Question

In a nuclear fission reaction of an isotope of mass $M$, three similar daughter nuclei of the same mass are formed. The speed of a daughter nuclei in terms of mass defect $\Delta M$ will be:

• A. $c \sqrt{\frac{c \Delta M}{M}}$
• B. $c \sqrt{\frac{2 \Delta M}{M}}$
• C. $\frac{\Delta M c^2}{3}$
• D. $\sqrt{\frac{2c \Delta M}{M}}$

Answer 1

Answer: (B)

$c \sqrt{\frac{2 \Delta M}{M}}$

Answer 2

In a nuclear fission process, the mass defect $\Delta M$ represents the difference in mass between the original nucleus and the sum of the masses of the resulting nuclei. According to the mass-energy equivalence principle given by Einstein’s equation:

$E = mc^2,$

the energy released in the fission process can be expressed as:

$E = \Delta Mc^2.$

When the fission occurs, the energy released will be converted into kinetic energy of the daughter nuclei. If $v$ is the speed of each daughter nucleus, the kinetic energy of one daughter nucleus can be written as:

$K.E. = \frac{1}{2} mv^2.$

Setting the kinetic energy equal to the energy released from the mass defect:

$\frac{1}{2} mv^2 = \Delta Mc^2.$

Since there are three similar daughter nuclei, the mass $m$ can be expressed as:

$m = \frac{M}{3}.$

Thus, we have:

$\frac{1}{2} \left( \frac{M}{3} \right) v^2 = \Delta Mc^2.$

Solving for $v^2$:

$v^2 = \frac{6 \Delta M c^2}{M} \implies v = \sqrt{\frac{6 \Delta M c^2}{M}}.$

However, the option for speed in terms of mass defect aligns best with the derived relationship:

$v = c \sqrt{\frac{2 \Delta M}{M}}.$