Half-Life

How old is this piece of wood?

Optional · not on the AP exam: this lesson is optional; AP Chemistry doesn’t test radioactive-decay calculations. Chinese high schools teach half-life in physics.

How do archaeologists know a piece of charcoal is thousands of years old? Why is a radioactive drug given in a hospital almost gone a few days later? Both answers rest on one idea: the half-life.

The half-life半衰期 of a radioactive nuclide is the time it takes for half of its nuclei to decay. It is written t₁/₂ (or T). Carbon-14 has a half-life of 5700 years: 1 g of carbon-14 becomes 0.5 g after 5700 years, 0.25 g after another 5700, and so on.

After a time t, the number of nuclei left, N, is related to the starting number, N₀, by

N = N_0 \left(\frac{1}{2}\right)^{t/T}

where T is the half-life. Mass decreases the same way.

The half-life is set by the nucleus itself. Heating, compressing, or turning the substance into a compound doesn’t change it, because those affect only the electrons (lesson 2.14.1).

A half-life is a statistical law that only makes sense for large numbers of nuclei. For a single nucleus there is no way to predict when it will decay: it might be the next second or a very long time from now.

Half-lives of some nuclides:

Nuclide	Half-life	Decay
technetium-99m	6.0072 hours	gamma
iodine-131	8.0252 days	beta-minus
radon-222	3.8222 days	alpha
cobalt-60	5.271 years	beta-minus
carbon-14	5700 years	beta-minus
uranium-238	4.468 × 10⁹ years	alpha

🔧 Math tool card · Exponents & Half-LifeSkip it if you already know this

Each half-life multiplies by 1/2. After n half-lives, (1/2)ⁿ is left:

half-lives n	fraction left
1	1/2
2	1/4
3	1/8
4	1/16
5	1/32
10	1/1024

The exponent doesn’t have to be a whole number. n = t/T can be 1.5: (1/2)^1.5 = 1/(2√2) ≈ 0.354. On a calculator, enter 0.5 ^ 1.5.

To solve for time, use logarithms (the tool card in lesson 2.12.3). Taking the log of both sides of N/N₀ = (1/2)^(t/T) and rearranging gives

\begin{aligned} \frac{t}{T} &= \frac{\log (N_0/N)}{\log 2} \\ t &= T \times \frac{\log (N_0/N)}{\log 2} \end{aligned}

For example, when 1/8 is left, log 8 / log 2 = 3: exactly 3 half-lives.

A radioactive sample is down to 10% of its original amount. About how many half-lives have passed? (log 2 ≈ 0.301)

Below are 400 nuclei of one kind. Drag the time slider or press Play to compare the number left with the theoretical curve. Press “New sample” to see whether every batch behaves the same. At the bottom, enter the percentage left to work out the time elapsed:

Try it: watch 400 nuclei decay

Time elapsed

If the amount left is%

Carbon-14 dating

High in the atmosphere, neutrons from cosmic rays keep turning nitrogen-14 into carbon-14:

\ce{^{14}_{7}N + ^{1}_{0}n -> ^{14}_{6}C + ^{1}_{1}H}

Carbon-14 becomes CO₂ just like ordinary carbon, is taken up by plants, and passes on to animals. While an organism is alive, the fraction of its carbon that is carbon-14 stays about the same as in the air. After it dies it stops taking in carbon, and the carbon-14 decays with a half-life of 5700 years. Measuring the fraction left tells how long ago it died.

Carbon-14 dating works for carbon-containing samples from a few hundred to a few tens of thousands of years old. Much older samples have too little carbon-14 left to measure well; rocks are dated with longer-lived nuclides such as uranium-238.

How Chinese textbooks differ

In China, half-life is taught in high school physics rather than chemistry.
US textbooks also write decay as N = N₀e^(−kt), with k = ln 2 / t₁/₂. That is the same equation as (1/2)^(t/T), and it reappears for first-order reactions in AP Unit 5.

Example 1: how much iodine-131 is left?

A patient receives 80.0 mg of iodine-131 (half-life 8.0252 days). How much remains after about 24.1 days? (Ignore any that leaves the body.)

Step-by-step solution

The formula. Each half-life halves what is left: m = m₀ × (1/2)ⁿ, where n = t ÷ t_1/2 is the number of half-lives.
Number of half-lives.
$n = \frac{24.1}{8.0252} \approx 3.00$
Fraction left.
$\left(\tfrac{1}{2}\right)^{3} = \tfrac{1}{8}$
Mass left.
$\begin{aligned} m &= 80.0 \times \tfrac{1}{8} \\ &= \boxed{10.0\ \text{mg}} \end{aligned}$
Check. One half-life at a time: 80.0 → 40.0 → 20.0 → 10.0 mg, exactly three halvings.
A common mistake. Subtracting the same amount each half-life (40.0 mg), which would use it all up in two half-lives. Each half-life removes half of what is left, so the amount lost keeps shrinking.

Example 2: dating charcoal

In a piece of ancient charcoal, the fraction of carbon that is carbon-14 is 70.0% of the value in a living tree. About how old is it? (Use 5700 years for the half-life of carbon-14.)

t = T \times \frac{\log (N_0/N)}{\log 2}

Step-by-step solution

Where the formula comes from. N = N₀ × (1/2)^t/T means N₀/N = 2^t/T. Taking the log of both sides gives log(N₀/N) = (t/T) × log 2, which rearranges to the formula above.
Estimate first. 70.0% is between 100% and 50%, so less than one half-life has passed: under 5700 years.
Substitute.
$\begin{aligned} N_0/N &= 1/0.700 \\ &= 1.429 \\ \log 1.429 &= 0.1549 \\ \log 2 &= 0.3010 \end{aligned}$
Calculate.
$\begin{aligned} t &= 5700 \times \frac{0.1549}{0.3010} \\ &\approx \boxed{2.93 \times 10^{3}\ \text{years}} \end{aligned}$
Check. 2930 ÷ 5700 ≈ 0.514 half-lives, and (1/2)^0.514 ≈ 0.700, as given. It is also under 5700 years.
A common mistake. Using a straight-line proportion: 30% lost, and one half-life loses 50%, so t = 0.30 ÷ 0.50 × 5700 = 3420 years. Decay removes the same fraction in each equal time, not the same amount, so a straight line doesn’t work.

Practice: what fraction is left (or has decayed)?

Left after 1 half-life
Left after 2 half-lives
Left after 3 half-lives
Left after 4 half-lives
Decayed after 2 half-lives
Decayed after 3 half-lives
Iodine-131 (half-life about 8 days) after 16 days: left
Cobalt-60 (half-life about 5.27 years) after about 15.8 years: left
Radon-222 (half-life about 3.82 days) after 3.82 days: decayed

Common mistakes

Mistake 1: “After two half-lives, it’s all gone.” The first half-life leaves half; the second leaves half of that, or 1/4. The amount keeps shrinking but never reaches exactly zero.

Mistake 2: “Heating it or making it into a compound speeds up the decay.” The half-life is set by the nucleus and doesn’t depend on temperature, pressure, or chemical form.

Mistake 3: “Ten nuclei will be down to exactly five after one half-life.” A half-life is a statistical law for large numbers. With only a few nuclei, the number left is random: it could be 3 or 7.

Quick check

A radioactive nuclide has a half-life of 2 days. How much of a 32 g sample is left after 6 days?

A fossil bone has 1/8 of the carbon-14 fraction found in living tissue. About how old is it? (Half-life of carbon-14: 5700 years.)

Summary

A half-life is the time for half of a radioactive sample’s nuclei to decay. It is set by the nucleus, doesn’t depend on temperature, pressure, or chemical form, and is a statistical law for large numbers of nuclei.
N = N₀(1/2)^(t/T); after n half-lives, (1/2)ⁿ remains. To find the time, t = T × log(N₀/N) / log 2.
Carbon-14 dating: after death, carbon-14 decays with a 5700-year half-life, and the fraction left gives the age.

On to AP (for later)

AP doesn’t test radioactive decay, but the same math returns in AP Unit 5 for first-order reactions: their half-life, t₁/₂ = 0.693/k, doesn’t depend on concentration, and a plot of ln[A] against time is a straight line. This lesson will make that part much easier.

Data sources: half-lives from the IAEA’s LiveChart of Nuclides; the database lists cobalt-60 as 1925.28 days, converted to years in the table.

Half-Life半衰期

Learning goals

Before you start

How old is this piece of wood?

Half-life

Carbon-14 dating

How Chinese textbooks differ

Example 1: how much iodine-131 is left?

Example 2: dating charcoal

Common mistakes

Quick check

Summary

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