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Question: A nucleus \({}_Z^AX\) has mass represented by m (A,Z). If \({m_p}\) and\({m_n}\)denote the mass of p...

A nucleus ZAX{}_Z^AX has mass represented by m (A,Z). If mp{m_p} andmn{m_n}denote the mass of proton and neutron respectively and BE the binding energy (in MeV) then.
A) BE=[m(A1Z)−Zmp−(A−Z)mn]C2\left[ {m\left( {{A_1}Z} \right) - Z{m_p} - \left( {A - Z} \right){m_n}} \right]{C^2}
B) BE=[Zmp+(A−Z)mn−m(A,Z)]C2\left[ {Z{m_p} + \left( {A - Z} \right){m_n} - m\left( {A,Z} \right)} \right]{C^2}
C) BE=[Zmp+Amn−m(A,Z)]C2\left[ {Z{m_p} + A{m_n} - m\left( {A,Z} \right)} \right]{C^2}
D) BE=m(A1Z)−Zmp−(A−Z)mnm\left( {{A_1}Z} \right) - Z{m_p} - \left( {A - Z} \right){m_n}

Explanation

Solution

Binding energy is required to separate particles from systems of particles. The binding energy is equal to the total mass of the constituent particles i.e. all the masses of proton and neutron subtracted by the given mass of nucleus. The mass of the nucleus must be less than the sum of the masses of the constituent particles.

Step by step solution:
Step 1:
The question is based on the binding energy. Have a look first on the definition of it.
The binding energy is the amount of energy required to separate a particle from a system of particles or to disperse all the particles of the system. Binding energy is especially applicable to subatomic particles in atomic nuclei, to electrons bound to nuclei in atoms, and to atoms and ions bound together in crystals.
Step 2:
Here we are given:
Number of protons is Z
The mass of the protons is MP{M_P}
Number of neutrons can be found by subtracting mass number A from number of protons i.e. (A−Z)
In the case of formation of nucleus the evolution of energy equals the binding energy of the nucleus takes place due to disappearance of a fraction of total mass. If the quantity of mass disappearing is Δm\Delta m then the binding energy is written as: BE=Δmc2\Delta m{c^2}….. (1)
Now the mass of the constituent particles are (ZMP+(A−Z)Mn)\left( {Z{M_P} + \left( {A - Z} \right){M_n}} \right) where, Mn{M_n} is the mass of neutron.
This implies that the total mass of the total constituent particles is equal to (ZMP+(A−Z)Mn)\left( {Z{M_P} + \left( {A - Z} \right){M_n}} \right)
The mass of the nucleus must be less than the sum of the masses of the constituent neutrons and protons. We can then write it as (ZMP+(A−Z)Mn−m(A,Z)\left( {Z{M_P} + \left( {A - Z} \right){M_n} - m(A,Z} \right)….. (2)
Equation (2) denoted Δm\Delta m
Putting equation (2) in (1) we get the binding energy in (MeV) is[(ZMP+(A−Z)Mn−m(A,Z)c2]\left[ {\left( {Z{M_P} + \left( {A - Z} \right){M_n} - m(A,Z} \right){c^2}} \right]

Hence option B is the correct answer.

Note:
The binding energy is used to determine whether fission or fusion will be a favorable process. The mass defect of a nucleus represents the mass of the energy binding the nucleus, and is the difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed.