20.4 Kinetics of Nuclear Decay
PRE-HEALTH PREP
Kinetics of Nuclear Decay
Half-Life and Radiocarbon Dating
General Chemistry 20.4
Lesson Overview
Nuclear decay kinetics follows first-order behavior exclusively, linking the half-life and rate constant through the relationship t1/2 = 0.693/k. Problems involving radioactive decay can be solved by repeated halving when whole numbers of half-lives are involved, or by applying the first-order integrated rate law for non-whole half-life scenarios. Radiocarbon dating extends these principles to determine the age of once-living materials by comparing their current carbon-14 activity to the steady-state level maintained during life, with an effective range of approximately 50,000–70,000 years.
This lesson is part of the General Chemistry series (Chapter 20.4). The first portion reviews the kinetics of nuclear decay, and the second applies that framework to radiocarbon dating. Because all spontaneous nuclear decay processes are first order, the mathematics is more straightforward than in the chemical kinetics chapter — there is no zero-order or second-order case to worry about.
Why All Nuclear Decay Is First Order
All spontaneous nuclear decay is first order. This is a significant simplification compared to the chemical kinetics chapter, where you had to contend with zero-order, first-order, and second-order processes. With nuclear decay, every pathway follows the same kinetic form, so the entire chapter reduces to mastering one equation and its implications.
Because nuclear decay is first order, it obeys the first-order integrated rate law:
where N is the amount of radioactive nuclide remaining at time t, N₀ is the initial amount, and k is the first-order rate constant.
Closely related to the rate constant is the half-life, t1/2. In practice, radioactive nuclides are almost always characterized by their half-life rather than their rate constant — but the two are interconvertible through a simple relationship derived below.
Deriving the Half-Life Equation
Starting from the first-order integrated rate law and rearranging, we can write:
The half-life is defined as the time required for exactly half of the radioactive nuclei in a sample to decay. At that moment, the amount remaining is N₀/2, so the ratio N/N₀ equals 1/2:
Solving for t1/2 and noting that ln(1/2) = −0.693 gives the half-life equation:
This compact equation lets you convert immediately between half-life and rate constant. Given a half-life, divide 0.693 by it to get k. Given a rate constant, divide 0.693 by it to get the half-life.
Beyond the Half-Life: The concept generalizes to any fraction. A "third-life" or "fifth-life" is the time for the amount to fall to 1/3 or 1/5 of its initial value, respectively. Professors occasionally ask about these on exams — simply substitute ln(1/3) or ln(1/5) in place of ln(1/2) in the derivation above and proceed identically.
Solving Problems with Whole Numbers of Half-Lives
Many exam problems are deliberately constructed so that the elapsed time corresponds to an exact, whole number of half-lives. Recognizing this pattern lets you solve the problem mentally, without reaching for a calculator.
The rule is simple: multiply by 1/2 for each half-life elapsed. Starting with 64 g, the amount remaining after each successive half-life is:
- After 1 half-life: 32 g
- After 2 half-lives: 16 g
- After 3 half-lives: 8 g
- After 4 half-lives: 4 g
Common Counting Mistake: When writing out a decay sequence (64 → 32 → 16 → 8 → 4), students often count the numbers and arrive at five. The correct count is four — count the arrows (the transitions), not the values. Each arrow represents one half-life.
Example: Finding Total Time from Half-Life
A 64-gram sample of a radioactive nuclide has a half-life of 5 hours. How long does it take for the mass to fall to 4 grams?
Going from 64 g to 4 g requires dividing by 2 exactly four times → 4 half-lives. At 5 hours per half-life: 4 × 5 = 20 hours.
Example: Finding Half-Life from Total Time
A researcher starts with 64 g of a radioactive nuclide. After 20 hours, only 4 g remains. What is the half-life?
64 → 4 g is again exactly 4 half-lives. If 4 half-lives span 20 hours, then one half-life = 20 ÷ 4 = 5 hours.
Non-Whole Half-Lives: Using the Integrated Rate Law
When the target amount does not correspond to a clean power of one-half, the first-order integrated rate law is required. The strategy is to convert the half-life into a rate constant and substitute directly into the equation.
Recall: k = 0.693 / t1/2
Example: Non-Whole Half-Life Calculation
A sample begins at 64 g with a half-life of 5 hours. How long does it take to reach 5 g?
Five grams falls between the 3-half-life mark (8 g, at 15 hours) and the 4-half-life mark (4 g, at 20 hours). Because 5 g is much closer to 4 g than to 8 g, the answer will be closer to 20 hours than to 15 hours. On a multiple-choice exam, this estimate alone may be sufficient.
For an exact answer, apply the integrated rate law with k = 0.693/5:
This confirms the estimate: 18.4 hours sits between 15 and 20, nearer to 20 — exactly as predicted.
Exam Strategy: On multiple-choice problems, always bracket the target amount between two consecutive whole-number half-life values first. If only one answer choice falls in that range, you can select it immediately without the full calculation. Save the plug-and-chug for cases where answer choices are clustered together.
4 Forms of Half-Life Problems: Mass, Percentage, Fraction, and Activity
Half-life problems can be presented in several equivalent forms. Recognizing the underlying pattern — successive halving — allows you to translate freely between them.
Percentages and Fractions
Starting at 100% (or a fraction of 1), each half-life cuts the remaining amount in half:
- Start: 100% / 1
- After 1 half-life: 50% / 1/2
- After 2 half-lives: 25% / 1/4
- After 3 half-lives: 12.5% / 1/8
- After 4 half-lives: 6.25% / 1/16
Asking "how long to reach 6.25% remaining?" or "how long to reach 1/16 remaining?" is identical to asking "how long to reach 4 g from 64 g?" — all three require four half-lives.
Remaining vs. Decayed — Don't Confuse Them: A problem might state how much has decayed rather than how much remains. If 75% has decayed, then 25% remains (2 half-lives). If 7/8 has decayed, then 1/8 remains (3 half-lives). If 15/16 has decayed, then 1/16 remains (4 half-lives). Always convert "amount decayed" to "amount remaining" before applying the integrated rate law.
Activity (Disintegrations per Minute)
Radioactivity is commonly measured in terms of activity — the number of nuclear disintegrations per unit time. Because activity is directly proportional to the number of radioactive nuclei present, it decreases by exactly half with each half-life, just like mass or moles do.
Example: Radioactive Decay Expressed as Activity
A sample initially registers 20 disintegrations per minute (dpm). Track its activity over successive half-lives:
- Start: 20 dpm
- After 1 half-life: 10 dpm
- After 2 half-lives: 5 dpm
- After 3 half-lives: 2.5 dpm
- After 4 half-lives: 1.25 dpm
Whether the problem gives you grams, percentage, fraction, or activity, the mathematical approach is identical.
How Radiocarbon Dating Works
The Carbon-14 Cycle in Living Organisms
Radiocarbon dating applies the kinetics of nuclear decay to determine the age of materials that were once living. The method is effective for organic remains up to approximately 50,000–70,000 years old. Beyond that window, so little ¹⁴C remains that accurate measurement becomes impossible.
The technique rests on a natural cycle involving carbon-14 (¹⁴C), a radioactive isotope produced in the upper atmosphere when cosmic radiation converts nitrogen-14 into carbon-14. This newly formed ¹⁴C continually undergoes beta decay back toward stability. The rates of production and decay have established a roughly constant equilibrium concentration of ¹⁴C in atmospheric CO₂.
Living organisms incorporate this atmospheric CO₂ throughout their lifetimes. Plants and other photosynthetic organisms absorb radioactive CO₂ directly; herbivores acquire it by eating those plants; carnivores acquire it by eating those herbivores. In every living organism, a dynamic steady state is maintained: ¹⁴C continually enters through food or photosynthesis and continually decays away, keeping the internal concentration equal to that of the atmosphere.
Death breaks that equilibrium. Once an organism dies, no new carbon-14 enters its remains. From that moment, the existing ¹⁴C decays without replenishment, and its concentration falls steadily over time. This provides a built-in clock: by measuring how much ¹⁴C remains in a sample and comparing it to the activity in a living organism today, scientists can calculate how long ago the organism died.
Half-Life of Carbon-14 and Practical Limitations
The accepted half-life of ¹⁴C is 5730 years (some sources cite 5715 years; 5730 years is the modern standard). This known half-life is what makes radiocarbon dating possible: once you know k = 0.693 / 5730, any current-to-initial activity ratio yields an age.
To appreciate the analytical challenge, consider the natural isotopic composition of carbon: approximately 98.9% is ¹²C and 1.1% is ¹³C — those two account for essentially 100% of all carbon. Carbon-14 exists at a vanishingly small trace level, far below 0.01%. After 10–12 half-lives of radioactive decay, so little ¹⁴C survives that modern instrumentation can no longer measure it reliably. This detection limit is why radiocarbon dating is restricted to the 50,000–70,000-year range.
Why Only Once-Living Materials? Radiocarbon dating works only on organic remains because only living organisms continuously cycle carbon-14 through their bodies during life. Minerals, rocks, and most geological materials do not incorporate atmospheric carbon-14 this way, so they have no "carbon-14 clock" to read. For geological timescales, scientists use other radiometric methods such as uranium-lead or potassium-argon dating.
Radiocarbon Dating: Worked Example
Problem: A piece of cloth recovered from an archaeological site has a carbon-14 activity of 4.1 disintegrations per minute. Living organisms today exhibit a carbon-14 activity of 16.4 disintegrations per minute. What is the approximate age of the cloth?
Step 1 — Check for a whole number of half-lives.
Does 16.4 divide evenly to 4.1 by repeated halving?
- 16.4 ÷ 2 = 8.2 dpm (after 1 half-life)
- 8.2 ÷ 2 = 4.1 dpm (after 2 half-lives)
Yes — exactly 2 half-lives have elapsed.
Step 2 — Calculate the age.
Each half-life of ¹⁴C is 5730 years:
If the numbers did not work out so cleanly: convert the half-life to a rate constant (k = 0.693 / 5730), substitute into the first-order integrated rate law with the initial activity (16.4 dpm) and current activity (4.1 dpm), and solve for t.
Key Principle: Activity is directly proportional to the number of radioactive nuclei present. This means activity can be used in place of mass, moles, or any other quantity representing "how much ¹⁴C" when plugging values into the integrated rate law. The ratio of current to initial activity equals the ratio of current to initial amount.
8 Key Takeaways
- All spontaneous nuclear decay is first order, governed by the integrated rate law ln(N/N0) = −kt.
- The half-life and rate constant are interconvertible via the half-life equation t1/2 = 0.693 / k. In nuclear problems you are almost always given the half-life; derive k from it as needed.
- When a problem involves an exact whole number of half-lives, solve by repeated halving — no calculator required. Count the transitions (arrows), not the data points.
- When the elapsed time does not correspond to a whole number of half-lives, use the first-order integrated rate law: compute k from the half-life, substitute into ln(N/N0) = −kt, and solve for the unknown.
- Half-life problems may be expressed in terms of mass, percentage, fraction, or activity (disintegrations per minute) — the mathematics is identical in every case because activity is directly proportional to the amount of radioactive nuclide.
- Watch for the "remaining vs. decayed" distinction: if 75% has decayed, 25% remains. Always work with the fraction or amount remaining when applying the integrated rate law.
- Radiocarbon dating exploits the steady-state concentration of ¹⁴C in living organisms (maintained by continuous dietary intake) and the known carbon-14 half-life of 5730 years. At death, intake ceases and the radioactive decay clock starts. Comparing the current activity of a sample to that of living tissue today gives the elapsed time since death.
- Radiocarbon dating is limited to materials that were once alive and to ages up to roughly 50,000–70,000 years, after which the residual ¹⁴C concentration is too low to measure accurately.