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Radioactive Decay and Half-Life

24 min Chemistry Math

Radioactive substances decay according to an exponential function — the same family you study in algebra and precalculus.

Exponential Decay

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

or equivalently $N(t) = N_0 e^{-\lambda t}$ where $\lambda = \frac{\ln 2}{t_{1/2}}$.

$N_0$ = initial amount, $t_{1/2}$ = half-life, $t$ = elapsed time.

After $n$ Half-Lives

$$N = N_0 \cdot \left(\frac{1}{2}\right)^n$$

After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%, etc.

Example 1

Iodine-131 has $t_{1/2} = 8$ days. Starting with 100 mg, how much remains after 24 days?

Number of half-lives: $n = 24/8 = 3$.

$N = 100 \times (1/2)^3 = 100/8 = 12.5$ mg.

Example 2

Carbon-14 has $t_{1/2} = 5{,}730$ years. A fossil has 25% of its original C-14. How old is it?

$0.25 = (1/2)^n$ → $(1/2)^2 = 0.25$ → $n = 2$ half-lives.

Age $= 2 \times 5{,}730 = 11{,}460$ years.

Example 3

A sample decays from 200 g to 50 g in 10 hours. Find the half-life.

$50/200 = 1/4 = (1/2)^2$, so 2 half-lives occurred in 10 hours.

$t_{1/2} = 10/2 = 5$ hours.

Practice Problems

1. A 400 g sample has $t_{1/2} = 6$ hours. How much remains after 18 hours?

2. After 4 half-lives, what fraction of the original sample remains?

3. Sr-90 has $t_{1/2} = 29$ years. How many years until only 12.5% remains?

4. A bone has 1/8 of its original C-14. How old is it?

5. Find the decay constant $\lambda$ for a substance with $t_{1/2} = 10$ days.

6. Using $N = N_0 e^{-\lambda t}$, find $N$ after 20 days if $N_0 = 500$ and $\lambda = 0.0693$.

Show Answer Key

1. $400/2^3 = 50$ g

2. $1/16 = 6.25\%$

3. 3 half-lives = 87 years

4. $1/8 = (1/2)^3$ → 3 half-lives → $3 \times 5730 = 17{,}190$ years

5. $\lambda = \ln(2)/10 \approx 0.0693$ per day

6. $500 e^{-0.0693 \times 20} = 500 e^{-1.386} \approx 125$ g