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The power obtained in a reactor using ${U^{235}}$ disintegration is $1000kW$. The mass decay of ${U^{235}}$ per hour is:
(A) $10\mu g$
(B) $20\mu g$
(C) $40\mu g$
(D) $1\mu g$

Last updated date: 03rd Mar 2024
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Hint To solve this problem, we have to remember the law of conservation of mass which states that for any system, the mass of the system must remain constant over time. And also Einstein's Mass-Energy relationship which says equivalent energy can be calculated by multiplying the mass by the square of the speed of light.
Formula Used
$E = m{c^2}$
Step-by-step solution:
Any substance having energy has a corresponding mass. Consequently, Equivalent energy is calculated using the Einstein mass-energy relation.
$E = m{c^2}$.................. $\left( 1 \right)$
According to this equation-
$E$ is the equivalent energy,
$m$ is the mass of which energy is calculated,
$c$ is the speed of light.

Complete Step by step solution
From the given data, we know that
Power obtained in the reactor using ${U^{235}}$ disintegration is
$P = 1000 \times {10^3}W$
We know that power is the rate of change of energy of the body.
$P = \dfrac{E}{t}$
$ \Rightarrow E = Pt$ ................ $\left( 2 \right)$
where $E$ is the energy obtained and $t$ is the time taken
Using the equation $\left( 1 \right)$, the formula for mass decay $\left( {\Delta m} \right)$ is,
$\Delta m = \dfrac{E}{{{c^2}}}$
Using equation $\left( 2 \right)$ we can write the formula as,
$\Delta m = \dfrac{{Pt}}{{{c^2}}}$
Since we know that the speed of light is $3 \times {10^8}m{s^{ - 1}}$ and the time taken is $t = 1hr = 3600\sec $ .
Substituting all the values in the above equation,
Mass decay of ${U^{235}}$per hour is,
$\Delta m = \dfrac{{\left( {1000 \times {{10}^3}} \right) \times \left( {3600} \right)}}{{{{\left( {3 \times {{10}^8}} \right)}^2}}}$
On solving further, we get
$\Delta m = 4 \times {10^{ - 8}}kg$
$ \Rightarrow \Delta m = 40\mu g$
So, the mass decay ${U^{235}}$ per hour is $40\mu g$ .

Hence, the correct option is (C) $40\mu g$ .

Additional information Einstein proposed the theory of rest mass-energy. This is the energy kept in any object due to its state of the rest position. The energy confined in this object is because it has a mass. Every object with mass has rest mass-energy. We recognize that the rest mass of the photon is zero. Therefore, the photon has no rest mass-energy. The term rest mass energy is essential to determine the kinetic energy of the objects with speed compared to the speed of light.

Note In mass-energy equivalence, the total mass of the system may change but the total energy and momentum do not, they remain constant. In this equation, Einstein states that when atoms fuse they produce a great amount of energy.