Counting Half-Lives

A lab sample contains $80$ mg of a substance with a half-life of $4$ hours. How much is left after $12$ hours?

Find the number of half-lives: $\frac{t}{T} = \frac{12}{4} = 3$ = 3 half-lives

Write the formula: $N(12) = 80 \left(\frac{1}{2}\right)^{3}$ = Formula set up

Compute the power: $\left(\frac{1}{2}\right)^{3} = \frac{1}{8}$ = $\frac{1}{8}$ of the sample is left

Multiply: $80 \times \frac{1}{8} = 10$ (or halve three times: $80 \to 40 \to 20 \to 10$) = $10$ mg

Answer: After 12 hours, **10 mg** of the substance is left.

Iodine-131 in a Hospital

Iodine-131 has a half-life of $8$ days. A hospital has $200$ mg of it. How much is left after $40$ days, and what percent is that?

Find the number of half-lives: $\frac{40}{8} = 5$ = 5 half-lives

Find the fraction left: $\left(\frac{1}{2}\right)^{5} = \frac{1}{2^5} = \frac{1}{32}$ = $\frac{1}{32}$

Find the amount: $200 \times \frac{1}{32} = 6.25$ = $6.25$ mg

Write it as a percent: $\frac{1}{32} = 0.03125 = 3.125\%$ = $3.125\%$

Answer: After 40 days, **6.25 mg** is left, which is **3.125%** of the original amount.

How Old Is the Bone? (Carbon-14)

A bone contains $\frac{1}{8}$ of the carbon-14 it had when the animal was alive. The half-life of carbon-14 is $5730$ years. How old is the bone?

Write the fraction as a power of $\frac{1}{2}$: $\frac{1}{8} = \frac{1}{2^3} = \left(\frac{1}{2}\right)^{3}$ = $n = 3$

Interpret the exponent: The exponent is the number of half-lives: $\frac{t}{5730} = 3$ = 3 half-lives have passed

Find the time: $t = 3 \times 5730 = 17190$ = $17190$ years

Answer: The bone is about **17190 years** old.

Finding the Half-Life from Data

A sample of $96$ g decays to $12$ g in $15$ hours. What is its half-life?

Find how many times smaller the sample is: $\frac{96}{12} = 8$ = The sample is 8 times smaller

Write 8 as a power of 2: $8 = 2^3$, so $96 \to 48 \to 24 \to 12$ = 3 half-lives

Divide the time by the number of half-lives: $T = \frac{15}{3} = 5$ = $T = 5$ hours

Check: $96 \left(\frac{1}{2}\right)^{15/5} = 96 \times \frac{1}{8} = 12$ ✓ = Correct

Answer: The half-life is **5 hours**.

Medicine Between Two Half-Lives

A patient takes $400$ mg of a medicine with a half-life of $6$ hours. How much is still in the bloodstream after $9$ hours? Use $\left(\frac{1}{2}\right)^{1.5} \approx 0.3536$ and round to one decimal place.

Find the number of half-lives: $\frac{9}{6} = 1.5$ = 1.5 half-lives (not a whole number)

Write the formula: $N(9) = 400 \left(\frac{1}{2}\right)^{1.5}$ = Formula set up

Use the given power: $400 \times 0.3536 = 141.44$ = $\approx 141.4$ mg

Check that the answer makes sense: After 6 hours there are $200$ mg, after 12 hours $100$ mg. $141.4$ is between them ✓ = Reasonable

Answer: About **141.4 mg** of the medicine is still in the bloodstream after 9 hours.

Mistake: Halving once for every hour (or day, or year) instead of once per half-life

Why: With a half-life of 4 hours, after 12 hours some students compute $80 \left(\frac{1}{2}\right)^{12}$. But the amount only halves every 4 hours, not every hour.

Correct: The exponent is the number of half-lives $\frac{t}{T} = \frac{12}{4} = 3$, so the answer is $80 \left(\frac{1}{2}\right)^{3} = 10$.

Mistake: Subtracting half of the ORIGINAL amount each half-life

Why: Thinking $100 \to 50 \to 0$ treats the decay as linear. Each half-life removes half of what is left NOW, not half of the starting amount.

Correct: Halve the current amount each time: $100 \to 50 \to 25 \to 12.5$. A sample never reaches exactly zero.

Mistake: Using the denominator as the number of half-lives

Why: If $\frac{1}{8}$ is left, some students say 8 half-lives passed. But 8 half-lives would leave $\frac{1}{256}$.

Correct: Write the denominator as a power of 2: $8 = 2^3$, so 3 half-lives have passed.

Mistake: Mixing up the fraction left and the fraction that decayed

Why: After 3 half-lives, $\frac{1}{8}$ is left, so $\frac{7}{8}$ has decayed. Questions can ask for either one.

Correct: Read the question carefully: fraction decayed $= 1 -$ fraction left. After 3 half-lives: $12.5\%$ left, $87.5\%$ decayed.

Carbon-14 Dating

Living things take in carbon-14. After they die, it decays with a half-life of 5730 years, so the fraction left shows the age of bones, wood and charcoal.

A piece of wood with $\frac{1}{4}$ of its original carbon-14 is $2 \times 5730 = 11460$ years old.

Medicine in the Bloodstream

Doctors use the half-life of a medicine to decide how often to give the next dose, so the amount in the body stays safe and effective.

A dose of $400$ mg with a half-life of $6$ hours leaves $400 \left(\frac{1}{2}\right)^{12/6} = 100$ mg after $12$ hours.

Iodine-131 Treatment

Iodine-131 is used to treat thyroid illness. Its half-life of 8 days tells the hospital how fast the radiation drops and when it is safe to store or dispose of leftovers.

$100$ mg of iodine-131 becomes $100 \left(\frac{1}{2}\right)^{16/8} = 25$ mg after $16$ days.

Q: Does a substance ever completely disappear?

A: Mathematically, no: $\left(\frac{1}{2}\right)^{n}$ is never zero. In practice, after about 10 half-lives less than $0.1\%$ is left ($\frac{1}{1024}$), which is usually treated as gone.

Q: Why is the exponent $\frac{t}{T}$ and not just $t$?

A: The factor $\frac{1}{2}$ applies once per half-life, not once per time unit. Dividing the time $t$ by the half-life $T$ counts how many half-lives have passed.

Q: What if the time is not a whole number of half-lives?

A: The formula still works, for example $400 \left(\frac{1}{2}\right)^{1.5}$. Use a calculator (type 400 × 0.5^1.5) or a given value like $\left(\frac{1}{2}\right)^{1.5} \approx 0.3536$. Finding an unknown time in that case needs logarithms, which come in a later lesson.

Q: How is half-life related to the decay rate in exponential decay?

A: In the Exponential Decay lesson, an amount is multiplied by $b = 1 - r$ every time unit. Half-life uses the same idea but chooses the time unit so that the factor is exactly $\frac{1}{2}$, which makes mental calculations much easier.