Half Life Chemistry Problems - Nuclear Radioactive Decay Calculations Practice Examples
Summary
TLDRThis 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.
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