Question

Assume a constant $1^{14}\mathrm{C}{/}^{12}\mathrm{C}$ ratio of 13.6 counts per minute per gram of living matter. A sample of a petrified tree was found to give 1.2 counts per minute per gram. How old is the tree? (For ${}^{14}\mathrm{C},t_{1/2}=5730$ years.)

Short answer

The petrified tree is approximately 18,350 years old.

Step-by-step solution

Find the decay constant

To find the decay constant, we will use the half-life formula: $$t_{1/2}=\frac{0.693}{\lambda }$$ where $t_{1/2}$ is the half-life and $\lambda$ is the decay constant. We are given the half-life value for ${}^{14}C$, which is 5730 years. We can find the decay constant as follows: $$\lambda =\frac{0.693}{5730}$$ Now let's compute the decay constant: $$\lambda \approx 1.21\times {10}^{-4}{\mathrm{year}}^{-1}$$

Determine the ratio of ${}^{14}C$ to ${}^{12}C$ in the sample

In the problem, we are given that the ${}^{14}C{/}^{12}C$ ratio in living matter is 13.6 counts per minute per gram, and that the ratio in the tree sample is 1.2 counts per minute per gram. We need to determine the ratio of the sample to that of living matter: $$\frac{{\mathrm{ratio}}_{\mathrm{sample}}}{{\mathrm{ratio}}_{\mathrm{living}}}=\frac{1.2}{13.6}$$ Now let's calculate the ratio: $$\frac{{\mathrm{ratio}}_{\mathrm{sample}}}{{\mathrm{ratio}}_{\mathrm{living}}}\approx 0.0882$$

Use decay formula to find the age of the tree

Now we will use the decay formula: $$N_t=N_0\times e^{-\lambda t}$$ where $N_t$ is the number of radioactive isotopes at time $t$, $N_0$ is the initial number of radioactive isotopes, $e$ is the base of the natural logarithm, $\lambda$ is the decay constant, and $t$ is the time elapsed (in years) since the tree died. We found the decay constant $\lambda$ in step 1 and the ratio of ${}^{14}C$ in the sample to that in living matter in step 2. We can use this information to rewrite the decay formula in terms of the ratios: $$\frac{{\mathrm{ratio}}_{\mathrm{sample}}}{{\mathrm{ratio}}_{\mathrm{living}}}=e^{-\lambda t}$$ Rearrange the formula to find $t$: $$t=-\frac{\ln (\frac{{\mathrm{ratio}}_{\mathrm{sample}}}{{\mathrm{ratio}}_{\mathrm{living}}})}{\lambda }$$ Now let's substitute the values we found in steps 1 and 2 to get the age of the tree: $$t=-\frac{\ln (0.0882)}{1.21\times {10}^{-4}{\mathrm{year}}^{-1}}$$ Finally, calculate the age of the tree: $$t\approx 18,350\mathrm{years}$$ So, the petrified tree is approximately 18,350 years old.

Key concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This occurs naturally in certain types of atoms, such as carbon-14, which are found in all living organisms.
As these atoms decay, they transform into more stable elements. The decay process follows a predictable pattern, allowing scientists to use it for dating ancient objects.
In the case of carbon-14, it decays into nitrogen-14, and this process can be tracked over time. This is the basis for radiocarbon dating, where the initial number of carbon-14 atoms decreases as the sample ages.
Understanding radioactive decay is crucial for interpreting the ratios in the given exercise and using them to calculate the age of the sample.

Half-Life

Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. For carbon-14, the half-life is 5730 years, meaning that every 5730 years, half of the carbon-14 in a sample will have decayed into nitrogen-14.
This concept allows for the calculation of the age of artifacts by measuring the remaining concentration of carbon-14. In the exercise, the known half-life of carbon-14 is used to determine the decay constant necessary for calculating the sample's age.
By understanding half-life, one can predict how much of a substance will remain after a particular number of years and thus deduce the time elapsed since the death of the organism.

Radiocarbon Dating

Radiocarbon dating is a method used to determine the age of ancient biological materials. It relies on measuring the amount of carbon-14 remaining in a sample.
Living organisms constantly exchange carbon with their environment, maintaining a steady amount of carbon-14. However, once they die, this exchange stops, and the carbon-14 begins to decay.
The process involves comparing the present carbon-14 levels in a sample with the expected levels in living matter, as seen in the exercise. By calculating this ratio and using the decay constant, we can estimate how long ago the organism died.
Radiocarbon dating is essential in archaeology and other fields to date ancient specimens accurately.

Decay Constant

The decay constant is a crucial factor in the equation of radioactive decay, representing the probability of a single nucleus decaying per unit of time.
Mathematically, it is related to the half-life through the formula: $$\lambda =\frac{0.693}{t_{1/2}}$$ For carbon-14, substituting the half-life gives a decay constant of approximately $1.21\times {10}^{-4}{\mathrm{year}}^{-1}$.
This constant helps determine how quickly a substance undergoes radioactive decay. In the exercise, it's used in the decay formula to calculate the age of the tree.
With the decay constant, scientists can link the ratio of remaining carbon-14 to the time elapsed, allowing for accurate dating of ancient materials.