Nuclei

The correct answer is (D) : A = 208; Z = 82
When one α−particle is emitted, then the mass number (A) of daughter nuclei decreases by 4 and the atomic number decreases by 2.

When one β−particle is emitted, then the mass number (A) of daughter nuclei increases by 1 and the atomic number remains the same.

Therefore, for the end nucleus, A = 208 : Z = 82

The correct answer is (D): 1.80 hours

Explanation:

- (-λ₁A) + ( -λ2A)
⇒ = - ( λ₁ + λ₂ )A
⇒ λ_eff = λ₁ + λ₂
⇒ = +
_{⇒ eff} = hours
= 1.8 hours

The number of moles of in is:

The number of nuclei in moles is:

The total energy released is:

Substitute:

Thus, the total energy released is .

The correct answer is 2.

1. During the first -decay: The mass number decreases by 4, and the atomic number decreases by 2. 2. During the -decay: The mass number remains the same, and the atomic number increases by 1. 3. During the -decay: There is no change in the mass number or the atomic number. 4. During the second -decay: The mass number decreases by 4, and the atomic number decreases by 2. 5. During the -decay: The mass number remains the same, and the atomic number decreases by 1. 6. During the final -decay: There is no change in the mass number or the atomic number.
Final Answer: The mass number and atomic number of are:

To solve this problem, we need to apply the formula relating the radius of a nucleus to its mass number. The radius of a nucleus is given by the formula:

where is a constant (~1.2-1.3 femtometers) and is the mass number.

We are given that the radius of the first nucleus is half of the radius of a nucleus with mass number 192, denoted as . Let be the mass number of the first nucleus, and .

1. For the nucleus with mass number 192, its radius is:
2. For the nucleus with mass number , its radius is:
3. According to the problem, the relation given is:
4. Canceling from both sides:
5. Cubing both sides to eliminate the cube root gives:
6. Calculate : (approximately)
7. Thus:

Rounding 24.68 to the nearest whole number gives 24. Therefore, the mass number of the nucleus having a radius equal to half of the radius of the nucleus with mass number 192 is 24.

Step 1: Energy Calculation using

The energy is calculated using Einstein's famous equation , where:

Simplifying, we get:

Step 2: Converting to kWh

To convert from kilowatt-seconds (kWs) to kilowatt-hours (kWh), we divide by 3600 (the number of seconds in an hour):

To find the electric potential at the surface of an atomic nucleus with atomic number and radius , we use the formula for the electric potential due to a point charge:

where is the Coulomb's constant, and is the total charge of the nucleus with . Thus, .

Substituting the known values into the formula:

Simplifying the expression:

The calculated potential is , which is approximately .

For a nucleus:

Volume:

Relationship for the radius:

Substitute in the volume equation:

Simplifying:

The question involves the disintegration of a nucleus into two smaller nuclei with their mass ratio given as 2:1. We are to determine the speed ratio with which these nuclei move following disintegration. Let's solve this using principles of conservation of momentum.

Concept Involved: Conservation of Momentum

The principle of conservation of momentum states that if no external force acts on a system of particles, the total momentum of the system remains constant. In this case, the system is the disintegrating nucleus, which is initially at rest, so its initial momentum is zero.

Step-by-Step Solution

1. Define the system: Initially, the momentum of the nucleus is zero because it is at rest.
2. Assign masses and speeds:
   • Let the masses of the two smaller nuclei be and , with and (since the mass ratio is 2:1).
   • Let their respective speeds be and .
3. Apply conservation of momentum:
   • The negative sign for implies that the second nucleus moves in the opposite direction.
   • This gives us the equation: .
4. Simplify the equation:
   • By canceling out the mass , we get: .
5. Determine speed ratio:
   • Rearranging, we find the speed ratio as .

Conclusion

Therefore, after disintegration, the two smaller nuclei will move in opposite directions with speeds in the ratio of 1:2.

The correct answer is: "In opposite directions with speed in the ratio of 1:2 respectively".

To determine which pair of nuclei are isobars, we first need to understand the term "isobars." Isobars are nuclei with the same mass number (A), but different atomic numbers (Z). This means they have different numbers of protons but the same total number of nucleons (protons + neutrons).

From the above table, we can see that option (c), and , are isobars. They have the same mass number (3) but different atomic numbers (1 and 2, respectively). This matches the definition of isobars.

Thus, the correct answer is and .

All other options are ruled out as they do not satisfy the condition of having the same mass number with different atomic numbers.