Nuclear Physics

Kinetic Energy and Electrostatic Potential Energy

At the closest distance of approach, the kinetic energy of the particle gets completely converted into electrostatic potential energy. This can be represented by the following equation:

Where:

• is the mass of the particle,
• is the velocity of the particle,
• is Coulomb's constant,
• and are the charges involved,
• is the closest distance of approach.

Rearranging the equation to solve for , we get:

Conclusion:

This means that the closest distance of approach is inversely proportional to the mass of the particle. As the mass increases, the closest distance of approach decreases.

To solve the problem of identifying the possible decay particles in the sequence, we need to analyze the changes in the atomic number (Z) at each step of decay:

1. Step 1:
   The atomic number decreases by 1, while the mass number remains the same. This suggests the emission of a positron, which corresponds to decay. In beta-plus decay, a proton is converted into a neutron, thus reducing the atomic number by one.
2. Step 2:
   The atomic number decreases by 2, and the mass number decreases by 4. This indicates an alpha particle ( ) emission. An alpha particle consists of 2 protons and 2 neutrons, so its emission decreases the atomic number by 2 and the mass number by 4.
3. Step 3:
   The atomic number increases by 1, while the mass number remains unchanged. This corresponds to beta-minus decay ( ), where a neutron is converted into a proton, increasing the atomic number by one.

Thus, following this step-by-step reasoning of decay processes, the sequence of decay particles is: , , .

Therefore, the correct answer is: β⁺, α, β⁻.

To determine the ratio of the radii of the two daughter nuclei, we use the formula for the nuclear radius: , where is a constant and is the mass number.

Let the mass numbers of the daughter nuclei be and .

The radius of the first nucleus is given by .

The radius of the second nucleus is given by .

We need the ratio , which is .

The terms cancel out, simplifying to .

This further simplifies to .

.

Thus, .

The ratio of the radii of the two daughter nuclei is .

The nuclear reaction described can be written in a general form as: . To determine the element X, we need to balance the atomic numbers and mass numbers on both sides of the equation.

Step 1: Balance Mass Numbers

The mass number on the left side is 22. Thus, the total mass number on the right side must also be 22. Let the mass number of X be A. Then, we have:

.

Step 2: Balance Atomic Numbers

The atomic number on the left side is 11 (Na). Thus, the total atomic number on the right side, which is the sum of the atomic number of X and the atomic numbers of the emitted particles, must also be 11. Knowing that the emitted particle (usually a neutrino or beta particle) has negligible atomic number, we have:

,

where Z is the atomic number of X. Solving gives:

.

Step 3: Determine Element X

With mass number 22 and atomic number 10, the element X is neon (Ne). Thus, the element X in the nuclear reaction is: .

The correct answer is .

Step 1: Effect of Alpha ( ) Decay

An alpha particle ( ) has a mass number of 4 and an atomic number of 2.

When element undergoes alpha decay:

$ has:

Mass number =

Atomic number =

Step 2: Effect of Positron ( ) Emission

Positron emission ( ) decreases the atomic number by 1 without changing the mass number:

\) has:

Mass number = 286 (unchanged)

Atomic number =

Step 3: Effect of Beta-Minus ( ) Decay

Beta-minus ( ) emission increases the atomic number by 1 without affecting the mass number:

\) has:

Mass number = 286 (unchanged)

Atomic number =

Step 4: Effect of Electron Capture ( )

Electron capture decreases the atomic number by 1, keeping the mass number unchanged:

\) $

Thus, has:

Mass number = 286 (unchanged)

Atomic number =

Conclusion

The final product has mass number 286 and atomic number 81, which corresponds to option (4).

Let's evaluate each statement:

A. Slow neutrons are more effective in causing nuclear fission in U than fast neutrons because they have a higher probability of interaction with the uranium nucleus. This statement is correct.

B. α-rays consist of two protons and two neutrons; they are identical to Helium nuclei. This statement is correct.

C. β-rays are high-speed electrons − or positrons⁺ emitted during radioactive decay. This statement is correct.

D. γ-rays are electromagnetic radiation with wavelengths shorter than X-rays. This statement is incorrect.

Therefore, statements A, B, and C are correct.