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:

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

Answer 2

To solve this problem, we need to understand the conversion of mass defect into the kinetic energy of the daughter nuclei resulting from a fission reaction.

Concept:

The principle at play is the conservation of energy in nuclear reactions, specifically the conversion of mass defect into kinetic energy.

According to Einstein's mass-energy equivalence, the energy released in a nuclear reaction is given by:

\(E = \Delta M c^2\)

where \(\Delta M\) is the mass defect and \(c\) is the speed of light.

Calculation:

In the given reaction, the mass defect \(\Delta M\) is converted into kinetic energy of the three daughter nuclei. If each daughter nucleus has the same kinetic energy, then:

\(K = \frac{\Delta M c^2}{3} \,\, \text{(for each nucleus)}\)

Also, the kinetic energy \(K\) for each nucleus can be expressed as:

\(K = \frac{1}{2} m v^2\)

where \(m\) is the mass of one daughter nucleus and \(v\) is the speed of the daughter nucleus. Given that \(m = \frac{M}{3}\) because three similar daughter nuclei are formed, we equate the expressions for kinetic energy:

\(\frac{1}{2} \times \frac{M}{3} \times v^2 = \frac{\Delta M c^2}{3}\)

Simplifying for \(v^2\), we have:

\(v^2 = \frac{2 \Delta M c^2}{M}\)

Thus, the speed \(v\) of the daughter nucleus is:

\(v = c \sqrt{\frac{2 \Delta M}{M}}\)

Conclusion:

The correct expression for the speed of a daughter nuclei in terms of mass defect \(\Delta M\) is \(c \sqrt{\frac{2 \Delta M}{M}}\). Therefore, the correct answer is option (b):

\(c \sqrt{\frac{2 \Delta M}{M}}\)

Tip:

Remember to check the number of daughter products and distribute the energy release accordingly when solving nuclear reaction problems involving multiple products.