Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

If a substance with half-life of 3 days is taken to another place in 12 days. What is the amount of substance left now? (a) \(1 / 8\) (b) \(1 / 32\) (c) \(1 / 4\) (d) \(1 / 16\)

Step-by-Step Solution

Answer

The amount of substance left is \( \frac{1}{16} \), which is option (d).

Step 1: Understand the concept of half-life

Half-life is the time required for a quantity to reduce to half of its initial amount. In this problem, the substance's half-life is 3 days.

Step 2: Calculate the number of half-life periods in 12 days

We need to determine how many half-life periods fit into 12 days. Since each half-life period is 3 days, we calculate: \[ \frac{12}{3} = 4 \] This means there are 4 half-life periods in 12 days.

Step 3: Determine the remaining fraction of the substance

The remaining amount of a substance after a number of half-life periods can be calculated using the formula:\[ \left(\frac{1}{2}\right)^N \] where \( N \) is the number of half-life periods. In this case:\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \] Thus, after 12 days, \( \frac{1}{16} \) of the substance remains.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. Over time, this results in the transformation of elements until a stable form is reached. During radioactive decay, particles such as alpha particles, beta particles, and gamma rays are often emitted. The process itself is random at the level of single atoms, making it impossible to predict when a particular atom will decay. However, for a large number of identical atoms, the decay rate becomes statistically predictable as their numbers decrease over time. This predictability allows us to use mathematical concepts to analyze the reduction of the substance's quantity.

Half-Life Period

The half-life period is a critical concept in the study of radioactive materials. It refers to the time needed for half of a substance to decay, which effectively reduces the amount of the substance by 50%. For example, if a substance with a half-life of 3 days begins with 10 grams, after 3 days, only 5 grams will remain. After another 3 days, only 2.5 grams will remain, and so on. This continuous halving of the original amount is a key idea in understanding how substances decay over time. It's important to note that the half-life is constant for a given substance, meaning that it doesn't change regardless of how much of the substance remains at any point in the decay process.

Exponential Decay

Exponential decay describes the manner in which the quantity of a substance decreases over time. This process follows the rule that a fixed percentage of the substance decays in each equal time period, leading to the characteristic smooth and continuous curve that starts with a rapid decrease that slows over time. Mathematically, this behavior is represented by the formula: \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T} \] where:

• \(N(t)\) is the remaining quantity of the substance after time \(t\),
• \(N_0\) is the initial quantity,
• \(T\) is the half-life period.

Understanding exponential decay provides insight into how the rate of decay decreases over time, showcasing the "half" nature of the decay curve.

Remaining Substance

To find out how much of a substance is left after a set period, especially in cases of decay, involves working with the exponential decay formulas. Once you know the number of half-life periods that have passed, you can calculate the remaining substance easily. In the case of half-life calculations, each half-life period effectively divides the remaining substance by two. For instance, if a substance goes through 4 half-life periods, the remaining substance is calculated as: \[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \] This means that only 1/16 of the original amount of the substance remains. Such calculations are vital in fields ranging from nuclear physics to environmental science, as they help quantify the persistence of materials and their potential impacts.