Problem 143 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radio isotope is four hours. If the initial mass of the isotope was \(200 \mathrm{~g}\) the mass remaining undecayed after 24 hours is (a) \(2.084 \mathrm{~g}\) (b) \(3.125 \mathrm{~g}\) (c) \(4.167 \mathrm{~g}\) (d) \(1.042 \mathrm{~g}\)

Step-by-Step Solution

Answer

(b) \(3.125 \mathrm{~g}\) is the remaining mass after 24 hours.

Step 1: Understand the Concept of Half-Life

The term 'half-life' refers to the time required for a quantity to reduce to half its initial value. For radioactive isotopes, this indicates the time it takes for half of the isotope to decay.

Step 2: Determine the Number of Half-Lives

Given that the half-life of the isotope is 4 hours, and we need to calculate the remaining mass after 24 hours, determine the number of half-lives. Divide the total time by the half-life:\[\text{Number of half-lives} = \frac{24 \text{ hours}}{4 \text{ hours/half-life}} = 6\]

Step 3: Apply the Exponential Decay Formula

Use the formula for exponential decay related to half-life, \(N = N_0 \left( \frac{1}{2} \right)^n\), where \(N_0\) is the initial mass, \(N\) is the mass remaining, and \(n\) is the number of half-lives. Substitute the given values:\[N = 200 \text{ g} \left( \frac{1}{2} \right)^6\]

Step 4: Calculate the Remaining Mass

Calculate \(\left( \frac{1}{2} \right)^6\) and then multiply by the initial mass:\[\left( \frac{1}{2} \right)^6 = \frac{1}{64}\]Then,\[N = 200 \times \frac{1}{64} = 3.125 \text{ g}\]

Step 5: Identify the Correct Answer

Based on the calculation, the remaining mass after 24 hours is \(3.125 \text{ g}\). Comparing this result to the options provided, the correct choice is (b) \(3.125 \text{ g}\).

Key Concepts

Half-Life

The concept of half-life is pivotal in understanding radioactive decay. It describes the period necessary for a substance to reduce to half of its initial amount. This is not only fundamental in radioactive decay but also finds applications in fields such as pharmacology and ecology. Understanding half-life helps us predict how a quantity of a radioactive isotope diminishes over time. For instance, if the half-life of a substance is 4 hours, like in the exercise, after 4 hours, only half of the substance remains. Another 4 hours elapse, and half of that remaining amount decays further. This process continues, each half-life resulting in a halving of the remaining substance. When dealing with half-life problems, remember:

• Time is crucial: knowing how many half-lives fit into a total time span is the first step.
• The reduction is not linear but exponential: each period halves the remaining material.

Exponential Decay Formula

Radioactive decay is a natural illustration of exponential decay, a process that reduces quantities at a rate proportional to their current value. The formula \[ N = N_0 \left( \frac{1}{2} \right)^n \] is used to compute the remaining quantity of a substance after a certain period, given its half-life.

• \(N_0\) represents the initial mass or quantity.
• \(N\) stands for the remaining mass after time \(t\).
• \(n\) signifies the number of half-lives that have passed.

To find \(n\), you divide the total time by the half-life of the substance. This allows us to understand the extreme efficiency of decay in hours or days, depending on the half-life. Substituting the values will directly give the remaining quantity as exemplified in the solution above. In this example, you have seen how a 24-hour period translates into 6 half-lives if each lasts 4 hours.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and processes that take place within the nucleus of an atom. This includes radioactivity, fission, fusion, and processes such as radioactive decay. Among these, radioactive decay is a crucial concept, enabling the transformation of elements and release of energy. In radioactive decay:

• Atoms of a particular isotope will transform into another element or a different isotope over time.
• This transformation emits radiation, which can be alpha particles, beta particles, or gamma rays.
• The decay rate is described by the half-life, a consistent measure across all conditions.

This field of chemistry not only illuminates the workings of decay but also opens doors to applications in energy production, medicine, and archeology. Understanding nuclear chemistry provides better insights into both the risk and utility of radioactive materials.