Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life period of a radioactive element is 140 days. After 560 days, one gram of the element will reduce to (a) \(1 / 2 \mathrm{~g}\) (b) \(1 / 4 \mathrm{~g}\) (c) \(1 / 8 \mathrm{~g}\) (d) \(1 / 16 \mathrm{~g}\)

Step-by-Step Solution

Answer

The correct answer is (d) \( \frac{1}{16} \mathrm{~g} \).

Step 1: Understand Half-Life

The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this problem, the half-life of the element is given as 140 days.

Step 2: Determine Number of Half-Lives

Calculate how many half-life periods (140 days each) are contained in 560 days. This is done by dividing 560 by 140, which results in 4 half-life periods.

Step 3: Apply Half-Life Decay Formula

Starting with 1 gram, use the fact that the amount of substance left after each half-life is half of the previous amount. After one half-life, it remains 0.5 grams, after two half-lives it becomes 0.25 grams, after three half-lives it is 0.125 grams, and after four half-lives it will be 0.0625 grams.

Step 4: Select Correct Option

Since after 560 days (i.e., four half-lives), the element reduces to 0.0625 grams or equivalently \( \frac{1}{16} \) grams, the correct option is (d) \( \frac{1}{16} \mathrm{~g} \).

Key Concepts

Half-Life

The term "half-life" is a key concept in understanding radioactive decay. It refers to the time required for half of the radioactive atoms in a sample to decay. This means that after each half-life period, the remaining quantity of the substance is reduced by half.
This concept is fundamentally important in predicting how long a radioactive element will remain active. For instance, if an element has a half-life of 140 days, then in 140 days, only half of the initial quantity will be left.
This halving process continues repeatedly, so it simplifies the complex decay into a series of predictable steps.

Nuclear Chemistry

Nuclear chemistry is the subfield of chemistry dealing with radioactivity and changes in the nucleus of atoms. Unlike chemical reactions, where electrons outside the nucleus are involved, nuclear chemistry focuses on the processes occurring within the nucleus itself.
This includes radioactive decay, a process where unstable atoms release energy in the form of radiation to become more stable. This release can occur in different forms, commonly in alpha, beta, or gamma decay.
Nuclear chemistry explains phenomena like radioactive decay, providing insight into how elements change into different elements over time due to nuclear reactions. It's vital for various applications, such as radioactive dating, energy production, and medical treatments.

Decay Formula

To predict the remaining quantity of a radioactive element over time, the decay formula is used. It is expressed mathematically as:\[N(t) = N_0 \times \left(\frac{1}{2}\right)^{(t/T_{1/2})}\]

• Here, \(N(t)\) is the quantity remaining after time \(t\).
• \(N_0\) is the initial quantity.
• \(T_{1/2}\) denotes the half-life of the element.

This formula comes in handy while solving problems related to nuclear decay. It allows us to calculate how much of a substance remains after a given period by simply knowing the half-life of the element.
This mathematical model helps simplify the understanding of radioactive decay, translating the half-life concept into a precise calculation tool easily applicable across various scenarios.