A 1.00-g sample of carbon from a modern
source gave 15.3 disintegrations per minute. A sample of carbon from an “old” source gave 920 disintegrations per hour. What is the age of the “old” sample of carbon? The half-life of carbon-14 is 5.73×103 year.
a. about 34,400 years
b. about 345,000 years
c. about 20 million years
d. less than a few years
Typo: The half-life is 5.7x10^3
Without even doing the math it the change in disintegrations is merely .03333, the sample must have decayed for
d. less than a few years
k = 0.693/t1/2
k = 0.693/5.7E3 = 1.22E-4
ln(15.3333/15.3) = 1.22E-4(t)
t = about 18 year.
vanden bout
actually drake was right on his first answer i had this question on my homework and it is d. less than a few years :)
To determine the age of the "old" sample of carbon, we can use the radioactive decay formula:
N = N0 * (1/2)^(t / T)
Where:
- N is the final amount of the radioactive substance
- N0 is the initial amount of the radioactive substance
- t is the time that has passed
- T is the half-life of the radioactive substance
Given that the half-life of carbon-14 is 5.73×10^3 years, we can solve for t using the information provided.
First, let's start with the modern source of carbon. From the given information, we know that a 1.00-g sample of carbon from a modern source gave 15.3 disintegrations per minute. We need to convert this rate to disintegrations per hour for consistency:
15.3 disintegrations/minute * 60 minutes/hour = 918 disintegrations/hour (approximately)
Now, let's calculate the initial amount of carbon (N0) for the modern source using the disintegrations per hour:
N0 = N / (1/2)^(t / T)
N0 = 918 / (1/2)^(0 / 5730)
Since the modern sample is from a current source, we assume that t is zero. Therefore:
N0 = 918
Now let's move on to the old source of carbon. From the given information, we know that a sample of carbon from the old source gave 920 disintegrations per hour.
We can use the same formula to calculate the time (t) for the old source:
N = N0 * (1/2)^(t / T)
920 = 918 * (1/2)^(t / 5730)
To solve for t, we can take the logarithm of both sides:
log2(920/918) = t / 5730 * log2(1/2)
Simplifying:
log2(1.00218) = t / 5730 * (-1)
Dividing both sides by log2(1/2):
t / 5730 = log2(1.00218) / log2(1/2)
t = (log2(1.00218) / log2(1/2)) * 5730
Using a calculator, we find:
t ≈ 1197 years
Therefore, the age of the "old" sample of carbon is approximately 1197 years, which is significantly less than any of the given answer choices (a few thousand years). So the correct answer is:
d. less than a few years.