Suppose you have a sample of clamshell at a Paleoindian site and you measure the 14C activity of a 100-gram sample of carbon as 320 disintegrations per minute (or 3.2 disintegrations per

gram of carbon per minute).

a.What was the activity (in disintegrations per minute) for the 100-gram sample at the time it formed?

b. How do you know this?

c. What is the age of the sample?

To answer these questions, we need to understand a concept related to radiocarbon dating. Radiocarbon dating is based on the fact that carbon-14 (14C) is present in the atmosphere and is absorbed by living organisms. Once an organism dies, it no longer takes in new carbon-14, and the level of carbon-14 in its remains gradually decreases over time through radioactive decay. By measuring the remaining 14C activity in a sample, we can determine its age.

Now let's go through each question:

a. What was the activity (in disintegrations per minute) for the 100-gram sample at the time it formed?
To calculate the original 14C activity, we need to consider the fact that the half-life of carbon-14 is approximately 5730 years. This means that after 5730 years, half of the original 14C in a sample will have decayed.

To determine the activity of the sample at the time it formed, we can apply the half-life concept. If the current activity of a 100-gram sample is 3.2 disintegrations per gram per minute, we can calculate the original activity as follows:

Original activity = current activity / (0.5)^n

Here, n represents the number of half-lives that have passed since the sample formed. As the sample is quite old (Paleoindian site), let's assume a conservative estimate of 10 half-lives:

Original activity = 3.2 / (0.5)^10

To calculate this, we can use a calculator or perform the calculations step by step: (0.5)^10 = 0.0009765625

Original activity = 3.2 / 0.0009765625

This gives us an original activity of approximately 3276.8 disintegrations per minute for the 100-gram sample at the time it formed.

b. How do you know this?
The calculation above is based on the understanding of the half-life of carbon-14 and the decay process. Scientists have conducted extensive research and experiments to determine the half-life of carbon-14 and measure its activity in various samples. These studies form the basis for our understanding of radiocarbon dating.

c. What is the age of the sample?
To determine the age of the sample, we can use the fact that the 14C activity decreases by half every 5730 years. By comparing the current activity (3.2 disintegrations per gram per minute) to the original activity (3276.8 disintegrations per minute), we can estimate the number of half-lives that have passed.

Age (in years) = number of half-lives * 5730 years

To calculate the number of half-lives, we can rearrange the equation used in question a:

Number of half-lives = log(current activity / original activity) / log(0.5)

Number of half-lives = log(3.2 / 3276.8) / log(0.5)

Using a calculator, we divide the logarithm of 3.2 divided by 3276.8 by the logarithm of 0.5, which gives us approximately 161.272.

Age = 161.272 * 5730 years

This gives us an estimated age of approximately 924,362.16 years for the clamshell sample.