Half-Life
Time for half a sample to decay.
Formula
Variables
How it works
Radioactive decay is modeled by an exponential law, meaning the rate of decay is proportional to how much undecayed material is still present. If you write the remaining amount as N(t)=N_0 e^{-\lambda t}, then the half-life is the time when N(t)=N_0/2, which gives 1/2=e^{-\lambda t_{1/2}} and therefore t_{1/2}=\ln 2/\lambda. The natural log appears because we are solving for the time hidden in the exponent.
When to use it
Use this formula when you know a decay constant and want the time required for a radioactive sample to fall to half its original amount. It is also useful for any process with exponential decay, such as certain chemical, biological, or environmental decay models.
Worked example, step by step
A radioactive isotope has a decay constant of \lambda = 0.028 day^{-1}. Find its half-life. Step 1: Write the formula: t_{1/2} = \frac{\ln 2}{\lambda} Step 2: Substitute the value of \lambda: t_{1/2} = \frac{\ln 2}{0.028} Step 3: Evaluate \ln 2 \approx 0.693 Step 4: Calculate: t_{1/2} = \frac{0.693}{0.028} \approx 24.75 Step 5: State the answer: the half-life is about 24.8 days
Common mistakes to avoid
- Using log base 10 instead of the natural logarithm
- Forgetting that the decay constant must be in inverse-time units so the result comes out in time units
- Confusing half-life with the time for the sample to decay to zero rather than to 50% remaining
Did You Know?
Carbon-14's 5,730-year half-life lets archaeologists date ancient wood, bone, and cloth.
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