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) \(1.042 \mathrm{~g}\)

Short answer

The remaining mass after 24 hours is 3.125 g, so the answer is (b).

Step-by-step solution

Understanding Half-life Concept

The half-life of a radioactive isotope is the time required for half of the isotope to decay. In this problem, the half-life is given as 4 hours.

Calculate Number of Half-lives

To find out how many half-lives occur in 24 hours, divide the total time by the half-life duration: \[ \text{Number of half-lives} = \frac{24 \text{ hours}}{4 \text{ hours/half-life}} = 6 \text{ half-lives} \]

Apply Exponential Decay Formula

The formula for exponential decay is: \[ \text{Remaining mass} = \text{Initial mass} \times \left( \frac{1}{2} \right)^n \]where \( n \) is the number of half-lives. Here, the initial mass is 200 g and \( n = 6 \).

Calculate the Remaining Mass

Substitute the values into the formula: \[ \text{Remaining mass} = 200 \text{ g} \times \left( \frac{1}{2} \right)^6 = 200 \times \frac{1}{64} = 3.125 \text{ g} \]

Select the Correct Answer

From the calculations, we find that the mass remaining after 24 hours is approximately 3.125 g. Therefore, the correct choice is (b) \(3.125 \mathrm{~g}\).

Key concepts

Half-Life Calculation

Half-life is a fundamental concept in nuclear chemistry, representing the period it takes for half of a radioactive substance to decay. This means each half-life reduces the amount of substance by 50%. Understanding how to calculate half-life helps us predict how long a radioactive material remains active, which can be useful in various fields such as nuclear medicine or radiometric dating.
For instance, if you have an isotope with a half-life of 4 hours, it will continue reducing its mass by half every 4 hours. After one half-life (4 hours), half of the original amount remains. After two half-lives (8 hours), a quarter of the original amount remains, and so on. To determine the amount left after multiple half-lives, repeatedly divide the remaining quantity by two until you've covered all the specified periods.

This concept is often used in calculations to find the mass remaining after a given time, as demonstrated in the exercise, where it's shown that you can use the half-life to piece together the remaining amount after several time intervals.

Exponential Decay Formula

The exponential decay formula is used to determine how much of a radioactive substance remains after a certain number of half-lives. It helps ascertain the decay process over time in a mathematical manner. The formula is expressed as:

• \[\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^n\]

where \( n \) stands for the number of half-lives that have elapsed.
To apply this, follow these steps:

• Determine the initial mass of the radioactive isotope.
• Calculate the number of half-lives by dividing the total elapsed time by the duration of one half-life.
• Substitute the initial mass and the number of half-lives into the formula to find the remaining mass.

This exponential function decreases rapidly at first, then slows over time, reflecting how nuclear decay works. In practice, calculating through this formula is essential for forecasting how full a sample of radioactive material will be after a set duration. As in the exercise, determining the half-life enables you to compute that, after 24 hours and 6 half-lives, 3.125 g remains from an initial 200 g.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and properties of atomic nuclei. It is vital for understanding the behavior of radioactive substances and involves studying aspects like radioactive decay, nuclear energy production, and the effects of radiation on materials.
Nuclear reactions differ from ordinary chemical reactions in that they involve changes in the atomic nucleus, often accompanied by the release of a significant amount of energy. These reactions are often categorized as fission (splitting of a nucleus) or fusion (joining of nuclei), both fundamental processes within nuclear reactors and stars.

• Radioactive Decay: It refers to the spontaneous breakdown of an unstable atomic nucleus, resulting in the release of energy and matter from the nucleus.
• Nuclear Stability: Nuclear chemistry extends into understanding why certain isotopes are stable while others are not, depending on the balance of protons and neutrons.

Through the study of nuclear chemistry, we gain insights into energy production and the decay rates of elements, proving essential for fields like nuclear energy production and radioactive dating. In the exercise, the application of nuclear chemistry principles allows one to manage isotopes in practical scenarios, such as calculating decay over time using the half-life and exponential decay formula.