Question

A radioactive element A decays by the sequence and with half-lives given below: \(\mathrm{A} \frac{\alpha}{30 \min }-\mathrm{B} \frac{2 \beta}{2 \text { days }} \rightarrow \mathrm{C}\) Which one of the following statement is correct? (a) after two hours, less than \(10 \%\) of the initial \(\mathrm{A}\) is left (b) maximum amount of B present at any time is less than \(50 \%\) of the initial amount of \(\mathrm{A}\). (c) stomic number of \(\mathrm{A}\) and \(\mathrm{C}\) are same (d) both (a) and (c) are correct

Short answer

The correct answer is (d): both (a) and (c) are correct.

Step-by-step solution

Determine Half-life of A

The half-life of element A is given as 30 minutes, meaning it takes 30 minutes for half of A to decay by emitting an alpha particle to form B.

Calculate A Remaining After 2 Hours

Since 2 hours is equivalent to 120 minutes, we can calculate the number of half-lives in 2 hours: \[ \frac{120 \text{ minutes}}{30 \text{ minutes}} = 4 \text{ half-lives} \]. After each half-life, the remaining fraction of A is halved, so after 4 half-lives: \[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]. This means \( \frac{1}{16} \) of A remains, which is \( 6.25\% \). Less than \(10\%\) of A remains after 2 hours, supporting statement (a).

Max Amount of B Formed

The maximum amount of B could possibly reach close to the initial amount of A due to conservation of mass until it starts decaying to C. Initially, as A decays to B, the maximum amount of B formed cannot exceed what was initially available, but it doesn't remain due to its decay to C.

Analyze Atomic Numbers of A and C

When A undergoes alpha decay to B, the atomic number decreases by 2 (as an alpha particle, \(^4_2\text{He}\), is lost). B undergoes beta decay (each beta decay increases the atomic number by 1), transitioning to C. After 2 beta decays, the atomic number of C is back to the original atomic number of A, making statement (c) correct.

Evaluate Correct Statements

Based on the above steps: (a) is correct because less than 10% of A remains after 2 hours, (b) is incorrect since B can be the majority initially until it starts decaying, and (c) is correct, as atomic numbers of A and C are the same. Therefore, (d) is correct.

Key concepts

Half-life Calculations

The concept of half-life is central to understanding radioactive decay. It's the time required for half of the radioactive nuclei in a sample to decay. This period varies between different substances. For element A, its half-life is 30 minutes. But what does this really mean?- Each half-life cycle reduces the quantity of the radioactive substance by 50%.- If you start with 100 grams of element A, after one half-life, only 50 grams remain. After two half-lives, just 25 grams remain, and so on.Mathematically, the amount of radioactive material left after a certain number of half-lives can be determined using \[ \left(\frac{1}{2}\right)^n \] where \( n \) is the number of half-lives.For example:- After 4 half-lives, as in the case of having 120 minutes elapse for element A, you have: \[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \] which equals 6.25% of the original material remaining. This is key to solving problems like determining how much of a substance is left over after a given time.

Alpha Decay

Alpha decay is a type of nuclear reaction where an unstable atom releases an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, equivalent to a helium nucleus.Here's what happens when an element undergoes alpha decay:- The atomic number decreases by 2.- The mass number decreases by 4.So, if element A undergoes alpha decay to become element B, it means:- If A originally had an atomic number \( Z \), B will have \( Z - 2 \).- Its mass number will reduce from \( A \) to \( A - 4 \).Alpha decay often occurs in heavy elements where the loss of mass can aid in achieving stability. It's a crucial concept because it changes both the identity and mass of the original atom significantly, as seen in our exercise.

Beta Decay

After alpha decay, element B may undergo beta decay. During beta decay, a neutron in the nucleus is transformed into a proton while emitting an electron (known as a beta particle) and an antineutrino.Key effects of beta decay:- The atomic number increases by 1.- The mass number remains unchanged.This is because:- A neutron turning into a proton leads to an increase in the atomic number, hence a new element.- The emitted beta particle has negligible mass, leaving the mass number constant.In our context:- Element B undergoes two beta decays.- Initially, B has an atomic number of \( Z - 2 \).- Two beta decays increase it back to \( Z \), restoring element C to the original atomic number of A (though the resulting element is different due to the combination of alpha decay followed by beta decay).Beta decay, therefore, illustrates nuclear transformation where identity shifts without affecting the mass figure, integral for nuclear equations and understanding elemental changes in decay sequences.