Half-Life Calculations Reference Cheat Sheet

Key Facts

• One half-life means half of the radioactive nuclei remain, so after $n$ half-lives the fraction remaining is $\left(\frac{1}{2}\right)^n$.
• The number of half-lives is found with $n = \frac{t}{t_{1/2}}$, where $t$ is elapsed time and $t_{1/2}$ is the half-life.
• The remaining amount after $n$ half-lives is $N = N_0\left(\frac{1}{2}\right)^n$, where $N_0$ is the starting amount.
• The percent remaining is $\%\text{ remaining} = 100\left(\frac{1}{2}\right)^n$.
• The percent decayed is $\%\text{ decayed} = 100 - \%\text{ remaining}$.
• Radioactive decay can also be modeled by $N = N_0e^{-\lambda t}$, where $\lambda$ is the decay constant.
• The decay constant and half-life are related by $\lambda = \frac{\ln 2}{t_{1/2}}$ and $t_{1/2} = \frac{\ln 2}{\lambda}$.
• For dating an object, solve for time using $t = \frac{\ln\left(\frac{N_0}{N}\right)}{\lambda}$ or $t = n t_{1/2}$ after finding $n$.

Vocabulary

Half-life

The time required for half of the radioactive nuclei in a sample to decay.

Parent isotope

The original radioactive isotope that undergoes nuclear decay.

Daughter product

The new atom or isotope formed when a parent isotope decays.

Decay constant

The value $\lambda$ that measures the probability of decay per unit time in the model $N = N_0e^{-\lambda t}$.

Activity

The rate of radioactive decay, often measured in becquerels, where $1\text{ Bq} = 1\text{ decay/s}$.

Carbon dating

A radiometric dating method that estimates the age of once-living material using the decay of carbon-$14$.