Half Life Chemistry Problems - Nuclear Radioactive Decay Calculations Practice Examples

Summary

This video provides a clear and practical guide to solving radioactive decay and half-life problems, commonly encountered in chemistry courses. It explains the concept of half-life, demonstrating how to calculate the remaining amount of a substance after a given time using both the step-by-step half-life method and the exponential decay equation. Several examples are covered, including iodine-131, sodium-24, and oxygen-15, highlighting how to find remaining fractions, decay times, and half-lives. The video emphasizes shortcuts, such as using powers of two for multiple half-lives, and offers detailed calculations for both conceptual understanding and precise results, making it accessible and engaging for learners.

Takeaways

• 😀 Half-life represents the time it takes for half of a substance to decay.
• 😀 You can calculate remaining substance using either conceptual counting of half-lives or a decay equation.
• 😀 The decay equation for first-order kinetics is A = A0 * e^(-k * t), where k is the decay constant.
• 😀 The decay constant k is calculated as ln(2) divided by the half-life.
• 😀 For conceptual calculations, repeatedly halve the sample for each half-life period to determine remaining amount.
• 😀 Time required for a certain amount to decay can be calculated using t = ln(A/A0) / -k.
• 😀 The fraction remaining after n half-lives is (1/2)^n, which can be converted to percentage or decimal form.
• 😀 When solving for half-life from decay data, first determine the rate constant k using k = ln(A/A0) / -t, then t1/2 = ln(2)/k.
• 😀 Conceptual methods are fast and intuitive for simple numbers, while the equation method is useful for non-integer or precise calculations.
• 😀 Converting percentages to decimals or fractions helps in comparing results to multiple-choice answers accurately.
• 😀 Counting half-lives provides a quick check and can be combined with equations to verify results.
• 😀 Understanding both conceptual and equation approaches ensures flexibility and accuracy in solving radioactive decay problems.

Q & A

What is the definition of half-life in radioactive decay?

Half-life is the time it takes for half of a radioactive substance to decay into another element or isotope.

How much of a 200 g sample of iodine-131 remains after 32 days if its half-life is 8 days?

After 32 days, 12.5 g of iodine-131 will remain. This is determined by successively halving the sample every 8 days or using the decay equation.

What is the decay constant (k) and how is it calculated for radioactive substances?

The decay constant k represents the rate of decay and is calculated as k = ln(2) / t₁/₂, where t₁/₂ is the half-life of the substance.

How can you determine the time it takes for a radioactive sample to decay to a certain amount using an equation?

Use the equation ln(A_t/A_0) = -k t, where A_0 is the initial amount, A_t is the final amount, k is the decay constant, and t is the time. Rearrange to solve for t: t = -ln(A_t/A_0)/k.

If 750 g of sodium-24 decays from an initial 800 g sample with a half-life of 15 hours, how long will it take?

It will take approximately 60 hours for 750 g to decay, which corresponds to 4 half-lives of sodium-24.

How do you calculate the fraction of a radioactive sample remaining after a number of half-lives?

The fraction remaining is calculated as (1/2)^n, where n is the number of half-lives that have passed.

After five half-lives of oxygen-15, what fraction of the sample remains?

After five half-lives, 1/32 of the sample remains, which is equivalent to 3.125% of the original amount.

How can you calculate the half-life of a substance if you know the initial and final amounts and the total time elapsed?

First, calculate the decay constant k using k = -ln(A_t/A_0)/t, then determine the half-life with t₁/₂ = ln(2)/k.

For a 512 g sample decaying to 4 g over 35 days, what is the half-life of the substance?

The half-life of the substance is approximately 5 days, determined either by counting the number of half-lives or using the decay equation.

Why is it sometimes more convenient to use stepwise halving instead of the decay equation?

Stepwise halving is easier and faster for 'nice' numbers or when the half-lives evenly divide the total time, allowing for quick estimates without complex calculations.

How can percentages be converted to fractions when calculating remaining radioactive material?

Divide the percentage by 100 to convert to a decimal, then simplify or compare with known fractions, such as 0.125 = 1/8 or 0.03125 = 1/32.

What is the importance of the negative sign in the radioactive decay equation A_t = A_0 e^{-k t}?

The negative sign indicates that the quantity of the substance is decreasing over time due to decay.