Hard Half-Life Calculations Practice Questions

Concept Explanation

Hard half-life calculations involve determining the time required for a substance to decrease to half of its initial value, typically requiring the use of logarithmic functions and first-order kinetic equations. While basic problems use whole-number multiples of half-lives, advanced calculations require the integrated rate law to find precise values for time, initial concentration, or remaining mass when the intervals are non-integer multiples. According to Wikipedia, the concept is fundamental in nuclear physics, pharmacology, and chemical kinetics.

For most radioactive decay and first-order chemical reactions, the relationship between the remaining amount ($N_t$), the initial amount ($N_0$), and time ($t$) is governed by the formula:

$N_t = N_0 \cdot (1/2)^{(t/t_{1/2})}$

In more complex scenarios, such as those found in reaction order practice, we use the decay constant ($\lambda$ or $k$). The relationship between the decay constant and half-life is defined by:

$k = \ln(2) / t_{1/2} \approx 0.693 / t_{1/2}$

By substituting this into the first-order integrated rate law, we get:

$\ln(N_t / N_0) = -kt$

Mastering these hard half-life calculations requires proficiency in manipulating natural logarithms ($\ln$) and exponents. This is particularly vital for students who need to study for exams in engineering school or medical programs where dosage decay and structural integrity are calculated to high precision.

Solved Examples

The following examples demonstrate how to handle non-integer half-lives and solve for variables other than the final amount.

Example 1: Solving for Time ($t$)

A sample of Iodine-131 (half-life of 8.02 days) has an initial activity of 500 MBq. How long will it take for the activity to drop to 35 MBq?

1. Identify the knowns: $N_0 = 500$, $N_t = 35$, $t_{1/2} = 8.02$.

2. Calculate the decay constant ($k$): $k = 0.693 / 8.02 = 0.0864 \text{ days}^{-1}$.

3. Use the integrated rate law: $\ln(35 / 500) = -(0.0864)t$.

4. Solve for the ratio: $\ln(0.07) = -2.659$.

5. Calculate $t$: $-2.659 = -0.0864t \Rightarrow t = 30.78 \text{ days}$.

Example 2: Solving for Initial Mass ($N_0$)

After 45 hours, a sample of a radioisotope with a half-life of 12 hours has decayed until only 4.2 grams remain. What was the starting mass?

1. Identify the knowns: $t = 45$, $t_{1/2} = 12$, $N_t = 4.2$.

2. Determine the number of half-lives ($n$): $n = 45 / 12 = 3.75$.

3. Apply the formula: $4.2 = N_0 \cdot (0.5)^{3.75}$.

4. Calculate $(0.5)^{3.75}$: $0.0743$.

5. Solve for $N_0$: $N_0 = 4.2 / 0.0743 = 56.51 \text{ grams}$.

Example 3: Solving for Half-Life ($t_{1/2}$)

A new synthetic element decays from 100g to 12.5g in exactly 14.2 minutes. However, a second impurity in the sample decays from 80g to 15g in the same time. Find the half-life of the impurity.

1. Identify knowns for the impurity: $N_0 = 80$, $N_t = 15$, $t = 14.2$.

2. Use the log form: $\ln(15 / 80) = -k(14.2)$.

3. Calculate the ratio: $\ln(0.1875) = -1.674$.

4. Solve for $k$: $-1.674 = -14.2k \Rightarrow k = 0.1179 \text{ min}^{-1}$.

5. Find $t_{1/2}$: $t_{1/2} = 0.693 / 0.1179 = 5.88 \text{ minutes}$.

Practice Questions

1. A medical tracer has a half-life of 6.0 hours. If a patient is injected with 200 mg, how much remains after 26 hours?

2. An archaeological artifact contains 15% of its original Carbon-14. If the half-life of C-14 is 5,730 years, how old is the artifact?

3. A substance decays by 99% in 40 days. Calculate its half-life in days.

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4. Bismuth-210 has a half-life of 5.0 days. If you start with 1.50 grams, how many milligrams are left after 17.2 days?

5. A technician finds that a sample of Phosphorus-32 (half-life 14.3 days) has decreased to 1/10th of its original activity. How much time has passed?

6. Two different isotopes, A and B, start with the same mass. Isotope A has a half-life of 10 years, and Isotope B has a half-life of 20 years. After 40 years, what is the ratio of the mass of A to the mass of B?

7. A chemical reaction follows first-order kinetics. If the concentration of a reactant drops from 1.20 M to 0.35 M in 45 minutes, what is the half-life of the reaction?

8. A radioactive source is considered safe when its activity drops to 0.1% of its initial level. If the half-life is 2.5 years, how long must the source be stored?

9. Radon-222 has a half-life of 3.8 days. If a basement has 400 Bq/m³ of Radon, how long will it take to reach the EPA action level of 148 Bq/m³?

10. A sample of Tritium ($t_{1/2} = 12.3$ years) was stored for 50 years. What percentage of the original sample has decayed?

Answers & Explanations

1. Answer: 9.92 mg. $N_t = 200 \cdot (0.5)^{(26/6)}$. $n = 4.333$. $N_t = 200 \cdot 0.0496 = 9.92$ mg.

2. Answer: 15,685 years. Use $\ln(0.15) = -k(t)$. $k = 0.693 / 5730 = 0.0001209$. $\ln(0.15) = -1.897$. $t = -1.897 / -0.0001209 = 15,685$.

3. Answer: 6.02 days. If 99% decayed, 1% remains ($N_t/N_0 = 0.01$). $\ln(0.01) = -k(40)$. $-4.605 = -40k \Rightarrow k = 0.1151$. $t_{1/2} = 0.693 / 0.1151 = 6.02$ days.

4. Answer: 140.6 mg. $N_t = 1.5 \cdot (0.5)^{(17.2/5)} = 1.5 \cdot 0.09375 = 0.1406$ g, which is 140.6 mg.

5. Answer: 47.5 days. $\ln(0.1) = -k(t)$. $k = 0.693 / 14.3 = 0.04846$. $-2.302 = -0.04846t \Rightarrow t = 47.5$ days.

6. Answer: 1:4. For A: $n = 40/10 = 4$ half-lives ($1/16$ remains). For B: $n = 40/20 = 2$ half-lives ($1/4$ remains). Ratio $1/16 : 1/4 = 1:4$.

7. Answer: 25.4 minutes. $\ln(0.35/1.20) = -k(45)$. $\ln(0.2916) = -1.232$. $k = 1.232 / 45 = 0.02738$. $t_{1/2} = 0.693 / 0.02738 = 25.4$ min.

8. Answer: 24.9 years. $\ln(0.001) = -k(t)$. $k = 0.693 / 2.5 = 0.2772$. $-6.908 = -0.2772t \Rightarrow t = 24.9$ years.

9. Answer: 5.45 days. $\ln(148/400) = -k(t)$. $k = 0.693 / 3.8 = 0.1824$. $\ln(0.37) = -0.994$. $t = -0.994 / -0.1824 = 5.45$ days.

10. Answer: 94.1%. $N_t/N_0 = (0.5)^{(50/12.3)} = (0.5)^{4.065} = 0.0596$. This means 5.9% remains, so $100 - 5.9 = 94.1$% has decayed.

Quick Quiz

Frequently Asked Questions

What is the difference between a physical half-life and a biological half-life?

Physical half-life is the time required for a radioactive isotope to decay to half its mass through nuclear processes. Biological half-life is the time it takes for a living organism to eliminate half of a substance through physiological processes like excretion or metabolism. For more on biological applications, check CDC radiation resources.

Can a half-life be zero or negative?

No, half-life must always be a positive value because it represents a duration of time. A zero half-life would imply instantaneous disappearance, while a negative half-life would imply the substance is growing exponentially rather than decaying.

Why is the number 0.693 used in half-life calculations?

The number 0.693 is the approximate value of the natural logarithm of 2 ($ln(2)$). It appears because the half-life is defined as the time when the ratio of remaining substance to initial substance is 0.5, and the inverse of $ln(0.5)$ is $-ln(2)$.

Does temperature affect the half-life of radioactive isotopes?

Unlike chemical reactions where temperature affects the rate constant as seen in the Arrhenius equation, radioactive half-life is an intrinsic nuclear property. It is not affected by temperature, pressure, or chemical bonding.

How do you calculate half-life for second-order reactions?

In second-order reactions, the half-life is not constant and depends on the initial concentration ($N_0$). The formula is $t_{1/2} = 1 / (k \cdot N_0)$, meaning the half-life increases as the substance is consumed.

What is the rule of 70 in relation to half-life?

The rule of 70 is a simplified way to estimate the doubling time or half-life of a quantity growing or decaying at a constant percentage rate. By dividing 70 by the percentage growth/decay rate, you get an approximation of the time required for the value to halve or double.

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