Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive isotope is 3 hours. If the initial mass of the isotope was \(300 \mathrm{~g}\), the mass which remained undecayed in 18 hours would be (a) \(4.68 \mathrm{~g}\) (b) \(2.34 \mathrm{~g}\) (c) \(1.17 \mathrm{~g}\) (d) \(9.36 \mathrm{~g}\)

Step-by-Step Solution

Answer

The remaining mass after 18 hours is approximately 4.68 g, so the answer is (a).

Step 1: Identify the problem components

First, note the given information: the half-life of the isotope is 3 hours, the initial mass is 300 g, and we need to find the remaining mass after 18 hours.

Step 2: Calculate the number of half-lives

Determine how many half-lives occur in 18 hours. Since each half-life is 3 hours, divide 18 hours by 3 hours per half-life: \[ \text{Number of half-lives} = \frac{18}{3} = 6 \]This means that 6 half-lives pass in 18 hours.

Step 3: Apply the half-life formula

With each half-life, the remaining mass is halved. Thus, the remaining mass after 6 half-lives is given by: \[ \text{Remaining Mass} = \text{Initial Mass} \times \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \]

Step 4: Compute the remaining mass

Substitute the values into the formula to find the remaining mass after 18 hours: \[ \text{Remaining Mass} = 300 \times \left( \frac{1}{2} \right)^{6} \]Calculating this results in: \[ \text{Remaining Mass} = 300 \times \frac{1}{64} = 4.6875 \text{ g} \]

Key Concepts

Half-Life Calculation

Understanding how to calculate half-life is critical when working with radioactive substances. The half-life of a radioactive isotope refers to the time it takes for half of the initial amount of the substance to decay. In this context, "decay" means the transformation of the material into a more stable form.
When dealing with half-lives, you can use a simple calculation. If you know the half-life of an isotope, and the time period over which you are observing the decay, you can determine how many half-life periods have passed. For example, if the half-life is 3 hours and you observe the decay over 18 hours, you divide 18 by 3. This reveals that 6 half-life periods have elapsed.

• A half-life period is fixed for each isotope and is characterized by a constant rate of decay.
• Knowing how to count half-lives can help you predict how much of a radioactive substance will remain undecayed after a given period.

The concept of half-life is at the heart of many calculations involving radioactive decay.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are atoms with unstable nuclei that decay over time. As they decay, they emit radiation and transform into other elements.
Each radioactive isotope has a characteristic half-life, which can range from fractions of a second to millions of years. This half-life is a reflection of the isotope's stability and influences how long it remains active or hazardous.

• Isotopes are variants of elements with the same number of protons but different numbers of neutrons.
• Radioactive isotopes are used in many fields, including medicine for imaging and treatment, as well as in dating materials through radiocarbon dating.

Understanding how these isotopes decay helps scientists to safely manage radiation and harness their properties for beneficial uses.

Mass Decay Formula

The mass decay formula is a powerful tool used to calculate the remaining mass of a radioactive substance after a certain number of half-lives. The formula is:
\[ \text{Remaining Mass} = \text{Initial Mass} \times \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \]
This formula essentially applies the halving effect of each half-life iteratively. Every time a half-life passes, the remaining mass of the isotope is halved.

• Start with the initial mass of the isotope.
• Determine the number of half-life periods that have occurred.
• Apply the formula to calculate the remaining mass.

Understanding this formula is essential for predicting how much of a radioactive substance will remain at any point in time, which is crucial in fields that deal with radioactive materials.