Question

Cosmic rays impinging on our atmosphere generate radioactive ${}^{14}\mathrm{C}$ from ${}^{14}\mathrm{N}$ nuclei. ${}^{78}$ These ${}^{14}\mathrm{C}$ atoms soon team up with oxygen to form ${\mathrm{CO}}_2$, so that plants absorbing ${\mathrm{CO}}_2$ from the air will have about one in a trillion of their carbon atoms in this form. Animals eating these plants ${}^{79}$ will also have this fraction of carbon in their bodies, until they die and stop cycling carbon into their bodies. At this point, the fraction of carbon atoms in the form of ${}^{14}\mathrm{C}$ in the body declines, with a half life of 5,715 years. If you dig up a human skull, and discover that only one-eighth of the usual one-trillionth of carbon atoms are ${}^{14}\mathrm{C}$, how old do you deem the skull to be?

Short answer

Answer: The age of the skull is approximately 17,145 years old.

Step-by-step solution

Identify initial and final fractions

The initial fraction of carbon atoms as ${}^{14}\mathrm{C}$ when the organism was alive is given to be one in a trillion (1/1,000,000,000,000). In the skull, the final fraction of carbon atoms as ${}^{14}\mathrm{C}$ is one-eighth of the initial fraction i.e., (1/8)*(1/1,000,000,000,000).

Calculate the ratio between initial and final fractions

We need to find the ratio between the final and initial fractions of ${}^{14}\mathrm{C}$. This ratio is equal to the fraction of carbon atoms left, after a certain number of half-lives. Ratio = (Final fraction) / (Initial fraction) = [(1/8)*(1/1,000,000,000,000)] / (1/1,000,000,000,000)

Ratio simplification

Now, let's simplify the ratio: Ratio = (1/8)*(1/1,000,000,000,000) / (1/1,000,000,000,000) = (1/8)

Calculate the number of half-lives

We will use the following formula for radioactive decay, which relates the remaining fraction of ${}^{14}\mathrm{C}$ and the number of half-lives (n): Remaining fraction = (1/2)^n Here, the remaining fraction is the ratio calculated in step 3. Let's plug in the values and solve for n: (1/8) = (1/2)^n Take the logarithm of both sides: log(1/8) = n * log(1/2) Therefore, n = log(1/8) / log(1/2)

Calculate the age of the skull

Now that we have the number of half-lives (n) that have passed, we can calculate the age of the skull using the following formula: Age = n * Half-life Plug in the values: Age = (log(1/8) / log(1/2)) * 5,715 years After solving the equation, we get the age of the skull: Age ≈ 17,145 years So, the age of the skull is approximately 17,145 years old.

Key concepts

Cosmic Rays and Isotope Formation

Cosmic rays are high-energy particles from space that travel at speeds close to the speed of light. When these particles reach Earth, they collide with atoms in the atmosphere. One significant outcome of these collisions is the formation of a variety of isotopes, including Carbon-14 ${(}^{14}C)$. This process begins when cosmic rays interact with the nuclei of nitrogen atoms ${(}^{14}N)$ present in the atmosphere.
The transformation happens as the cosmic rays cause the ejection of protons from nitrogen nuclei, turning them into Carbon-14. Once formed, these ${}^{14}C$ atoms combine with oxygen to create carbon dioxide $(C{O}_2)$. This $C{O}_2$ is absorbed by plants during photosynthesis. When animals consume these plants, they incorporate ${}^{14}C$ into their tissues, maintaining a balance of isotopes as long as they are alive. Understanding this process is crucial for grasping how Carbon-14 becomes part of the natural carbon cycle and sets the stage for radioactive carbon dating.

Carbon-14 Decay Process

The decay of Carbon-14 is a slow and steady process, which is at the heart of radioactive carbon dating. Once an organism dies, it ceases to absorb Carbon-14 from the environment. Therefore, the amount of ${}^{14}C$ in the body begins to decrease through radioactive decay.
Radioactive decay involves the transformation of ${}^{14}C$ to nitrogen over time, emitting a beta particle in the process. This decay process follows a predictable pattern. With each passing half-life, half of the original ${}^{14}C$ content remains unchanged, while the other half converts to nitrogen. This predictable behavior allows scientists to calculate the approximate time since death by examining the remaining ${}^{14}C$ in an organism's remains.
In the exercise, the measurement of ${}^{14}C$ shows that only one-eighth of the original ${}^{14}C$ amount remains, indicating how much time has passed since the organism's death.

Half-life Calculation in Radioisotopes

The concept of a half-life is fundamental to understanding radioactive decay and is a key component in calculating dates using radioisotopes. A half-life is the time it takes for half of a given amount of a radioactive isotope to decay. For Carbon-14, this period is approximately 5,715 years.
In the provided problem, we need to determine the time elapsed since the organism's death by calculating how many half-lives have passed. Given the ratio of remaining ${}^{14}C$ to its initial amount as one-eighth, we can deduce that three half-lives have occurred because $(1/2{)}^3=1/8$.
Using the known half-life of $5,715$ years, we multiply the number of half-lives (3) by this duration to estimate the age. Hence, the skull is approximately $5,715$ years per half-life times three half-lives, resulting in an age of around $17,145$ years. This calculation method helps archaeologists and scientists accurately date ancient biological materials.