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

The Correct Option is B

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:

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