Question

Find $BE$ per nucleon of $^{56}Fe$ where $m{{(}^{56}}Fe)=55.936u$ ${{m}_{n}}=1.00727u,\,{{m}_{p}}=1.007274u.$
$\begin{aligned} & A.\,\,477.45MeV \\\ & B.\,\,8.52MeV \\\ & C.\,\,577MeV \\\ & D.\,\,10.52MeV \\\ \end{aligned}$

Answer 1

Iron ($Fe$) has a total 26 protons and 30 neutrons in its nucleon. So, while calculating the Binding Energy ($BE$) for Iron, take the number of protons and neutrons into consideration and also after the final result is obtained, divide it with the total number of elements in the nucleon, to get the binding energy per nucleon.

Formula used:
$Binding\,Energy\,(BE)=\Delta m{{c}^{2}}$, where $\Delta m$ is the change in mass, and $c$ is the speed of light.

Complete step by step answer:
For calculating the binding energy of Iron according to the above formula, first we need to calculate the change in mass of the element, which can be calculated using the below formula:
$\Delta m=Mass\,of\,Nucleon-Mass\,of\,Nucleus$
According to the question, this formula can be re-written as:
$\Delta m=26{{m}_{p}}+30{{m}_{n}}-m{{(}^{56}}Fe)$
where, ${{m}_{p}}=1.007274u,$
${{m}_{n}}=1.00866u,$and
$m{{(}^{56}}Fe)=55.936u$, as per the information given in the question.
Substituting the given values into the above equation, we get:
$\begin{aligned} & \Delta m=(26\times 1.007274+30\times 1.00866-55.936)\,u \\\ & \Delta m=(26.189124+30.2598-55.936)\,u \\\ & \Delta m=0.512924\,u \\\ \end{aligned}$
Now,
Substitute the value of $\Delta m$and ${{c}^{2}}$, in the binding energy formula mentioned above, we get:
$\begin{aligned} & BE=(0.512924\times 931)MeV \\\ & BE=477.532\,MeV \\\ \end{aligned}$
So, the binding energy per nucleon for Iron is:
$\begin{aligned} & Binding\,Energy\,per\,Nucleon=\dfrac{477.532}{56}MeV \\\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=8.52MeV \\\ \end{aligned}$

So, the correct answer is “Option B”.

Note:
The question has explicitly mentioned to find out the binding energy per nucleon of the element, but many students make the error of just finding out the binding energy of the element and since there is an option that matches the wrong answer, the student don’t read the question again and mark the wrong answer and move forward, due to which they do not get marks for this answer.