Question

Number of Uranium-235 nuclei required to undergo fission to give $9 \times {10^{12}}\,J$ of energy is
A. $2.8125 \times {10^{24}}$
B. $28.125 \times {10^{24}}$
C. $281.25 \times {10^{24}}$
D. $28125 \times {10^{24}}$

Answer 1

Hint:We know that the energy released in the fission of one uranium nucleus is $200\,MeV$. If we take the number of nuclei required to produce energy of $9 \times {10^{12}}\,J$ as $n$ . Then n times the energy released in the fission of one uranium nucleus will be equal to $9 \times {10^{12}}\,J$.
By using this we can find the value of n by dividing the energy $9 \times {10^{12}}\,J$ with the value of energy produced in the fission of one uranium nucleus.

Complete step by step solution:
We know that the energy produced by fission of one uranium nucleus is
$E = 200\,MeV$
We need to find the number of uranium nuclei required to undergo fission to give $9 \times {10^{12}}\,J$ of energy.
Let us find the energy released in the fission of one uranium nucleus in Joule.
We know that,
$1\,MeV = {10^{6\,}}eV$
$ \Rightarrow E = 200\,MeV = 200 \times {10^{6\,}}eV$
In order to convert energy in electron volt to joule we need to multiply it by $1.6 \times {10^{ - 19}}$.
$\because 1\,eV = 1.6 \times {10^{ - 19}}J$
Then we get
$ \Rightarrow E = 200 \times {10^{6\,}}eV = 200 \times {10^6} \times 1.6 \times {10^{ - 19}}\,J$
$ \Rightarrow E = 200 \times 1.6 \times {10^{ - 13}}J$
This is the energy per fission of uranium 235 nuclei.
Let us suppose that $n$ molecules of uranium undergo fission to give $9 \times {10^{12}}\,J$ of energy.
That is n times the energy produced per fission is equal to $9 \times {10^{12}}\,J$ .
$ \Rightarrow n \times E = 9 \times {10^{12}}\,J$
From this we can calculate the value of n as
$ \Rightarrow n = \dfrac{{9 \times {{10}^{12}}}}{E}\,$
$ \Rightarrow n = \dfrac{{9 \times {{10}^{12}}\,J}}{{200 \times 1.6 \times {{10}^{ - 13}}J}}\,$
$\therefore n = 2.8125 \times {10^{24}}\,$
This is a number of uranium nuclei required to undergo fission to give $9 \times {10^{12}}\,J$ joules of energy.

So, the correct answer is option A.

Note:Fission is the process by which a heavy nucleus splits into smaller lighter nuclei. During the fission less energy is required than the energy needed to bind them together so energy will be released in fission reaction. The energy released in the case of one uranium nucleus is $200\,MeV$.