suppose you have a sample of clamshell at a paleoindian site and you measure the 14c activity of aa 100-gram sample of carbon as 320 disintegretions per scond (or 3.2 disingtegrations per gram of carbon per second).
a. what was the activity(in disitegrations per second) for the 100 gram sample at the time it formed.
b. how do you know this?
c. What is the age of the sample?
a. To determine the activity of the 100-gram sample at the time it formed, we first need to calculate the initial activity using the decay equation for radioactive decay. The decay equation for carbon-14 is:
N(t) = N(0) * e^(-λt)
Where:
N(t) = the activity at time t
N(0) = the initial activity
λ = the decay constant of carbon-14 (dependent on the half-life)
t = time
We know the current activity (320 disintegrations per second) for a 100-gram sample. We also know that carbon-14 has a half-life of approximately 5730 years. Using this information, we can solve for the initial activity (N(0)).
First, convert the current activity to disintegrations per gram of carbon per second. Given that we have 3.2 disintegrations per gram of carbon per second, dividing by 100 grams gives us 0.032 disintegrations per gram of carbon per second.
Now, substitute the values into the decay equation:
0.032 = N(0) * e^(-0.693 / 5730 * t)
Here, we assume t = 0, as that represents the time when the sample formed. Rearranging the equation:
N(0) = 0.032 / e^(-0.693 / 5730 * 0)
Since e^0 is equal to 1, we can simplify the equation to:
N(0) = 0.032
Therefore, the activity of the 100-gram sample at the time it formed was 0.032 disintegrations per second.
b. We know this because the decay of carbon-14 is a well-documented process with a known half-life. By applying the radioactive decay equation and considering the current activity of the sample, we can determine the initial activity. This allows us to estimate the activity of the sample at the time it formed.
c. The age of the sample can be determined by using the decay equation once again. Rearranging the equation, we have:
t = -(5730 / 0.693) * ln(N(t) / N(0))
Substituting the known values, where N(t) is the current activity (320 disintegrations per second) and N(0) is the initial activity (0.032 disintegrations per second), we can calculate the age:
t = -(5730 / 0.693) * ln(320 / 0.032)
Using a calculator, compute the natural logarithm of (320 / 0.032) and divide it by 0.693, then multiply the result by 5730. This gives you the age of the sample in years.
Note: The above calculations assume the carbon-14 half-life remains constant throughout time and provide an approximate age of the sample.