Question

The mass of deuteron $\left( {{}_1{H^2}} \right)$ nucleus is $2.013553$ \[a.m.u.\] if the masses of the proton and neutron are $1.007275$ \[a.m.u.\] and $1.008665$ \[a.m.u.\] respectively. Calculate the mass defect, the packing fraction, binding energy and binding energy per nucleon.

Answer 1

Hint:Mass defect can be defined as the difference between the calculated mass and the actual mass of an atomic nucleus. Binding energy is the energy required for separation of substituents like proton and neutron from the nucleus. It can be positive or negative. We will find out the value of binding energy using Einstein's famous mass energy equivalence relation. We will find out the packing fraction using $M,A$ .

Formula used:
$\Delta m$=$\left[ {Z{m_P} + (A - Z){m_n}} \right] - M$
Where, $\Delta m$= mass defect ,${m_p}$= mass of proton,${m_n}$= mass of neutron,$M$= mass of nucleus,$A$= mass number $(Z + N)$= total number of protons and neutrons and $Z = $ Atomic number = number of protons.
Einstein famous mass energy equivalence relation:
$E = m{c^2}$
To find out nuclear binding energy m is replaced by$\Delta m$and the expression changes to $E = \Delta m{c^2}$
To find out binding energy per nucleon we divide the binding energy by mass number that is ${E_{bn}} = \dfrac{{{E_b}}}{A}$
Packing fraction of the nucleus is the ratio of difference of exact mass and atomic mass number to the Atomic mass number. It is written as $f = \dfrac{{(M - A)}}{A}$
M=exact mass and A= Atomic mass number
Using these famous formulas we can calculate mass defect, binding energy and binding energy per nucleon etc.

Complete step by step answer:
First, to calculate mass defect, we should have the values of $A,Z,N$. In our case, we have deuteron. Mass number of the deuteron is $2$ and the number of protons$ = 1$. Therefore, \[A = 2,{\text{ }}Z = 1,{\text{ }}N = \left( {A - Z} \right) = \left( {2 - 1} \right) = 1\]
\[
M = 2.013553{\text{ }}a.m.u.\;\; \\
\Rightarrow{m_p}\; = 1.007275a.m.u. \\
\Rightarrow{m_n}\; = 1.008665a.m.u.\;\;\; \\ \]
Putting all the values in expression of mass defect, we get
$\Delta m = \left[ {1 \times 1.007275 + (2 - 1) \times 1.008665} \right] - 2.013553 \\
\Rightarrow\Delta m = \left[ {1.007275 + 1.008665} \right] - 2.013553 \\
\Rightarrow\Delta m = 2.01594 - 2.013553 \\
\Rightarrow\Delta m = 0.002387a.m.u. \\ $
Now we will find out the binding energy. To find out binding energy, we will use,
$E = \Delta m{c^2}$
$
1a.m.u. = 931MeV \\
\Rightarrow {c^2} = 931MeV \\ $
Putting all the values in this expression, we get
$
{E_b} = 0.002387 \times 931 \\
\Rightarrow {E_b} = 2.22MeV \\ $
To find out the binding energy per nucleon, we will use ${E_{bn}} = \dfrac{{{E_b}}}{A}$
Putting all the values, we will get
$
{E_{bn}} = \dfrac{{{E_b}}}{A} \\
\Rightarrow{E_{bn}} = \dfrac{{2.22}}{2} \\
\Rightarrow{E_{bn}} = 1.11MeV \\ $
To find out the packing fraction, we will use $f = \dfrac{{(M - A)}}{A}$
Putting all the values, we get
$
f = \dfrac{{\left( {2.013553 - 2} \right)}}{2} \\
\Rightarrow f = \dfrac{{0.013553}}{2} \\
\therefore f = 0.0067765 \\ $
So, mass defect, packing fraction, binding energy, binding energy per nucleon of deuteron are $0.002387,0.0067765,2.22MeV,1.11MeV$ respectively.

Note: keep in mind that if the values of packing fraction comes positive then the nucleus is unstable and if it comes negative then the nucleus is stable. From the binding energy concept we can determine whether fission will occur or fusion will occur. Don’t confuse A and Z because most of the students get confused. Therefore, Remember that A is always determined by the atomic number and Z is the mass number.