the activity of a sample of carbon has one-eighth of the initial activity. carbon-14 has a half-life of 5730 yr. how old is the sample in yr?
one eighth is three half-lives.
Now the hard way:
amount left=original amount(1/2)^t/thalflife
1/8=(1/2)^t/th
t/th=3
or even harder:
amountleft=original*e^-.693t/th
1/8=e^-.693 t/th
ln of each side
ln (1/8)=-.693 t/th
-2.0793=-.693 t/th
t/th= 3
the question ask how old is the sample in yr. so that mean t/th is 3 yrs.
To determine the age of the sample, we can use the concept of radioactive decay and the half-life of carbon-14. The half-life is the amount of time it takes for half of the radioactive substance to decay.
Given that the activity of the sample is one-eighth of the initial activity, we can infer that the remaining carbon-14 is one-eighth of the original amount.
Since the half-life of carbon-14 is 5730 years, after each half-life, the amount of carbon-14 remaining is halved. So, we can set up the following equation:
(1/2)^n = 1/8
Here, 'n' represents the number of half-lives. We want to find the value of 'n'. We can simplify the equation:
1/2^n = 1/8
To solve for 'n', we can take the logarithm of both sides. I will use the base 2 logarithm (log2):
log2(1/2^n) = log2(1/8)
-n = log2(1/8)
Using the property of logarithms that states log(a/b) = log(a) - log(b):
-n = log2(1) - log2(8)
Since log2(1) = 0, we can simplify further:
-n = -log2(8)
Using the fact that log2(8) = 3:
-n = -3
We can solve for 'n' by multiplying both sides by -1:
n = 3
Therefore, the number of half-lives 'n' is equal to 3. The age of the sample can be calculated by multiplying the number of half-lives by the half-life of carbon-14:
Age = n * half-life
Age = 3 * 5730 years
Age ≈ 17,190 years
Hence, the sample is approximately 17,190 years old.