To determine the age of the fossil bone based on the given carbon isotope ratio, we can use the concept of radioactive decay, specifically focusing on carbon-14 (14C). The ratio of 14C to 12C in living organisms is relatively constant, but once an organism dies, the 14C begins to decay at a known rate, characterized by its half-life. In this case, the half-life of 14C is 5730 years.
Understanding the Ratio
The statement that the 14C : 12C ratio in the fossil bone is (1/16) of that in a living animal bone means that the fossil bone has significantly less 14C compared to a living organism. This reduction in the ratio indicates that a considerable amount of time has passed since the organism died.
Calculating the Age of the Fossil
To find the age of the fossil, we can follow these steps:
- Initial Ratio: Let's denote the original ratio of 14C to 12C in a living organism as 1 (or 100%).
- Current Ratio: The fossil bone has a ratio of (1/16) of the original, meaning it retains only 1/16 of the original 14C.
- Decay Calculation: The decay of 14C follows a half-life pattern. After each half-life, the amount of 14C is halved.
To find out how many half-lives have passed to reach a ratio of (1/16), we can express this mathematically:
Half-life Calculation
Starting from the original amount (1), we can see how many times we need to halve it to reach (1/16):
- After 1 half-life: 1/2
- After 2 half-lives: 1/4
- After 3 half-lives: 1/8
- After 4 half-lives: 1/16
This shows that 4 half-lives have passed since the organism died.
Calculating the Total Time
Now, we can calculate the total time that has elapsed:
Total time = Number of half-lives × Half-life duration
Total time = 4 × 5730 years = 22920 years
Final Result
The age of the fossil bone is approximately 22,920 years. This method of using the decay of carbon-14 is a common technique in archaeology and geology for dating ancient organic materials.
Clarifying the Underlined Line
The underlined line refers to the significant decrease in the 14C : 12C ratio, indicating that the fossil bone has undergone extensive radioactive decay since the death of the organism. This decay process is what allows scientists to estimate the age of the fossil accurately.
In summary, by understanding the principles of radioactive decay and the specific half-life of carbon-14, we can effectively determine the age of ancient biological materials. This method is crucial for dating fossils and understanding the timeline of life on Earth.
