Question

What is (a) mass defect, and (b) binding energy in Oxygen $_8^{16}O$, whose nuclear mass is 15.995 amu. (${m_p} = 1.0078amu;{\text{ }}{m_n} = 1.0087amu$)

Answer 1

Hint: We are given an oxygen isotope. If formulas of its mass defect and binding energy are known to us then we can calculate them easily as we are given the mass number, atomic number and the masses of proton and neutron.

Formula used:
(1). Mass defect of a nuclei is given as
$\Delta M = \left[ {Z{M_p} + \left( {A - Z} \right){M_n} - M} \right]{\text{ }}...{\text{(i)}}$
where $\Delta M$ is called the mass defect of the given nuclei.
Z is the atomic number while A is the atomic number of the given nuclei
${M_p}$ is used to signify the mass of proton given as ${M_p} = 1.0078amu$

while ${M_n}$ signifies the mass of the neutron which is given as ${M_n} = 1.0087amu$
(2). Binding energy of a nuclei is given as follows:
$B.E. = \Delta M{c^2}{\text{ }}...{\text{(ii)}}$
where $\Delta M$ is the mass defect of the given nuclei.

Complete step by step answer:
It is observed that the mass obtained by adding the total mass due to the nucleons is not equal to the observed mass of that nucleus. The difference in mass of a nucleus is called the mass defect signified by the symbol $\Delta M$. We can calculate it for a nucleus using equation (i).
The mass defect signifies that some force is holding together the nucleons that is reducing the mass of the nucleus leading to the difference in theoretical and observed mass. This energy which is binding the nucleons in the nucleus is called the binding energy of a nucleus. It is given as the energy equivalent of mass defect as given in equation (ii).
We need two calculate the mass defect and binding energy for oxygen $_8^{16}O$. We have the following information with us.
$M = 16 \\\ Z = 8 \\\ A = 16 \\\ {m_p} = 1.0078amu \\\ {m_n} = 1.0087amu \\\$
Mass defect:
$\Delta M = \left[ {Z{M_p} + \left( {A - Z} \right){M_n} - M} \right]$
Substituting the known values, we get
$\Delta M = \left[ {8 \times 1.0078 + \left( {16 - 8} \right) \times 1.0087 - 16} \right] \\\ = 0.132amu \\\$
Binding energy:
$B.E. = \Delta M{c^2}$
Substituting the known values, we get
$B.E. = 0.132 \times 931.5MeV \\\ = 122.958MeV \\\$
These are the required answers.

Note: The nucleus of an atom contains protons and neutrons which are collectively called the nucleons. The mass number is defined as the total number of nucleons in a nucleus and the atomic number is defined as the total number of protons in a nucleus.