Half Life Chemistry Problems - Nuclear Radioactive Decay Calculations Practice Examples
TLDRThe video script provides an in-depth exploration of the concept of half-life, particularly in the context of chemistry and radioactive decay. It begins with a practical example involving iodine-131, demonstrating how to calculate the remaining mass after a given number of half-lives, both through direct calculation and using the formula involving the rate constant (k). The script then covers similar calculations for sodium-24 and oxygen-15, highlighting the use of equations to find decay times and remaining fractions after specific numbers of half-lives. The video concludes with determining the half-life of an unknown element, element X, from decay data. Throughout, the script emphasizes the importance of understanding half-life as a fundamental concept in chemistry and physics, and it offers both conceptual and mathematical approaches to tackle related problems.
Takeaways
- 🧪 The half-life of a substance represents the time it takes for half of the substance to decay into another element.
- ⏳ Iodine-131 has a half-life of 8 days, meaning that after 8 days, half of the initial amount will have transformed into another element.
- 📉 To calculate the remaining amount after a certain period, you can use either a simple step-by-step decay calculation or a mathematical equation involving the rate constant (K).
- 🔢 The rate constant (K) is found by dividing the natural logarithm of 2 by the half-life of the substance.
- 📌 The formula for radioactive decay is AF = Ao * e^(-KT), where AF is the final amount, Ao is the initial amount, K is the rate constant, and T is time.
- 🕒 For the example of Sodium-24, it takes 60 hours for 750 grams to decay to 50 grams, which is four half-lives of 15 hours each.
- 🕗 The half-life of Oxygen-15 is 2 minutes, and after five half-lives, which is 10 minutes, 1/32 of the original sample remains.
- 🔧 To convert a percentage to a decimal, divide by 100 or move the decimal point two places to the left.
- 🔎 The fraction remaining after a certain number of half-lives can be found by dividing 1 by 2 raised to the power of the number of half-lives.
- 🕒 For Element X, which decays from 512 grams to 4 grams in 35 days, the half-life can be calculated by dividing the total time by the number of half-lives it took to reach the final amount.
- 📐 The half-life can also be determined using the formula half-life = ln(2) / K, where K is the rate constant calculated from the decay equation.
Q & A
What is the half-life of iodine-131 mentioned in the video?
-The half-life of iodine-131 is 8 days.
How much iodine-131 will remain after 32 days if you start with 200 grams?
-After 32 days, 12.5 grams of iodine-131 will remain.
What is the formula for calculating the remaining amount of a substance after decay?
-The formula is AF = Ao * e^(-kt), where AF is the final amount, Ao is the initial amount, k is the rate constant, and t is time.
What is the rate constant (K) for iodine-131 given its half-life?
-The rate constant K for iodine-131 is 0.08664 when calculated using the natural log of 2 divided by the half-life of 8 days.
How many half-lives does it take for 800 grams of sodium-24 to decay to 50 grams?
-It takes four half-lives for 800 grams of sodium-24 to decay to 50 grams.
What is the half-life of sodium-24?
-The half-life of sodium-24 is 15 hours.
How long will it take for 750 grams of sodium-24 to decay?
-It will take 60 hours for 750 grams of sodium-24 to decay.
What fraction of a sample of oxygen-15 will remain after five half-lives?
-After five half-lives, 1/32 of a sample of oxygen-15 will remain.
What is the formula to convert a percentage to a decimal?
-To convert a percentage to a decimal, divide by 100 or move the decimal point two units to the left.
How many half-lives does it take for a 512 gram sample of element X to decay to 4 grams?
-It takes seven half-lives for a 512 gram sample of element X to decay to 4 grams.
What is the half-life of element X given its decay from 512 grams to 4 grams over 35 days?
-The half-life of element X is approximately 5 days.
How can you calculate the rate constant (K) for radioactive decay?
-The rate constant (K) can be calculated using the formula K = ln(2) / half-life of the substance.
Outlines
🔬 Understanding Half-Life: Iodine-131 Example
This paragraph introduces the concept of half-life in the context of radioactive decay, specifically using iodine-131 with a half-life of 8 days as an example. It explores two methods to determine the remaining amount of iodine-131 after 32 days: a straightforward conceptual approach and a mathematical equation involving the rate constant K. The conceptual approach involves sequentially dividing the initial amount by two for each half-life period, resulting in 12.5 grams remaining after 32 days. The mathematical approach calculates the rate constant K as the natural logarithm of 2 divided by the half-life, then uses this in an equation to find the final amount (AF), which closely matches the conceptual method's result.
⏳ Time to Decay for Sodium-24
The second paragraph discusses the decay time for a sample of sodium-24 with a half-life of 15 hours. It starts with 800 grams of sodium-24 and asks how long it will take for 750 grams to decay, leaving 50 grams unchanged. The explanation involves counting the number of half-lives (four in this case) and multiplying this by the half-life duration to get a total time of 60 hours. An alternative method uses the natural logarithm to find the rate constant K and then solve for time (t), which confirms the 60-hour duration.
📉 Fraction Remaining After Multiple Half-Lives: Oxygen-15
The third paragraph deals with calculating the fraction of a sample of oxygen-15 that remains after five half-lives, given a half-life of 2 minutes. It explains a step-by-step reduction of the sample size by half for each half-life, resulting in 3.125% remaining after five half-lives. The paragraph also demonstrates how to convert this percentage to a decimal and then to a fraction, identifying 1/32 as the correct fraction remaining. Additionally, it presents a faster method for such calculations by realizing that the fraction remaining after 'n' half-lives is 1 divided by 2 to the power of 'n'.
⚗️ Determining the Half-Life of Element X
The final paragraph is about finding the half-life of an element referred to as 'Element X'. Given that it takes 35 days for a 512-gram sample to decay to 4 grams, the paragraph outlines a method to determine the number of half-lives that occur in this process (seven half-lives). It then calculates the half-life by dividing the total time by the number of half-lives, resulting in a half-life of five days. The paragraph also shows how to use the natural logarithm to find the rate constant K and subsequently the half-life, which corroborates the initial finding.
Mindmap
Keywords
💡Half-Life
💡Radioactive Decay
💡Rate Constant (k)
💡First Order Kinetics
💡Natural Logarithm (ln)
💡Sodium-24
💡Iodine-131
💡Exponential Decay
💡Half-Life Equation
💡Sample Decay
💡Element X
Highlights
The video discusses the concept of half-life, particularly in the context of chemistry and radioactive decay.
Iodine-131 has a half-life of 8 days, and the problem involves calculating the remaining amount after 32 days.
Two methods are presented for solving decay problems: one conceptual and one using an equation.
The conceptual method involves understanding that half-life is the time for half of a substance to decay.
Using the conceptual method, it's determined that 12.5 grams of Iodine-131 will remain after 32 days.
The rate constant K is introduced as a way to solve decay problems using an equation, where K = ln(2) / half-life.
The equation AF = Ao * e^(-KT) is used to find the final amount of a substance after decay, where AF is the final amount, Ao is the initial amount, K is the rate constant, and T is time.
The rate constant K for Iodine-131 is calculated to be approximately 0.08664, which is then used in the equation to find the final amount remaining.
An example with Sodium-24 is used to illustrate how to find the time it takes for a certain amount of a substance to decay.
The half-life of Sodium-24 is given as 15 hours, and the problem is to find out how long it takes for 750 grams to decay, leaving 50 grams.
By using the half-life and the decay equation, it's shown that it takes 60 hours for 750 grams of Sodium-24 to decay to 50 grams.
Another example involves Oxygen-15 with a half-life of 2 minutes, and the problem is to find the fraction remaining after five half-lives.
The fraction remaining after five half-lives is found to be 1/32 by both conceptual reasoning and mathematical calculation.
A faster method for finding the fraction remaining after multiple half-lives involves dividing 1 by 2 raised to the power of the number of half-lives.
A problem involving Element X with a sample size of 512 grams decaying to 4 grams in 35 days is used to determine the half-life of Element X.
By calculating the rate constant K and using the decay equation, the half-life of Element X is found to be approximately 5 days.
The video concludes with a summary of the methods and equations used for solving radioactive decay problems.
Transcripts
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