Medium Half-Life Calculations Practice Questions

Concept Explanation

Medium half-life calculations involve determining the time required for a substance to decrease to half of its initial amount, often requiring the use of logarithmic formulas for non-integer intervals.

In nuclear physics and chemistry, the half-life ($t_{1/2}$) is a constant that describes the rate of decay for a specific radioactive isotope or a first-order chemical reaction. While basic problems might involve simple doubling or halving of time intervals, medium-level problems usually require the integrated rate law or the specific half-life formula: $N(t) = N_0(1/2)^{t/h}$, where $N(t)$ is the remaining amount, $N_0$ is the initial amount, $t$ is the elapsed time, and $h$ is the half-life. To solve for time or the decay constant ($k$), we use the relationship $k = 0.693 / t_{1/2}$. Understanding these calculations is essential for students learning reaction order practice questions, as half-life is a defining characteristic of first-order kinetics.

According to the Environmental Protection Agency (EPA), isotopes like Iodine-131 have specific half-lives that dictate how long they remain hazardous in the environment. Mastery of these calculations allows scientists to predict the safety of medical treatments and the age of archaeological finds through carbon dating. If you find yourself struggling with the mathematical rigor of these problems, learning how to study for exams with poor memory can help you internalize the necessary logarithmic identities and constants.

Solved Examples

The following examples demonstrate how to manipulate the half-life formula to find different variables.

1. Finding the Remaining Mass: A sample of Phosphorus-32 has a half-life of 14.3 days. If you start with 100 grams, how much remains after 42.9 days?

   1. Identify the variables: $N_0 = 100g$, $t = 42.9$, $h = 14.3$.

   2. Calculate the number of half-lives ($n$): $n = t / h = 42.9 / 14.3 = 3$.

   3. Apply the formula: $N(t) = 100 \u00d7 (1/2)^3$.

   4. Solve: $100 \u00d7 0.125 = 12.5$ grams.

2. Calculating the Half-Life: A radioactive substance decays from 80g to 10g over the course of 24 hours. What is its half-life?

   1. Determine the fraction remaining: $10 / 80 = 1/8$.

   2. Relate the fraction to half-lives: $(1/2)^n = 1/8$, so $n = 3$.

   3. Set up the time equation: $3 = 24 \text{ hours} / h$.

   4. Solve for $h$: $h = 24 / 3 = 8$ hours.

3. Solving for Time with Logarithms: Sodium-24 has a half-life of 15 hours. How long will it take for a 50.0g sample to decay to 12.0g?

   1. Use the formula: $12.0 = 50.0 \u00d7 (0.5)^{t/15}$.

   2. Divide by initial amount: $0.24 = (0.5)^{t/15}$.

   3. Take the natural log of both sides: $\ln(0.24) = (t/15) \u00d7 \ln(0.5)$.

   4. Isolate $t$: $t = 15 \u00d7 [\ln(0.24) / \ln(0.5)] \approx 15 \u00d7 2.0588 = 30.88$ hours.

Practice Questions

1. A sample of Radon-222 has a half-life of 3.8 days. If a basement contains 200 mg of Radon-222, how many milligrams will remain after 11.4 days?

2. The half-life of Carbon-14 is approximately 5,730 years. An ancient piece of wood is found to have only 25% of the Carbon-14 found in living trees. How old is the wood?

3. Barium-131 has a half-life of 11.5 days. If a laboratory has 40.0g of Barium-131, how much will be left after 46 days?

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4. A 64g sample of a radioactive isotope decays to 2g in 25 days. What is the half-life of this isotope?

5. Technetium-99m, used in medical imaging, has a half-life of 6.0 hours. If a patient is injected with 30 mg, how much remains in their body after 15 hours?

6. A substance has a decay constant ($k$) of 0.03465 day\u207b\u00b9. Calculate the half-life of this substance.

7. If the half-life of a material is 50 years, what percentage of the original material will remain after 125 years?

8. A sample of Iodine-131 (half-life = 8.02 days) has an initial activity of 500 Bq. How many days will it take for the activity to drop to 100 Bq?

9. Cobalt-60 is used in cancer therapy and has a half-life of 5.27 years. If a source is 10 years old, what fraction of its original activity remains?

10. An unknown isotope decays from 150 mg to 120 mg in 5 hours. What is the half-life of the isotope in hours?

Answers & Explanations

Review the detailed solutions below to check your work and understand the logic behind each calculation.

• 1. Answer: 25 mg. 11.4 days divided by 3.8 days equals exactly 3 half-lives. $(1/2)^3 = 1/8$. $200 \text{ mg} / 8 = 25 \text{ mg}$.

• 2. Answer: 11,460 years. 25% remaining means 2 half-lives have passed ($0.5 \u00d7 0.5 = 0.25$). $2 \u00d7 5,730 = 11,460$ years.

• 3. Answer: 2.5g. 46 days / 11.5 days = 4 half-lives. $40.0 \u00d7 (1/2)^4 = 40.0 / 16 = 2.5g$.

• 4. Answer: 5 days. The fraction remaining is $2/64 = 1/32$. Since $(1/2)^5 = 1/32$, 5 half-lives have passed. $25 \text{ days} / 5 = 5$ days.

• 5. Answer: 5.3 mg. Use $N = N_0(0.5)^{t/h}$. $N = 30(0.5)^{15/6} = 30(0.5)^{2.5}$. $30 \u00d7 0.1767 = 5.30$ mg.

• 6. Answer: 20 days. Using the formula $t_{1/2} = 0.693 / k$, we get $0.693 / 0.03465 = 20$.

• 7. Answer: 17.68%. $n = 125 / 50 = 2.5$ half-lives. $(0.5)^{2.5} = 0.17677$, which is $17.68\%$.

• 8. Answer: 18.6 days. $100 = 500(0.5)^{t/8.02}$. $0.2 = (0.5)^{t/8.02}$. $\ln(0.2) = (t/8.02)\ln(0.5)$. $t = 8.02 \u00d7 [-1.609 / -0.693] = 18.6$ days.

• 9. Answer: 0.268 (or 26.8%). $n = 10 / 5.27 = 1.8975$ half-lives. $(0.5)^{1.8975} = 0.268$.

• 10. Answer: 15.53 hours. $120 = 150(0.5)^{5/h}$. $0.8 = (0.5)^{5/h}$. $\ln(0.8) = (5/h)\ln(0.5)$. $-0.2231 = (5/h)(-0.6931)$. $h = (5 \u00d7 -0.6931) / -0.2231 = 15.53$ hours.

Quick Quiz

Frequently Asked Questions

What is the difference between a decay constant and a half-life?

The decay constant is the probability of a nucleus decaying per unit time, while the half-life is the actual time it takes for half of the sample to decay. They are inversely related by the factor of the natural log of 2 (approximately 0.693).

Can half-life be affected by temperature or pressure?

For radioactive decay, the half-life is a nuclear property and is not affected by external physical conditions like temperature or pressure. However, for chemical reactions, the half-life can change if the temperature affects the reaction rate constant, as discussed in the Arrhenius equation practice questions.

How is half-life used in carbon dating?

Scientists measure the ratio of Carbon-14 to Carbon-12 in organic remains and compare it to the ratio in the atmosphere. Since Carbon-14 decays with a known half-life of 5,730 years, the difference indicates how long ago the organism died.

Does a half-life ever reach zero?

Mathematically, the amount of a substance never reaches zero because it is an asymptotic relationship where you are always halving the remainder. In reality, once only a few atoms are left, the statistical nature of decay means the sample will eventually disappear entirely.

Why is half-life important in pharmacology?

Health professionals use half-life to determine how long a medication stays active in a patient's bloodstream. This information is critical for establishing dosing schedules to maintain therapeutic levels without causing toxicity.

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