𝐸� � is positive. (D) 𝐸� � increases if the nucleus undergoes a beta decay emitting a positron.

on March 21, 2025

We analyze the problem step by step to determine which statements are correct.

Step 1: Understanding binding energy

The binding energy of a nucleon (proton or neutron) in a nucleus is the energy required to remove that nucleon from the nucleus.
Protons experience both the nuclear force (attractive) and the Coulomb repulsion (due to other protons), while neutrons only experience the nuclear force.

Step 2: Coulomb repulsion effect

The Coulomb repulsion between protons increases the binding energy of protons compared to neutrons.
The Coulomb energy for a nucleus with $Z$ protons is proportional to $Z (Z - 1)$ , as each proton repels every other proton.

Step 3: Analyzing the statements

(A) $E_{p} - E_{n}$ is proportional to $Z (Z - 1)$ .

The difference in binding energy between a proton and a neutron, $E_{p} - E_{n}$ , is primarily due to the Coulomb repulsion between protons.
The Coulomb energy is proportional to $Z (Z - 1)$ , so $E_{p} - E_{n}$ is also proportional to $Z (Z - 1)$ .
Statement (A) is correct.

(B) $E_{p} - E_{n}$ is proportional to $A^{2 / 3}$ .

The Coulomb energy depends on the size of the nucleus, which is proportional to $A^{1 / 3}$ . However, the difference $E_{p} - E_{n}$ is not directly proportional to $A^{2 / 3}$ .
Statement (B) is incorrect.

(C) $E_{p} - E_{n}$ is positive.

Protons experience additional Coulomb repulsion, which increases their binding energy compared to neutrons. Thus, $E_{p} - E_{n}$ is positive.
Statement (C) is correct.

(D) $E_{p}$ increases if the nucleus undergoes a beta decay emitting a positron.

In positron emission, a proton is converted into a neutron, reducing the number of protons $Z$ and increasing the number of neutrons $N$ .
This reduces the Coulomb repulsion, leading to a decrease in the binding energy of the remaining protons. Thus, $E_{p}$ decreases, not increases.
Statement (D) is incorrect.

Final Answer:

Statements (A) and (C) are correct.

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