Problem 67 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive substance having a half-life of 3 days was received in 12 days. It was found that there was only \(3 \mathrm{~g}\) of the isotope in the container. The initial weight of the isotope when packed was (a) \(12 \mathrm{~g}\) (b) \(24 \mathrm{~g}\) (c) \(48 \mathrm{~g}\) (d) \(96 \mathrm{~g}\)

Step-by-Step Solution

Answer

The initial weight of the isotope was 48 g (option c).

Step 1: Understand the Problem

We need to find the initial amount of a radioactive substance given its half-life, the time elapsed, and the remaining quantity. The half-life is 3 days, 12 days have passed, and 3 g of the isotope remains.

Step 2: Calculate Number of Half-lives

Determine the number of half-lives that have passed in the given time:\[ \text{Number of half-lives} = \frac{12 \text{ days}}{3 \text{ days per half-life}} = 4 \] Four half-lives have passed from the initial time to when the measurement was taken.

Step 3: Apply the Half-life Formula

The amount remaining of a substance after a certain number of half-lives is given by the formula:\[ N = N_0 \left(\frac{1}{2}\right)^n \]where \(N\) is the remaining quantity, \(N_0\) is the initial amount, and \(n\) is the number of half-lives. Substitute the known values:\[ 3 = N_0 \left(\frac{1}{2}\right)^4 \]

Step 4: Solve for the Initial Amount

Rearrange and solve for \(N_0\):\[ N_0 = 3 \times 2^4 = 3 \times 16 = 48 \]The initial amount of the isotope was 48 g.

Key Concepts

Half-life

The concept of half-life is crucial in understanding how radioactive substances decay over time. Half-life is defined as the time required for half of a radioactive substance to decay into a more stable form. In simpler terms, it is the amount of time it takes for half of the atoms in a given sample to transition into a different state or element.

For example, if you start with 100 grams of a radioactive substance that has a half-life of 3 days, after 3 days, you would expect to have 50 grams of the original substance left, with the other 50 grams having decayed. After another 3 days (a total of 6 days), you would have 25 grams remaining, and so on.

• Half-life is not affected by external conditions such as temperature or pressure
• It is unique to each radioactive element
• Understanding half-life helps in dating archaeological findings, managing nuclear waste, and even in medical treatments

Isotopes

Isotopes are atoms of the same element that contain the same number of protons but a different number of neutrons. This difference in neutron count results in a variance in atomic mass but the chemical properties remain largely similar. Isotopes can either be stable or unstable (radioactive).

Radioactive isotopes undergo decay over time, eventually transforming into a different element or state. This transformation is what we observe in processes like radioactive decay, where isotopes emit radiation as they decay.

• Isotopes often have applications in diverse fields like medicine (for diagnostic imaging) and archaeology (for radiocarbon dating).
• Each isotope of an element has its own specific half-life. For example, carbon has a stable isotope, Carbon-12, and a radioactive isotope, Carbon-14.
• Understanding isotopes helps scientists and researchers understand complex natural processes and historical events.

Radioactivity

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation. This emission occurs because the isotopes are trying to reach a more stable state. The process can emit energy in the form of alpha particles, beta particles, or gamma rays.

Radioactivity was first discovered by scientist Henri Becquerel and further studied by Marie and Pierre Curie. It plays a vital role in many technological and scientific applications, including power generation and medical treatments.

• Natural radioactivity is found in many elements on Earth, including uranium, thorium, and radon.
• Artificial radioactivity can be induced by bombarding a stable element with particles in a reactor or particle accelerator.
• Radioactivity is a natural phenomenon but requires careful handling due to the potential health risks associated with exposure to radiation.