Archaeologists can determine the age of artifacts made of wood or bone by measuring the amount of the radioactive isotope 14C present in the object. The amount of isotope decreases in a first-order process. If 27.6% of the original amount of 14C is present in a wooden tool at the time of analysis, what is the age of the tool? The half-life of 14C is 5,730 yr. Give your answer in scientific notation.
k = 0.693/half life. Substitute and solve for k.
Then ln(No/N) = kt
No = 100
N = 27.6
k = from above
Solve for time in years. Remember to change to scientific notation for the answer. Post your work if you get stuck.
log(.275)/ log(.5)=1.862496476
1.862496476*5730=10672.10481
1.07x10^4
To determine the age of the wooden tool, we can use the concept of radioactive decay and the given information about the amount of remaining 14C isotope.
The first step is to identify the fraction of the original 14C isotope that is still present. We are told that 27.6% of the original amount remains. To convert this fraction to a decimal, we divide it by 100: 27.6% = 0.276.
Next, we need to relate the remaining fraction to the amount of time that has passed since the artifact was last alive (i.e., when it stopped taking in new carbon from the environment).
The decay of 14C follows a first-order process, which means its rate of decay is proportional to the amount of 14C remaining. The half-life of 14C is given as 5,730 years, which is the time it takes for half of the isotope to decay.
We can use the equation for exponential decay:
N(t) = N₀ * e^(-λt)
Where:
N(t) is the amount of 14C remaining at time t
N₀ is the initial amount of 14C
e is the base of natural logarithms (approximately 2.71828)
λ is the decay constant
t is the time that has passed
Since we are given the remaining fraction (0.276), we can substitute it into the equation:
0.276 = e^(-λt)
Now, we can rearrange the equation to solve for the age of the tool (t):
e^(-λt) = 0.276
Taking the natural logarithm (ln) of both sides allows us to isolate t:
ln(e^(-λt)) = ln(0.276)
Using the property of logarithms, we can bring down the exponent:
-λt * ln(e) = ln(0.276)
Since ln(e) is equal to 1, we can simplify further:
-λt = ln(0.276)
Finally, we can solve for t:
t = -ln(0.276) / λ
Substituting the value of the half-life (5,730 years) for λ, we can calculate the age of the tool:
t = -ln(0.276) / 5,730
Evaluating this expression will give us the age of the tool in years.