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How much energy is released when the mass of $_8{O^{16}}$ nucleus is completely converted into an energy? The binding energy per nucleon of $_8{O^{16}}$ is $7.97$$M$$ev$ and ${m_p} = 1.0078$$u$ and ${m_n} = 1.0087u$.

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Last updated date: 17th Jun 2024
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Answer
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Hint: To find the energy released when the nucleus is completely converted into an energy, first we need to find the total energy of nucleons then find the energy released using the formula of mass of energy binding of nucleus.

Formula used:
Nuclear binding energy,
$E = \Delta m{c^2}$
Where,
$m$ is the mass of the defect
$c$ is the speed of the light

Complete step by step solution:
Nuclear binding energy is an energy which splits the nucleus into an atom of its components. It can determine the fission and fusion of the process.
The binding energy of the nuclei is always in positive number and all the nuclei will have the net energy to separate into individual protons and the neutrons.
By converting the mass to energy we can calculate the nuclear binding energy,
$E = \Delta m{c^2}$
The mass of energy binding of the nucleus is the difference between the mass of the nucleus and the sum of mass of the nucleons.
$\Delta m$$ = $ mass of nucleus – (mass of proton + neutron)
The data given In the question,
Number of nucleons $ = $ $16$
Energy per nucleons $ = $$7.97$$M$$eV$
Total energy ($E$) $ = $$16 \times 7.97$
$\Rightarrow$ = $127.52$ $MeV$
The mass of the proton is given in the question as,
${m_p} = 1.0078u$
and also the mass of the nucleons are given as,
${m_n} = 1.0087u$
We know that the nuclear binding energy is given by,
$ \Rightarrow $ $E = \Delta m{c^2}$
on putting the values we have,
$ \Rightarrow $ $E = 8(1.0078 + 1.0087) \times 931$
on further solving, we get
$ \Rightarrow $ $E = 15067.288$$MeV$
Then the energy released is,
$ \Rightarrow $ ${E_0} = 127.52 - 15018.892$
$ \Rightarrow $ ${E_0} = - 14899.438$ $MeV$

Therefore the energy is released when the mass of the $_8{O^{16}}$ nucleus is completely converted into energy is $ - 14899.438$ $MeV$.

Note: For the lighter elements then the iron, the fusion releases the energy because of the nuclear binding energy with increase of mass. If the element is heavier than iron it will release the energy upon the fission. The binding energy is available as nuclear energy and this can be used to produce electricity, nuclear weapons, nuclear power etc.