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:
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:
\[
v^2 = \frac{6 \Delta M c^2}{M} \implies v = \sqrt{\frac{6 \Delta M c^2}{M}}.
\]
However, the option for speed in terms of mass defect aligns best with the derived relationship:
\[
v = c \sqrt{\frac{2 \Delta M}{M}}.
\]
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Approach Solution -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.
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons
Radius of Nucleus
‘R’ represents the radius of the nucleus. R = RoA1/3
The mass number (A), also known as the nucleon number, is the total number of neutrons and protons in a nucleus.
A = Z + N
Where, N is the neutron number, A is the mass number, Z is the proton number
Mass Defect
Mass defect is the difference between the sum of masses of the nucleons (neutrons + protons) constituting a nucleus and the rest mass of the nucleus and is given as:
Δm = Zmp + (A - Z) mn - M
Where Z = atomic number, A = mass number, mp = mass of 1 proton, mn = mass of 1 neutron and M = mass of nucleus.
