Question

MP denotes the mass of a proton and Mn that of a neutron. A given nucleus, of binding energy B, contains Z protons and N neutrons. The mass M(N, Z) of the nucleus is given by (c is the velocity of light )

• A. M(N, Z) = NMn + ZMP + Bc²
• B. M(N, Z) = NMn + ZMP – \(\frac{B}{c^2}\)
• C. M(N, Z) = NMn + ZMP +\(\frac{B}{c^2}\)
• D. M(N, Z) = NMn + ZMP – Bc²

Answer 1

The Correct Option is B

To solve this problem, we need to understand the concept of nuclear binding energy and its relation to the mass of a nucleus.

The binding energy \( B \) of a nucleus is the energy required to disassemble the nucleus into its constituent protons and neutrons. It is a measure of the stability of the nucleus. According to Einstein's mass-energy equivalence principle, the binding energy can be expressed in terms of mass as:

E = \frac{B}{c^2}

Here, \( c \) is the speed of light in a vacuum.

The mass of a nucleus \( M(N, Z) \) can be calculated by subtracting the equivalent mass of the binding energy from the total mass of its individual protons and neutrons. Therefore, this is given by:

• \( N \) represents the number of neutrons, and each neutron has a mass \( M_n \).
• \( Z \) represents the number of protons, and each proton has a mass \( M_p \).

Hence, the total mass of protons and neutrons (without considering binding energy) is:

\( Total\_mass = N M_n + Z M_p \)

The actual mass of the nucleus is less than this total mass by the amount corresponding to the binding energy. Therefore, the correct formula for the mass of the nucleus is:

\( M(N, Z) = N M_n + Z M_p - \frac{B}{c^2} \)

Thus, among the given options, the correct answer is:

\( M(N, Z) = N M_n + Z M_p - \frac{B}{c^2} \)

This choice reflects the fact that the binding energy reduces the total mass of the nucleus compared to the isolated nucleons.