Question

Carbon dating The half-life of $C-14$ is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only $77\mathrm{\%}$ of the $C$ -14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has $6.2\mathrm{\%}$ of the $\mathrm{C}-14$ that it had when it was alive. Estimate when the wood was cut.

Short answer

Question: Estimate the time elapsed since (a) a cloth was painted, given that the dye has 77% C-14 remaining, and (b) a wood was cut, given that it has 6.2% C-14 remaining. Answer: (a) The cloth was painted approximately 1,682 years ago. (b) The wood was cut approximately 18,241 years ago.

Step-by-step solution

Setup the Decay Equation

We need to find the time $t$ when the cloth was painted, knowing that 77% of the C-14 remains. We can setup the decay equation: $$\frac{N(t)}{N_0}=0.77$$ Which can be written as: $$(1/2{)}^{t/5730}=0.77$$ To solve for $t$, we'll need to use logarithms.

Solve for Time $t$

Taking the logarithm (base 2) of both sides, we have: $$\frac{t}{5730}=lo{g}_2(0.77)$$ Now, we can isolate $t$ by multiplying both sides by 5730: $$t=5730\cdot lo{g}_2(0.77)$$ Finally, calculate the value of $t$: $$t\approx -1682.16$$ The negative sign means the cloth was painted approximately 1682 years before the present time. #Part b - Wood#

Setup the Decay Equation

We need to find the time $t$ when the wood was cut, knowing that only 6.2% of the C-14 remains. We can setup the decay equation: $$\frac{N(t)}{N_0}=0.062$$ Which can be written as: $$(1/2{)}^{t/5730}=0.062$$ To solve for $t$, we'll need to use logarithms again.

Solve for Time $t$

Taking the logarithm (base 2) of both sides, we have: $$\frac{t}{5730}=lo{g}_2(0.062)$$ Now, we can isolate $t$ by multiplying both sides by 5730: $$t=5730\cdot lo{g}_2(0.062)$$ Finally, calculate the value of $t$: $$t\approx -18241.29$$ The negative sign means the wood was cut approximately 18,241 years before the present time.

Key concepts

Radioactive Decay Equation

Understanding the radioactive decay equation is essential for interpreting carbon dating results. Radioactive decay is a random process where an unstable atomic nucleus loses energy by emitting radiation. In the context of carbon dating, we're dealing with the decay of Carbon-14 ((C-14)). This isotope has a fixed rate of decay, which is expressed through its half-life. The basic form of the radioactive decay equation that relates the remaining number of radioactive atoms (N(t)) at time t to the original number of atoms (N_0) is given by: $$\frac{N(t)}{N_0}=(\frac{1}{2}{)}^{\frac{t}{\text{half-life}}}$$In this exercise, knowing that only a certain percentage of (C-14) remains in an object, we can use this equation to solve for the time t, which tells us when the object stopped exchanging carbon with the environment—typically when it died or was manufactured.

Logarithmic Functions

Logarithmic functions are indispensable tools when working with exponential relationships, such as those found in the radioactive decay equation. A logarithm answers the question: to what exponent do we need to raise a base number to obtain another number? Mathematically, for a base b and a number y, the logarithm log_b(y) is the number x such that b^x = y. In carbon dating, we often deal with base 2 logarithms because the decay equation involves the factor of (1/2)^{t/\text{half-life}}, which corresponds to the half-life concept. To solve for tt in our decay equation, we take the logarithm of both sides which allows us to untangle t from the exponent.

For example, if (1/2)^{t/5730} = 0.77, applying the base 2 logarithm would yield $$\frac{t}{5730}=lo{g}_2(0.77)$$Then by simplifying, we can find the value of t and learn about the age of the archaeological find.

Half-life Calculation

Half-life is a critical concept in radioactive decay and carbon dating. It's the time required for half the quantity of a radioactive element to undergo decay. Carbon-14 has a half-life of approximately 5730 years. Using this information, we can calculate how long ago a piece of organic material stopped absorbing (C-14) - when it was cut down or used by humans, for instance.

The half-life calculation is embedded in the decay equation we've discussed. By solving the equation for t after inserting the half-life of the isotope, we can work out when the material was last active. In our cloth example, only 77% remains, meaning it's been less than a half-life since the cloth was painted. Conversely, for the wood, only 6.2% of the (C-14) is left, indicating many half-lives have passed. This is evidenced by the much older calculated age of 18,241 years.