Question

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). $1\mu g$

Answer 1

As the Uranium undergoes radioactive decay, it gives out different particles due to which its composition as well as energy changes. Power is the energy absorbed or energy released per unit time. While in radioactivity the energy changes are equal to the product of mass decay and speed of light.

Formulae Used:
$E=\Delta m\times {{c}^{2}}$
$E=P\times t$

Complete step by step solution:
Uranium, ${{U}^{235}}$ is a radioactive material which gives out alpha, beta and gamma particles during its decay. After decay, the overall composition of the atom changes as well as energy changes take place.
The energy changes can be given by-
$E=\Delta m\times {{c}^{2}}$ - (1)
Here,$E$ is the energy released
$\Delta m$ is the difference in mass between the parent and daughter atoms, also known as the mass decay.
$c$ is the speed of light in air
Energy in terms of power is-
$E=P\times t$ - (2)
Here,$P$ is the power
$t$ is time taken
Comparing eq (1) and eq (2), we get,
$P\times t=\Delta m\times {{c}^{2}}$
Given, $P=1000\times {{10}^{3}}W$, $t=1\,hr=1\times 3600s$
In the above equation, we substitute values to get,
$\begin{aligned} & 1000\times {{10}^{3}}\times 3600=\Delta m{{(3\times {{10}^{8}})}^{2}} \\\ & \Delta m=\dfrac{1000\times {{10}^{3}}\times 3600}{{{(3\times {{10}^{8}})}^{2}}} \\\ & \Delta m=\dfrac{36\times {{10}^{8}}}{9\times {{10}^{16}}} \\\ & \Delta m=4\times {{10}^{-8}}kg=40\times {{10}^{-6}}gm \\\ \end{aligned}$
$\therefore \Delta m=40\mu g$

The mass decay is $40\mu g$, therefore, the correct option is (C).

Additional Information:
In a radioactive decay, the radioactive substances give out alpha, beta and gamma particles after a given time interval. The half life period of a radioactive substance is the time taken for it to reduce to half its mass. The average life period of a radioactive substance is the time taken by it to completely vanish.

Note:
During a radioactive decay, an unstable atom gives out energy and particles to transform into a new atom called the daughter atom. Two protons and two neutrons are bound together in an alpha particle, similar to a helium nucleus. Beta particles are high speed electrons or positrons. Gamma rays are electromagnetic rays which have photons carrying energy.