How to Solve Half-Life Problems

Half-life doesn’t care what you do to the sample

The half-life of a radioactive isotope is one of the most stubborn quantities in chemistry. Heat it, freeze it, compress it, dissolve it in acid, lock it in a vault — carbon-14 decays with a half-life of 5,730 years regardless. Half-life is a property of the nucleus, not the sample. That’s what makes radiometric dating work and what makes nuclear waste hazardous on geological timescales: there’s nothing you can do chemically to speed it up or slow it down.

Once you accept the invariance, the math reduces to a single first-order exponential and four variables (initial amount, remaining amount, elapsed time, half-life). Tell me three, I’ll find the fourth.

The equations you actually need

The half-life formula

N = N₀ × (1/2)^(t / t₁/₂)

• N: amount remaining after time t
• N₀: initial amount
• t: elapsed time
• t₁/₂: half-life

The decay constant form

If you prefer working with rate constants:

λ = ln(2) / t₁/₂ = 0.693 / t₁/₂

N = N₀ × e^(−λt)

Same physics, equivalent algebra. Use whichever fits the form of your data.

The procedure

Step 1: List what you know

Of N₀, N, t, and t₁/₂, which three are given? The fourth is your unknown.

Step 2: Get t and t₁/₂ into matching units

If half-life is in years and elapsed time is in days, convert one before you divide. Mismatched units are the single most common source of catastrophically wrong answers in this topic.

Step 3: Solve for the unknown

• Find N: N = N₀ × (1/2)^(t / t₁/₂)
• Find t: t = t₁/₂ × ln(N₀/N) / ln(2)
• Find N₀: N₀ = N / (1/2)^(t / t₁/₂)
• Find t₁/₂: t₁/₂ = t × ln(2) / ln(N₀/N)

Step 4: Sanity-check against the decay table

If t / t₁/₂ rounds to an integer n, the remaining fraction should be 1/2ⁿ. Use that to catch order-of-magnitude errors.

Worked examples

Example 1: Whole-number half-lives

Iodine-131 has a half-life of 8.02 days. You start with 100.0 mg. How much remains after 24.06 days?

24.06 / 8.02 = 3.000 → exactly 3 half-lives.

N = 100.0 × (1/2)³ = 100.0 × 0.125 = 12.5 mg

Example 2: Non-integer half-lives

Phosphorus-32, t₁/₂ = 14.3 days. Starting from 80.0 mg, what’s left after 20.0 days?

20.0 / 14.3 = 1.399.

N = 80.0 × (1/2)^1.399 = 80.0 × 0.3789 = 30.3 mg

Example 3: Solving for time

A strontium-90 sample (t₁/₂ = 28.8 yr) has decayed to 25% of its original activity. How long ago did it start?

N/N₀ = 0.25 → t = 28.8 × ln(4) / ln(2) = 28.8 × 1.386 / 0.693 = 57.6 years

That’s exactly 2 half-lives, which matches the 25% remaining (1/2 → 1/4).

Example 4: Carbon-14 dating

A wooden artifact has 62.5% of the C-14 found in living wood. C-14’s half-life is 5,730 years. How old is it?

t = 5730 × ln(1/0.625) / 0.693 = 5730 × 0.4700 / 0.693 = 3,890 years

Example 5: Solving for half-life

A sample drops from 120.0 mg to 15.0 mg in 36.0 minutes. What’s its half-life?

N/N₀ = 15.0 / 120.0 = 0.125 = 1/8 → 3 half-lives.

t₁/₂ = 36.0 × ln(2) / ln(120/15) = 36.0 × 0.693 / 2.079 = 12.0 minutes

Check: 36.0 / 12.0 = 3 half-lives, (1/2)³ = 1/8, 120 × 1/8 = 15. Confirmed.

The decay table you should memorize

Half-lives elapsed	Fraction remaining	Percent remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%
5	1/32	3.125%
10	1/1024	~0.1%

After ten half-lives, less than 0.1% of the original sample remains. That’s the practical rule of thumb for “essentially decayed” — though for very long-lived isotopes, “essentially decayed” can still mean the activity is dangerous.

Common errors

Mismatched time units. Half-life in years, elapsed time in days, and you forget to convert? Your answer is off by a factor of 365. Always check that t and t₁/₂ are in the same units before dividing.

Confusing fraction remaining with fraction decayed. “75% has decayed” means 25% remains. Plug 0.25 into the formula, not 0.75.

Treating half-life as variable. Half-life is constant for a given isotope. It doesn’t shorten as the sample shrinks. Each half-life cuts the current amount in half, not the original amount.

Using log base 10 without rebasing. The formula uses natural log with ln(2) in the denominator, or log base 2 directly. Substituting log₁₀ without adjusting gives a wildly wrong answer.

Assuming all decay is first-order. Radioactive decay always is — that’s why half-life is well-defined. Chemical reaction kinetics are not always first-order, so don’t transfer half-life intuition to general kinetics without checking the order.

Use our Half-Life Calculator to solve for any of the four variables with the steps spelled out and the decay curve plotted.