Nuclear Physics

To find the half-life period of the radioactive material, we will use the basic principle of radioactive decay that relates the remaining quantity of a substance to its initial amount over time, governed by the formula:

Where:

• is the final quantity of the substance.
• is the initial quantity of the substance.
• is the time period.
• is the half-life of the substance.

According to the problem, the activity drops to of its initial value in 30 years. We will plug these values into the formula:

Note that . Hence, we set up the equation:

Solving for the half-life gives:

Thus, the half-life period of the radioactive material is 7.5 years which matches with the given option.

The correct answer is (A) : 174 and 71
Equivalent reaction can be written as
D → D₄ + 2α + β^– + γ
⇒ Mass number of D₄ = Mass number of D – 2 × 4
= 182 – 8 = 174
⇒ Atomic number of D₄
= Atomic number of D – 2 × 2 + 1
= 74 – 4 + 1
= 71

The correct answer is (D) : (K_p + Q) > 0
K_p> 0
If Q is released ⇒Q > 0
⇒ K_p + Q> 0
If Q is absorbed ⇒Q < 0
Even then particle has to be given kinetic energy greater than magnitude of Q to maintain momentum conservation.
⇒ K + Q > 0

Given:

• Ratio of nuclear sizes:
• Ratio of speeds:

Step 1: Apply Conservation of Linear Momentum

The total momentum before disintegration is equal to the total momentum after disintegration. Assuming the nucleus is initially at rest, the total momentum before disintegration is zero. Thus, the momenta of the two fragments after disintegration must be equal and opposite:

where:

• and are the masses of the two fragments
• and are their respective speeds.

Rearranging for the velocity ratio:

Step 2: Relate Mass and Radius

Assuming the fragments have the same density, their masses are proportional to the cubes of their radii:

We are given . Substituting this into the mass ratio equation:

Step 3: Find the Ratio of Velocities

Using the velocity ratio equation from Step 1:

Therefore, , where .

Final Answer:

The value of is 2.

Assertion (A): The binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range 30 to 170. This is true because for medium and heavy nuclei, the binding energy per nucleon remains approximately constant and stable in this mass range.

Reason (R): Nuclear force is short-ranged. This is true because nuclear forces act only within a very short range (of the order of a few femtometers) and are responsible for binding nucleons together. Since the stability of the binding energy per nucleon is a consequence of the short-ranged nature of nuclear forces, the reason (R) correctly explains the assertion (A).

Thus, the correct answer is .

To solve this problem, we need to calculate the energy released during the reactions given. The reactions occur as follows:

1. The reaction of lithium with a neutron:
2. The reaction of deuterium with tritium:

We are given the following atomic masses:

• Neutron mass
• Deuterium mass

The energy released in a nuclear reaction is calculated using the mass defect, which is the difference between the mass of products and the mass of reactants. This difference is then converted to energy using the relation:

Calculate for First Reaction:

Reactants:

Products:

Mass defect:

Energy released:

Calculate for Second Reaction:

Reactants:

Products:

Mass defect:

Energy released:

Total Energy Released:

The total energy released by both reactions is the sum of the energies from each reaction:

Therefore, the energy released during the explosion is 22.22 MeV.

To determine the energy required to remove a neutron from , we need to calculate the difference in binding energy between the isotopes and . Here are the steps and calculations:

1. The formula for binding energy (BE) of a nucleus is given by: where is the number of protons, is the number of neutrons, is the mass of a proton, is the mass of a neutron, and is the atomic mass of the nuclide.
2. For , the atomic mass , , .
3. For , the atomic mass , , .
4. We need to find the energy required to remove a neutron. Thus, calculate the binding energy difference:
5. Express the difference in masses:
   
6. Simplify the mass difference:
   
7. Convert the mass difference to energy:
   Using , we calculate the energy change:
   

Therefore, the energy required to remove a neutron from is 4.95 MeV.

To solve the problem, we need to determine the number of triple alpha processes per second, denoted as .

Firstly, calculate the energy released in a single triple alpha process. The reaction is:

The masses of the participating isotopes are:
- Mass of = 4.0026 u
- Mass of = 12 u

The initial total mass of the three nuclei:

The mass defect is the difference in mass before and after the reaction:

The energy released can be calculated using :

Convert the energy to joules (1 MeV = J):


Now, find the rate using the given power output of the star:
Power . Since power is energy per unit time,


Solve for :



Thus, the value of is 5. The computed value fits within the expected range of 5,5, confirming the solution is correct.

Analyze the Fission Reaction:
The nuclear fission of typically occurs when it absorbs a neutron, forming in an excited state, which then undergoes fission.
The reaction can be written as:

Check for Conservation of Mass Number and Atomic Number:
For each option, verify that the total mass number (A) and atomic number (Z) on the right side of the reaction matches the total on the left side.
Total Mass Number (A):
Total Atomic Number (Z):

Evaluate Each Option:
Option 1:
Mass number:
Atomic number:
This satisfies the mass and atomic number balance.

Option 2:
Mass number:
Atomic number:
This does not satisfy the balance.

Option 3:
Mass number:
Atomic number:
This does not satisfy the balance.

Option 4:
Mass number:
Atomic number:
This satisfies the mass and atomic number balance.

Conclusion:
Only Option 4 satisfies the conservation of both mass number and atomic number in the nuclear fission process.

To solve the problem of finding the difference between the total mass of all the nucleons and the nuclear mass of a given nucleus, we need to apply the concept of mass-energy equivalence. The binding energy of the nucleus is given as .

According to Einstein's mass-energy equivalence principle, the binding energy ( ) is related to the mass defect ( ) by the equation:

where is the speed of light in vacuum, approximately .

We can rearrange the formula to compute the mass defect ( ):

Substituting the given values:

Since the options are given in micrograms, we can convert kilograms to micrograms:

1 kilogram = micrograms, so:

Thus, the difference between the total mass of all the nucleons and the nuclear mass of the given nucleus is 20 micrograms.

The correct answer is: .

To find the nuclear binding energy of the isotope ^{12}_{5}B, we need to first understand the components involved. The nuclear binding energy is the energy required to separate an atomic nucleus into its individual protons and neutrons. It can be calculated using the mass defect, which is the difference between the sum of the masses of the individual nucleons and the actual mass of the nucleus.

1. We have the isotope , which means it consists of 5 protons and neutrons.
2. The total mass of the separated nucleons would be , where is the mass of a proton, and is the mass of a neutron.
3. The mass defect is then given by the difference: where is the mass of the isotope nucleus.
4. The nuclear binding energy is found using Einstein's mass-energy equivalence principle, , where is the speed of light.
5. Thus, substituting the expression for mass defect into the equation, the binding energy becomes:

Therefore, the correct option that represents the nuclear binding energy of the isotope is:

To determine the identity of the emitted particles (R) in the given nuclear fission reaction, we need to first understand the principle of nuclear reactions, where both mass numbers and atomic numbers are conserved. The given reaction is:

1. Mass number balance: The total mass number on both sides of the reaction must be equal. On the left, the mass number of is 236. On the right, the total mass number is the sum of mass numbers of , , and 3R.
2. Calculating:
3. Atomic number balance: The total atomic number must also be conserved. The atomic number of is 92, and for the products, it should add up to 92.
4. Calculating:
5. Identity of R: The particle with mass number 1 ( ) and atomic number 0 ( ) is the neutron.

Therefore, the correct answer is the particle "Neutron." This identification fits with common fission reactions where neutrons are usually emitted.

Given reaction:

Mass defect:

Calculating:

Energy released:

Step 1: Understand the process of beta decay. In beta decay, a neutron decays into a proton, emitting an electron (beta particle) and an electron antineutrino.
Conclusion: The correct nuclear process is beta decay, which corresponds to option (3).

To solve this question, we need to understand how the mass density of a nucleus relates to its mass number and its radius .

The mass density of a nucleus is given by the formula:

We know that the mass of a nucleus is approximately proportional to its mass number , because the mass number represents the sum of protons and neutrons, which have nearly equal mass.

The volume of a spherical nucleus is calculated using the formula:

For a nucleus, it is empirically found that the radius is related to the mass number by the equation:

where is a constant. Thus, the volume can be expressed as:

Substituting this into the formula for density, we get:

Upon simplification, it becomes:

From this expression, we can see that the density is independent of since the terms cancel each other out.

Therefore, the correct answer is:

Independent of

The question involves evaluating two statements: an assertion about nuclear binding energy and a reason related to nuclear forces.

1. Understanding the Assertion (A):
   • The assertion states that the binding energy per nucleon is found to be practically independent of the atomic number for nuclei with mass numbers between 30 and 170.
   • For nuclei within this mass range, the binding energy per nucleon is relatively constant and then begins to decrease for heavier nuclei. This is validated by the semi-empirical mass formula which indicates this plateau in binding energy values due to competing nuclear forces.
2. Understanding the Reason (R):
   • The reason states that nuclear force is long-range. This is incorrect because nuclear forces are actually short-range forces, effective only over distances on the order of a few femtometers ( ).
   • Nuclear force being short-range explains why it strongly influences nucleons tightly packed within an atomic nucleus, but not between widely separated nuclei.

In conclusion, the assertion (A) about the nature of binding energy per nucleon for the specified mass range is correct. However, the reason (R) provided is a false statement about the nature of nuclear forces. Thus, the correct answer is:

(A) is true but (R) is false

Let's examine the types of reactions listed in LIST-I and match them with their corresponding descriptions in LIST-II.

1. Reaction A:
   • This is a nuclear fission reaction. It involves the splitting of a large nucleus into smaller fragments, usually producing free neutrons and lighter nuclei. This matches with "III. Fission".
2. Reaction B:
   • This is a chemical reaction where hydrogen gas reacts with oxygen gas to form water. This matches with "I. Chemical Reaction".
3. Reaction C:
   • This is a nuclear fusion reaction with a positive Q value. Helium is produced along with a neutron, releasing energy. This matches with "II. Fusion with +ve Q value".
4. Reaction D:
   • This is a nuclear fusion reaction but has a negative Q value because it absorbs energy, indicated via gamma radiation . This matches with "IV. Fusion with -ve Q value".

Thus, the correct matching from LIST-I with LIST-II is as follows:

• A-III: The reaction is a nuclear fission process.
• B-I: The reaction is a chemical reaction.
• C-II: The reaction is a fusion reaction with a positive Q value.
• D-IV: The reaction is a fusion reaction with a negative Q value.

Therefore, the correct answer is:

A-III, B-I, C-II, D-IV

Given: - Binding energy per nucleon of = 1.1 MeV
- Binding energy per nucleon of = 7.0 MeV
The energy released is the difference between the binding energy of the reactants and products: Thus, the energy released is:

Using the equation for rolling motion, the time taken for the solid sphere to reach the bottom will be less than the time taken by the hollow sphere, as the solid sphere has a greater rotational inertia compared to the hollow one.

Final Answer: .

The time taken by an object to roll down an incline depends on its moment of inertia. The solid sphere has a smaller moment of inertia compared to the hollow sphere, which allows it to accelerate more quickly and reach the bottom in less time. Using the equation for rolling motion, the time taken for the solid sphere to reach the bottom will be less than the time taken by the hollow sphere, as the solid sphere has a greater rotational inertia compared to the hollow one.
Final Answer: .

Step 1: At the point of closest approach, the kinetic energy of the -particle is completely converted into electrostatic potential energy.
Step 2: For an -particle, Using the relation (in MeV–fm units):
Step 3: Distance of closest approach:
Step 4: Convert into nanometres: Closest matching option: