Problem 39 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive nuclide is 10 months. The fraction of the substance left behind after 40 months is (a) \(1 / 2\) (b) \(1 / 4\) (c) \(1 / 8\) (d) \(1 / 16\)

Step-by-Step Solution

Answer

The answer is (d) \(1/16\).

Step 1: Understanding Half-life

Half-life is the time required for a quantity to reduce to half of its initial value. For this problem, the half-life of the substance is given as 10 months.

Step 2: Calculate Number of Half-lives

To find out how many half-lives have passed in 40 months, we divide the total time by the duration of one half-life.\[ \text{Number of half-lives} = \frac{40\, \text{months}}{10\, \text{months/half-life}} = 4 \]

Step 3: Apply the Half-life Formula

The fraction of a substance remaining after a number of half-lives is given by the formula: \[ \left(\frac{1}{2}\right)^n \]where \( n \) is the number of half-lives. Substituting the value of \( n \) from the previous step:\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

Step 4: Select the Correct Answer

The fraction of the substance left after 40 months is \( \frac{1}{16} \), which corresponds to option (d).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This can include particles such as alpha or beta particles or even electromagnetic radiation like gamma rays. As nuclei decay, they transform into different elements or different isotopes of the same element. Radioactive decay is a random process at the level of single atoms, meaning we cannot predict exactly when a particular atom will decay. However, when dealing with a large number of atoms, the decay rate is predictable and measured by the half-life. The process is fundamental in nuclear physics and has applications in fields such as radiometric dating, nuclear medicine, and energy generation.

Nuclide

A nuclide is a specific type of atom characterized by its number of protons, neutrons, and the specific energy state of its nucleus. Each nuclide is associated with specific properties, including half-life and types of decay. For instance, carbon-14 is a well-known radioactive nuclide used in radiocarbon dating because of its well-defined half-life. Nuclides are classified by:

• Their atomic number (number of protons)
• Their mass number (total number of protons and neutrons)
• Their nuclear energy state

Understanding nuclides is crucial when studying radioactive substances because it helps you determine how the element will behave under certain conditions.

Fraction Remaining

The fraction of a radioactive substance remaining after a given number of half-lives is critical in understanding how much of the material is still active or undecayed. This fraction is calculated using the formula \[ \left(\frac{1}{2}\right)^n \]where \( n \) is the number of half-lives that have elapsed. Each half-life reduces the substance to half of its previous quantity. For example:

• After 1 half-life: \( \frac{1}{2} \)
• After 2 half-lives: \( \frac{1}{4} \)
• After 3 half-lives: \( \frac{1}{8} \)
• After 4 half-lives: \( \frac{1}{16} \)

This exponential decay pattern makes it possible to predict the amount of material remaining at any given time.

Number of Half-Lives

To determine how much of a radioactive substance remains, you need to know how many half-lives have passed. The number of half-lives is calculated by dividing the total time of decay by the half-life period of the substance:\[ \text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life period}} \]For example, if the half-life of a substance is 10 months and 40 months have passed, then the number of half-lives is:\[ \frac{40}{10} = 4 \]After finding the number of half-lives, you can then determine the fraction remaining of the substance using the half-life formula. Calculating the number of half-lives helps in forecasting how substances like medication or radioactive tracers will behave over time.