Nuclei

Step 1: State the conservation laws.
The parent nucleus is at rest, so the total initial momentum is zero. Due to conservation of momentum, the total final momentum must also be zero. Let be the mass number of the parent nucleus. The mass of a nucleus is proportional to its mass number.

Step 2: Define the masses and velocities of the final products.
The products are two alpha particles ( ) and a daughter nucleus ( ). Mass number of an alpha particle: . Mass number of the daughter nucleus: . Let be the velocities, and be the kinetic energy of each alpha particle.

Step 3: Apply the conservation of momentum.
Since the parent nucleus is at rest, the initial momentum is zero. The two particles are emitted, and their kinetic energies are equal. Since , and their masses are equal, their momenta must be equal in magnitude: . For momentum to be conserved, the two particles must be emitted in opposite directions, and the daughter nucleus must recoil to balance the net momentum of the two 's.

Step 4: Analyze the momentum of the two alpha particles.
Since , and the total momentum is zero, the net momentum of the two 's can be in any direction. The lowest kinetic energy for the daughter nucleus happens when the two 's are emitted in opposite directions. The highest kinetic energy for the daughter nucleus happens when the two 's are emitted in the same direction, which violates conservation of momentum. The total momentum of the two alpha particles is . By conservation of momentum, .
The magnitude of the momentum of an alpha particle is .
The possible range of is . For maximum recoil, and must be parallel. The momentum of the daughter nucleus is .
The kinetic energy of the daughter nucleus is .
For the maximum recoil, .

Step 5: Calculate the total kinetic energy.
The total kinetic energy is the sum of the kinetic energies of the two alpha particles and the daughter nucleus. Using the value (which corresponds to the maximum possible recoil). This corresponds to option (C).

Let be the initial amount of the radioactive substance.
The law of radioactive decay is given by , where is the decay constant.
We are given that the substance decays by 10% in 16 hours. This means 90% remains.
Let hours. Then .
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We need to find the percentage that remains after 2 days.
Time days = hours = 48 hours.
We need to find .
We can rewrite this using the information we have:
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From the first part, we know .
So, the fraction remaining after 48 hours is .
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The percentage remaining is .

Let the parent nucleus P have mass number A and atomic number Z. We can represent it as .
Step 1: Alpha decay.
An alpha particle is a helium nucleus, represented as .
When P undergoes alpha decay, its mass number decreases by 4 and its atomic number decreases by 2. Let the intermediate nucleus be X.
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Step 2: Two beta-minus decays.
A beta-minus ( ) particle is an electron, represented as .
In a decay, a neutron in the nucleus converts into a proton and an electron. This increases the atomic number by 1 and leaves the mass number unchanged.
The intermediate nucleus X undergoes two successive decays to become the final nucleus Q.
First decay: .
Second decay: .
Step 3: Compare the parent nucleus P and the final nucleus Q.
Parent nucleus: .
Final nucleus: .
Nuclei that have the same atomic number (Z) but different mass numbers (A) are called isotopes.
Since both P and Q have the same atomic number Z, they are isotopes.
% Quick tip

Step 1: Understanding the Concept:
Power is the rate at which energy is produced. The total energy produced in a given time is the product of the power and the time duration. This total energy is also equal to the number of fissions multiplied by the energy released per fission.
Step 2: Key Formula or Approach:
1. Total Energy, .
2. Total Energy, .
Equating these gives: .
Step 3: Detailed Explanation:
First, convert all given quantities to SI units.
- Power, .
- Time, .
- Number of fissions (nuclei) in this time, .
Now, calculate the total energy produced in one hour:
This total energy is released from fissions. We can now find the energy released per fission:
Step 4: Final Answer:
The energy released per fission is J. Therefore, option (C) is correct.

Step 1: Understanding the Concept:
This problem involves radioactive decay and the concept of half-life. The activity (or radiation level) of a radioactive sample decreases exponentially with time. We need to find the time it takes for the activity to reduce to the safe level.
Step 2: Key Formula or Approach:
The formula for radioactive decay relates the initial activity ( ), the activity at time ( ), and the half-life ( ):
Alternatively, we can write , where is the number of half-lives that have passed, and .
Step 3: Detailed Explanation:
Let the safe radiation level be .
The initial radiation level is given as 32 times the safe level, so .
The final radiation level, when the material can be handled safely, is .
The half-life is hours.
We need to find the time .
Using the decay formula:
We set this equal to the decay factor:
Since , we can write:
By comparing the exponents, we find that the number of half-lives that must pass is .
Now, we can calculate the total time :
Step 4: Final Answer:
The material can be handled safely after 12.5 hours. Therefore, option (D) is correct.

• Fraction remaining after time : .
• At 10 days: , disintegrated: .
• At 15 days: , disintegrated: .
• Ratio: .
• Hence ratio = 6:7.

• Energy per fission: MeV = J = J.
• Required power MW = W = J/s.
• Number of fissions per second: .
• Hence correct answer = }.