Final answer:
To find the age t of a sample using carbon-14 dating, measurements of the sample's total carbon mass mc, activity a, the atmospheric carbon-14 to carbon ratio r, and the decay constant λ are used to determine the initial number of carbon-14 atoms, from which the sample's age can be calculated.
Step-by-step explanation:
To calculate the age t of a sample using carbon-14 dating, we apply the principle that the measured activity in the sample, a, is due to the decay of carbon-14 atoms, understanding that the initial ratio of carbon-14 to total carbon atoms in the sample was equal to the current atmospheric ratio r. Given that the decay constant for carbon-14 is λ, and the mass of a carbon-14 atom is ma, we can set up an equation using the known decay rate formula N(t) = N_0e^{-λ t} where N(t) is the current number of carbon-14 atoms and N_0 is the initial number. The total mass of carbon in the sample is mc, and the activity a relates to the current number of carbon-14 atoms by a = λN(t). From this, we can solve for the age t of the sample.
The total number of carbon atoms in the sample can be calculated by dividing the total mass mc by the mass of an individual carbon atom ma. The initial number of carbon-14 atoms can then be found by multiplying the total number of carbon atoms by the atmospheric ratio r.