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Carbon-14 decay is a natural process that helps us date ancient materials. Carbon-14 (or \({}^{14}C\) ) is a radioactive isotope of carbon. It decays into nitrogen-14 over time by releasing a beta particle. This decay process is useful in radiocarbon dating.
When an organism dies, it stops absorbing carbon. The \({}^{14}C\) in its body begins to decay at a known rate which is characterized by its half-life. The half-life of \({}^{14}C\) is about 5730 years, meaning that half of the carbon-14 atoms will have decayed after this time period.
In our problem, we are calculating the probability of a \({}^{14}C\) atom decaying in your lungs during a single breath. This involves understanding the rate at which \({}^{14}C\) decay happens.

Radioactive isotopes are atoms that have an unstable nucleus. This instability causes them to lose energy by emitting radiation in a process called radioactive decay. Each radioactive isotope decays at a characteristic rate which is quantified by its half-life.
In our given scenario, \({}^{14}C\) is a radioactive isotope of carbon. It decays over time, allowing scientists to measure the amount of \({}^{14}C\) remaining in an object to determine its age. The decay constant (λ) of \({}^{14}C\) can be calculated using the relationship: \[ λ = \frac{ \text{ln}(2) }{ \text{half-life} } \] For \({}^{14}C\), this decay constant is very small, showing that \({}^{14}C\) atoms decay very slowly over time.
In the context of the exercise, we're interested in how often these decays might happen within a quick 3.5-second breath.

The ideal gas law is a fundamental equation in chemistry and physics that describes how gases behave. It combines several gas laws into one formula: \[ \text{PV} = \text{nRT} \] Where:

• P is the pressure of the gas,
• V is the volume of the gas,
• n is the number of moles of the gas,
• R is the ideal gas constant (0.0821 Lâ‹…atm/(molâ‹…K)),
• T is the temperature in Kelvin.

In our exercise, the ideal gas law helps calculate the number of moles of \( \text{CO}_2 \) in a volume of air you breathe in. Under standard conditions (1 atm pressure and 298K temperature), we can determine the number of moles and then use Avogadro's number to find the number of molecules. This step is crucial for finding out how many \({}^{14}C\) atoms are in that breath.

Probability calculation is a branch of mathematics that helps predict how likely events are to occur. In our exercise, several probability calculations are needed to find out the chances of a \({}^{14}C\) atom decaying in a short period.
We start with finding the fraction of \( \text{CO}_2 \) molecules that contain \({}^{14}C \). Given the tiny fraction of \({}^{14}C\) in the atmosphere, this part involves multiplication by a very small number.
Next, we use the decay constant \(λ\) to compute the probability that a \({}^{14}C\) atom will decay during a specific time frame (in our problem, one breath taken over 3.5 seconds):

• The formula used is \[ P = 1 - e^{-λt} \] .
• For small values of λt, the probability can be approximated with \ P ≈ λt \

Finally, multiplying the number of \({}^{14}C\) atoms by the single atom decay probability gives the total probability of any \({}^{14}C\) atom decaying during that breath. This combination of small probabilities and large numbers shows how fundamental principles can predict everyday scenarios.