(a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years.
(b)How much Carbon-14 will be present in the sample 100 years from now?
Need easily understandable steps to get my head around.
Thanks in advance
For half-life questions, it is usual to use a base of 1/2 or .5
amount = 12 (1/2)^(t/5700) , where t is the number of years
if the amount = 10
10 = 12 (.5)^(t/5700)
.8333... = .5^(t/5700)
take logs of both sides and use log rules
log .8333... = (t/5700) log .5
t/5700 = log .8333... / log .5
see if you can get t = appr 1499 years
for your second part, evaluate
amount = 12 (.5)^(100/5700)
let me know what you got
in general, for half-life questions:
amount = a (.5)^(t/k)
where a is the starting amount, and k is the half-life time , t and k of the same units of time
For "doubling" type of questions, use a base of 2
To determine the age of the fossil in question (a), we can use the concept of half-life. The half-life of Carbon-14 is 5700 years, which means that after 5700 years, half of the Carbon-14 atoms in a sample will have decayed.
(a) Let's start by calculating the number of half-lives that have occurred between the time the fossil died and the present time:
Step 1: Calculate the fraction of Carbon-14 remaining in the fossil:
Fraction Remaining = (Current Carbon-14 / Initial Carbon-14)
Given that the current Carbon-14 content is 10 grams and the initial Carbon-14 content was 12 grams, the fraction remaining is:
Fraction Remaining = (10 g / 12 g) = 0.833
Step 2: Calculate the number of half-lives (n) using the equation:
Fraction Remaining = (1/2)^n
Where n is the number of half-lives.
0.833 = (1/2)^n
Taking the logarithm of both sides:
log(0.833) = log[(1/2)^n]
Using the logarithmic property that log(a^b) = b * log(a):
log(0.833) = n * log(1/2)
Rearranging the equation to solve for n:
n = log(0.833) / log(1/2) = 0.186 / (-0.301) = 0.618
Step 3: Calculate the age of the fossil:
Age = Number of half-lives * Half-life
Using the value of n we calculated:
Age = 0.618 * 5700 years = 3520.26 years
Therefore, the age of the fossil is approximately 3520.26 years.
(b) To determine the amount of Carbon-14 that will be present in the sample 100 years from now, we can use the half-life concept again:
Step 1: Calculate the number of half-lives that will occur in 100 years:
Number of Half-lives = Time / Half-life = 100 years / 5700 years = 0.01754
Step 2: Calculate the fraction remaining in the sample:
Fraction Remaining = (1/2)^(Number of Half-Lives) = (1/2)^(0.01754) ≈ 0.986
Step 3: Calculate the amount of Carbon-14 that will be present:
Current Carbon-14 content * Fraction Remaining = 10 g * 0.986 ≈ 9.86 grams
Therefore, approximately 9.86 grams of Carbon-14 will be present in the sample 100 years from now.