AP Physics 2 Exam Question - Nuclear Reactions

A nuclear reaction involves the alpha decay of a radioactive isotope. Consider a hypothetical nucleus that undergoes alpha decay according to the following reaction:

^? X --> ^{A-4}Y + ^4_2He

Where ^? X represents the original radioactive isotope, ^{A-4}Y represents the daughter nucleus, and ^4_2He represents an alpha particle.

1. In order for this nuclear reaction to occur, what condition must be satisfied regarding the total sum of the mass numbers and atomic numbers on both sides of the reaction? Explain why this condition is necessary.

Provide a step-by-step explanation, including relevant formulas and concepts.

Answer:

The condition that must be satisfied regarding the total sum of the mass numbers and atomic numbers is known as the law of conservation of mass and charge.

According to this law, the total sum of mass numbers (A) and atomic numbers (Z) must be conserved before and after the nuclear reaction, ensuring that there is no creation or destruction of mass or charge.

For the given nuclear reaction:

^? X --> ^{A-4}Y + ^4_2He

The mass number (A) and atomic number (Z) are as follows:

Original isotope: ^? X (mass number = ? and atomic number = ?) Daughter nucleus: ^{A-4}Y (mass number = A-4 and atomic number = Z) Alpha particle: ^4_2He (mass number = 4 and atomic number = 2)

Considering the conservation of mass number (A) on both sides of the reaction, we can write the equation as:

? = (A-4) + 4

Simplifying the equation:

? = A

This equation shows that the total mass number on both sides of the reaction is the same, indicating the conservation of mass number.

Similarly, considering the conservation of atomic number (Z) on both sides of the reaction, we can write the equation as:

? = Z + 2

Simplifying the equation:

? = Z

This equation shows that the total atomic number on both sides of the reaction is the same, indicating the conservation of atomic number.

Therefore, the condition that must be satisfied is that the total sum of mass numbers and atomic numbers must be conserved before and after the nuclear reaction. This condition ensures that there is no violation of the law of conservation of mass and charge.

Note: In the question, the given radioactive isotope (^? X) should be replaced with a specific isotope to evaluate the specific values of mass number and atomic number.