Half Life Formula with Worked Examples

Key Points

• The half life formula is $N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$, where $t_{1/2}$ is the half-life and $N_0$ the starting amount.

• The US Environmental Protection Agency defines radioactive half-life as the time required for half of the radioactive atoms present to decay.

• The exponent $\frac{t}{t_{1/2}}$ is simply the number of half-lives elapsed, so it need not be a whole number.

• Written with the natural exponential instead, the law is $N = N_0 e^{-\lambda t}$ with the decay constant $\lambda = \frac{\ln 2}{t_{1/2}}$.

• Each half-life removes half of what remains rather than a fixed amount, so the quantity approaches zero without ever reaching it.

What Is the Half Life Formula?

The half life formula gives how much of a decaying quantity remains after a given time:

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

The EPA states that radioactive half-life is the time required for half of the radioactive atoms present to decay, which is exactly what the formula encodes.

Why Decay Never Reaches Zero

Each half-life removes half of whatever is left, not a fixed quantity. Starting with 100 grams, the first halving removes 50 grams, the second removes 25, the third removes 12.5.

The amounts removed shrink in step with the amount remaining, so there is always something left to halve. After $n$ half-lives the fraction remaining is $\left(\frac{1}{2}\right)^n$, which gets arbitrarily small but is never zero.

That is the structural difference from linear decay. A quantity losing 10 grams per year from 100 grams is gone in exactly ten years, while a quantity halving every year still has a measurable trace after twenty.

Variable Key

Examples of the Half Life Formula

Example 1

A 100 g sample has a half-life of 5 years. How much remains after 15 years?

The number of half-lives is $\frac{15}{5} = 3$:

$$N = 100 \left(\frac{1}{2}\right)^3 = 100 \times \frac{1}{8} = 12.5$$

Final answer: 12.5 g

Example 2

For the same sample, how much remains after 10 years, assuming it is all gone by then?

A half-life of 5 years suggests half is gone after 5 years and the other half after another 5, so nothing should remain at 10 years.

That reasoning treats the second halving as removing the original half rather than half of what is left. After 5 years 50 g remain, and the next 5 years removes half of that 50, not the full amount.

$$N = 100 \left(\frac{1}{2}\right)^2 = 25$$

Final answer: 25 g remains after 10 years. Two half-lives leave a quarter, never nothing.

Example 3

A substance has a half-life of 20 minutes. Find its decay constant.

$$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.6931}{20} \approx 0.03466 \text{ per minute}$$

Final answer: about $0.0347$ per minute

Example 4

A sample falls from 80 g to 10 g in 30 years. Find the half-life.

The fraction remaining is $\frac{10}{80} = \frac{1}{8} = \left(\frac{1}{2}\right)^3$, so three half-lives have passed:

$$t_{1/2} = \frac{30}{3} = 10$$

Final answer: 10 years

Recognising the fraction as a power of one half avoids any logarithms here.

Example 5

Carbon-14 has a half-life of 5,730 years. A sample retains 25 per cent of its original carbon-14. How old is it?

A quarter remaining means two half-lives:

$$t = 2 \times 5730 = 11460$$

Final answer: about 11,460 years

This is the basis of radiocarbon dating, where the measured fraction gives the number of half-lives and therefore the age.

The last row is a distinct quantity. The mean lifetime is longer than the half-life, since $\tau = \frac{t_{1/2}}{\ln 2}$, and the two are not interchangeable.

Half Life, Where Things Go Sideways

Mistake 1: Expecting the quantity to reach zero

Where it slips in: Two halves make a whole, so two half-lives sound as though they should account for everything.

Don't do this: Assume nothing remains after two half-lives.

The correct way: Each half-life halves what remains, so two leave a quarter and three leave an eighth. The fraction left is always $\left(\frac{1}{2}\right)^n$.

Students who write out the running amounts for the first three half-lives, as 100, 50, 25, 12.5, never make this error, because the pattern of shrinking removals is visible in the list. Reasoning about halves in the abstract is what produces the zero.

Mistake 2: Treating decay as linear

Where it slips in: A constant half-life sounds like a constant rate of loss.

Don't do this: Subtract the same amount for each equal period of time.

The correct way: Multiply by one half for each half-life. The amount lost per period falls as the quantity falls, which is what makes the graph a curve rather than a straight line.

Mistake 3: Confusing the half-life with the decay constant

Where it slips in: Both describe how fast something decays, and both appear in the two forms of the law.

Don't do this: Substitute $t_{1/2}$ where the exponential form expects $\lambda$.

The correct way: Convert with $\lambda = \frac{\ln 2}{t_{1/2}}$. They are reciprocally related, so a long half-life means a small decay constant.

Key Takeaways

• The half life formula is $N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$, with the exponent counting half-lives.

• Each half-life removes half of what remains, so the quantity never reaches zero.

• The exponential form $N = N_0 e^{-\lambda t}$ is the same law, linked by $\lambda = \frac{\ln 2}{t_{1/2}}$.

• When the remaining fraction is a power of one half, the number of half-lives can be read off without logarithms.

• To work through exponential relationships with a teacher, explore a high school math tutor or math classes online.

Practice These Three Before Moving On

A 240 g sample has a half-life of 8 days. Find what remains after 24 days. Then find the half-life of a sample falling from 64 g to 4 g in 36 hours, and convert a half-life of 50 years into a decay constant.

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Read More

• Logarithms

• Logarithm Rules

• Exponential Equations

• Exponential Form

• Exponents