Question

The masses of a neutron and a proton are 1.0087 and 1.0073 amu respectively. If the neutrons and protons combine to form helium nucleus of mass 4.0015 amu. The binding energy of the helium nucleus will be:
A. 28.4 MeV
B. 20.8 MeV
C. 27.3 MeV
D. 14.2 MeV

Answer 1

Hint: The lost mass contributes to the binding energy of the nucleus. A stable atom has more binding energy. Binding energy is the amount of energy, nucleons release, when they are brought together to form the atom.

Formula used:
Mass defect of an atom is given by,

$\Delta m=[Z{{m}_{p}}+(A-Z){{m}_{n}}]-{{m}_{atom}}$………………………….(1)
Where,
Z is the atomic number(number of protons),
A is the mass number (number of nucleons),

${{m}_{p}}$ is the mass of a proton,
${{m}_{n}}$ is the mass of a neutron.

Now, the binding energy can be found using Einstein’s famous equation,
$E=m{{c}^{2}}$………………………….(2)
Where,
m is the mass,
and c is the velocity of light.

Complete step by step answer:
The symbol of helium is

$He_{2}^{4}$

So, the number of protons in the nucleus is 2
And the number of neutrons in the nucleus is (4-2) = 2.

Using equation (1), the mass defect is,

$\Delta m=[2\times {{m}_{p}}+2\times {{m}_{e}}]-{{m}_{atom}}$
$\Rightarrow \Delta m=[2\times 1.0073+2\times 1.0083]-4.0015$
$\Rightarrow \Delta m=4.0312-4.0015$
$\Rightarrow \Delta m=0.0297$

First, let us calculate the binding energy for a mass defect of 1 amu.
Using equation (2) we can write,

$E=m{{c}^{2}}$
$=(\dfrac{1.6606\times {{10}^{-27}}kg}{1amu}){{(2.998\times {{10}^{8}}\dfrac{m}{s})}^{2}}$ J
$=(1.4924\times {{10}^{-10}}J)(\dfrac{1MeV}{1.6022\times {{10}^{-13}}J})(2.998\times {{10}^{8}}\dfrac{m}{s})$
$=931.5$MeV

So, 1 amu mass defect will generate 931.5 MeV.
Hence the binding energy of the helium nucleus is,
$BE=0.0297\times 931.5$ MeV
$BE=27.3$
The correct answer is (C).

Note:
You need to know the following conversion factors,
1 amu =$1.667\times {{10}^{-27}}$
1 N = 1$\dfrac{kg-m}{{{s}^{2}}}$
1J = 1 Nm
1MeV = $1.6022\times {{10}^{-13}}$
Remember the following formula to finish a problem like this faster next time,
$BE=\Delta m\times (\dfrac{931.5MeV}{1amu})$
If you don’t remember the conversion formula in the previous equation that can hamper your speed. So, remember the conversion rate. Consider the symbol of elements to determine the atomic number and mass number correctly. Always remember that mass number is the summation of the number of protons and number of neutrons.