Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# ${{\text{M}}_{\text{p}}}$​ denotes the mass of a proton and ​ ${{\text{M}}_{\text{n}}}$that of a neutron. A given nucleus, of binding energy ${\text{B}}$, contains ${\text{Z}}$ protons and ${\text{N}}$ neutrons. The mass ${\text{M}}\left( {{\text{N,Z}}} \right){\text{}}$ of the nucleus is given by (c is the velocity of light)A.${\text{M}}\left( {{\text{N,Z}}} \right){\text{ = N}}{{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}}{\text{ + B}}{{\text{c}}^{\text{2}}}$B.${\text{M}}\left( {{\text{N,Z}}} \right){\text{ = N}}{{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}}{\text{ - }}\dfrac{{\text{B}}}{{{{\text{c}}^{\text{2}}}}}{\text{ }}$C.${\text{M}}\left( {{\text{N,Z}}} \right){\text{ = N}}{{\text{M}}_{\text{n}}}{\text{+ Z}}{{\text{M}}_{\text{p}}}{\text{ + }}\dfrac{{\text{B}}}{{{{\text{c}}^{\text{2}}}}}$D.${\text{M}}\left( {{\text{N,Z}}} \right){\text{ = N}}{{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}}{\text{ - B}}{{\text{c}}^{\text{2}}}$

Last updated date: 06th Sep 2024
Total views: 427.5k
Views today: 8.27k
Verified
427.5k+ views
Hint: To answer this question, recall the concept of Einstein's theory of relativity. We shall find the mass defect in terms of the binding energy of an atom. Substitute both the equations and rearrange the terms in the formula to answer this question.
Formula used:
1)${\text{B = m}}{{\text{c}}^{\text{2}}}$ where ${\text{B}}$ is the binding energy, ${\text{c}}$ is the speed of light and
$\Delta {\text{m}}$ is the mass defect ---(i)
2) ${{\Delta m = [N}}{{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}}{\text{ - M}}\left( {{\text{N,Z}}} \right){\text{]}}$ where ${\text{N}}$ is the no. of neutrons,${\text{Z}}$ is the atomic no. Or no. of protons,
${{\text{M}}_{\text{n}}}$is the mass of a neutron, ${{\text{M}}_{\text{p}}}$ is the mass of a proton and ${\text{M}}$ is the practical mass ---(ii)
3) ${\text{N}} = {\text{A}} - {\text{Z}}$ where ${\text{N}}$ is the no. of neutrons, ${\text{A}}$ is the mass number and, ${\text{Z}}$ is the atomic no.

From the question, we know that we need to calculate the mass defect in terms of the binding energy of an atom.
For this, we need to put the value of equation (ii) in equation (i)
We get,
${\text{B = }}\left[ {{\text{N}}{{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}}{\text{ - M}}\left( {{\text{N,Z}}} \right)} \right]{{\text{c}}^{\text{2}}}$
Now arranging this equation using ${\text{N}} = {\text{A}} - {\text{Z}}$ we now have
$\dfrac{{\text{B}}}{{{{\text{c}}^{\text{2}}}}}{\text{ = }}\left( {{\text{A - Z}}} \right){{\text{M}}_{\text{n}}}{\text{ + Z}}{{\text{M}}_{\text{p}}} - {\text{M}}\left( {{\text{N,Z}}} \right)$
Rearranging this further by grouping the common terms
$\Rightarrow {\text{M}}\left( {{\text{N,Z}}} \right){\text{ = Z}}{{\text{M}}_{\text{p}}}{\text{ + }}\left( {{\text{A - Z}}} \right){{\text{M}}_{\text{n}}}{\text{ - }}\dfrac{{\text{B}}}{{{{\text{c}}^{\text{2}}}}}$
We reach the desired equation
$\Rightarrow {\text{M}}\left( {{\text{N,Z}}} \right){\text{ = Z}}{{\text{M}}_{\text{p}}}{\text{ + N}}{{\text{M}}_{\text{n}}}{\text{ - }}\dfrac{{\text{B}}}{{{{\text{c}}^{\text{2}}}}}$

Therefore, we can conclude that the correct answer to this question is option B.