Please help me solve this question:
A layer of peat beneath the glacial sediments of the last ice age had a carbon-14 content of 25% of that found in living organisms. How long ago was this ice age?
You will need k and you don't have information to get it. Look up the half life of C-14 which I think is in the neighborhood of 5800 years. Use the exact number when you find it.
Then k = 0.693/t1/2
ln(No/N) = kt
Substitute 100 for No.
N = 25
k from the half life
Solve for t.
To determine how long ago the ice age occurred, we need to understand the concept of carbon-14 dating and its decay rate.
Carbon-14 is a radioactive isotope that is naturally present in living organisms, and its half-life is approximately 5730 years. This means that after 5730 years, only half of the original amount of carbon-14 remains.
In the question, it is mentioned that the peat layer had a carbon-14 content of 25% of that found in living organisms. This implies that the remaining 75% has decayed since the time of the ice age.
Let's denote the original amount of carbon-14 in the peat layer as 100%. After the decay, we are left with 25% of the original amount. Using this information, we can set up an equation:
(100%) * (0.5)^n = (25%)
In this equation, 'n' represents the number of half-lives that have passed since the ice age.
To solve for 'n,' we can take the logarithm of both sides of the equation to eliminate the exponent:
log(0.5^n) = log(0.25)
n * log(0.5) = log(0.25)
Using the properties of logarithms, we can simplify the equation further:
n = log(0.25) / log(0.5)
Evaluating this expression using a calculator, we get:
n ≈ 2.3219
Since 'n' represents the number of half-lives, we can round this value to 2, as we cannot have a fraction of a half-life. Therefore, approximately 2 half-lives have passed since the ice age.
To find the time elapsed, we multiply the number of half-lives by the half-life of carbon-14:
Time elapsed ≈ 2 * 5730 years
Calculating this, we find:
Time elapsed ≈ 11,460 years
Therefore, the ice age occurred approximately 11,460 years ago.
To solve this question, we need to understand the concept of carbon-14 dating. Carbon-14 (or radiocarbon) dating is a method used to determine the age of organic materials based on the decay rate of carbon-14 isotopes.
The first step is to recognize that living organisms have a normal amount of carbon-14 in their bodies. When an organism dies, it no longer takes in any new carbon-14, and over time, the carbon-14 in its remains undergoes radioactive decay with a known half-life of about 5730 years.
Now, let's approach the question.
Given that the layer of peat beneath the glacial sediments had a carbon-14 content of 25% of that found in living organisms, it means that after the ice age, there was still some carbon-14 left in the peat, albeit reduced to 25% compared to a living organism.
To determine how long ago the ice age happened, we can use the half-life of carbon-14 to calculate the number of half-lives that occurred to reach 25% of its original content.
Since the radioactive decay of carbon-14 occurs with a half-life of 5730 years, one half-life would reduce the carbon-14 content by half, and two half-lives would reduce it to a quarter (25%).
Let's calculate the number of half-lives that occurred:
Number of Half-lives = Logarithm(Base 0.5)(Final amount/Initial amount)
Number of Half-lives = Logarithm(0.5)(0.25/1)
Number of Half-lives ≈ 2
Therefore, two half-lives have occurred for the carbon-14 content in the peat layer to reach 25% of its original amount.
To determine the time it took for these two half-lives, we multiply the half-life duration by the number of half-lives:
Time = Half-life x Number of Half-lives
Time = 5730 years x 2
Time ≈ 11,460 years
Therefore, the ice age occurred approximately 11,460 years ago.