Question

An atom bomb, weighing ${{1kg}}$, exploded. It releases $9 \times {10^{13}}{{J}}$ of energy. What percentage of mass is converted into energy?

Answer 1

Hint:Atom bombs can be included as a nuclear weapon. It is based on the principle nuclear fission which produces massive explosions. Mass-energy equivalence is the basis for solving this problem. It was introduced by Einstein.

Complete answer:
Einstein introduced an equation which relates the mass and energy. It states that the mass and energy are similar or almost the same.
As we know, using Einstein’s equation, energy, ${{E}} = {{m}}{{{c}}^2}$, where ${{m}}$ is the mass and ${{c}}$ is the velocity of light.
As the ${{{c}}^2}$ term used in the above equation has a very high value. This indicates that even a small amount of mass is related to a large amount of energy.
It is given that the energy, ${{E = }}9 \times {10^{13}}{{J}}$
We know that the velocity of light, ${{c = 3}} \times {{1}}{{{0}}^8}{{m}}{{{s}}^{ - 2}}$
Substituting these values in the equation, we get
$9 \times {10^{13}}{{J}} = {{m}}{\left( {{{3}} \times {{1}}{{{0}}^8}} \right)^2}$
On simplification, we get
${{m}} = \dfrac{{9 \times {{10}^{13}}}}{{{{\left( {{{3}} \times {{1}}{{{0}}^8}} \right)}^2}}} = \dfrac{{9 \times {{10}^{13}}}}{{9 \times {{10}^{16}}}} = {10^{ - 3}}{{kg = 0}}{{.001kg}}$
We need to find the percentage of mass which is converted to energy. Any percentage measurement can be done by dividing a portion by a part of total over the total.
Thus the percentage by mass can be calculated by dividing the mass after explosion divided by the mass taken initially, which is multiplied by $100$
i.e. $\% = \dfrac{{0.001{{kg}}}}{{1{{kg}}}} \times 100 = 0.1\% $

So only a $0.1\% $ of mass is converted to energy.
Note:
The explosion of an atomic bomb comes under nuclear fission. During nuclear fission, the energy released is from the nuclear bond change between subatomic particles. In the Einstein equation, the mass will be the rest mass. Rest mass is the mass of a stationary object. It is not the mass of atoms or subatomic particles which are involved in the reaction.