The radioactive decay of carbon-14 is first-order and the half-life is 5800 years. While a plant

or animal is living, it has a constant proportion of carbon-14 (relative to carbon-12) in its
composition. When the organism dies, the proportion of carbon-14 decreases as a result of
radioactive decay and the age of the organism can be determined if the proportion of carbon14
in its remains is measured. If the proportion of carbon-14 in an ancient piece of wood is
found to be one quarter that in living trees, how old is the sample?

k = 0.693/t1/2

Then ln(No/N) = kt
If you call No 100 (as in 100%), then N will be 25%, you know k and you olve for t in years. Post your work if you get stuck.

To determine the age of the sample, we can use the half-life formula for radioactive decay. The half-life formula is given by:

N = N0 * (1/2)^(t / T)

Where:
N is the final amount of carbon-14 in the sample (one quarter of the initial amount)
N0 is the initial amount of carbon-14 in the sample (amount in living trees)
t is the time that has passed (unknown)
T is the half-life of carbon-14 (5800 years)

Given that the proportion of carbon-14 in the ancient sample is one quarter of that in living trees, we can say:

N = (1/4) * N0

Plugging the values into our equation, we get:

(1/4) * N0 = N0 * (1/2)^(t / T)

Now, let's solve for t:

(1/4) = (1/2)^(t / T)

Taking the logarithm of both sides (base 2) to eliminate the exponent:

log2(1/4) = log2((1/2)^(t / T))

-2 = (t / T)

Simplifying further:

t = -2 * T

Substituting the value of T (half-life of carbon-14 = 5800 years):

t = -2 * 5800

t = -11600 years

Since time cannot be negative, we can ignore the negative sign. Therefore, the ancient piece of wood is approximately 11600 years old.

To determine the age of the ancient wood sample, we need to understand the concept of radioactive decay and how the half-life of carbon-14 is related to its decay.

Radioactive decay is a process in which unstable atoms lose energy or particles in order to become more stable. Carbon-14 is a radioactive isotope of carbon that undergoes this decay. It decays at a constant rate, and its half-life is 5800 years. This means that after 5800 years, half of the carbon-14 atoms in a sample will have decayed.

By measuring the proportion of carbon-14 in the ancient wood sample relative to carbon-12, we can determine how much decay has occurred and therefore estimate the age of the sample.

Let's assume that the proportion of carbon-14 in living trees is represented by 100% or 1.0. In the ancient wood sample, if the proportion of carbon-14 is only one quarter (1/4) that of living trees, it means that 75% (0.75) of the carbon-14 has decayed.

Since each half-life represents a 50% reduction in carbon-14, we can calculate the number of half-lives that have passed since the ancient wood was alive.

To find the number of half-lives, we can use the following formula:

Number of Half-Lives = ln(Ratio of Remaining Carbon-14) / ln(0.5)

Where "ln" represents the natural logarithm.

Let's substitute the given values into the formula:

Number of Half-Lives = ln(0.25) / ln(0.5)

Using a calculator, we can find the natural logarithm of 0.25 divided by the natural logarithm of 0.5. This gives us approximately 2.7726.

Therefore, the number of half-lives that have passed is approximately 2.7726.

Now, to calculate the age of the ancient wood sample, we multiply the number of half-lives by the half-life of carbon-14:

Age = Number of Half-Lives * Half-Life

Age = 2.7726 * 5800 years

Using a calculator, we find that the age of the ancient wood sample is approximately 16,027 years.

So, based on the proportion of carbon-14 in the sample being one quarter that of living trees, the age of the wood sample is estimated to be approximately 16,027 years.