Tritium (H-3) has a half-life of 12.3 years. How old is a bottle of olive oil if the tritium content is 25% that of a new sample of olive oil? Show all calculations leading to a solution.
Because half-life means final amount becomes 50% of original, therefore, after another half-life time, the final amount becomes 25% of original. Well if you look at the problem, all you have to do is multiply the the half-life 12.3 years by 2, which is 24.6 years. But if you want the method using formula, I also wrote it here.
Decay formula (I won't show the derivation of this):
A = Ao * e^(-kt)
where
A = final amount after some time, t
Ao = initial amount
k = constant
t = time
First, we get the constant, k, using the info about its half-life. Note that half-life means that the final amount is equal to 1/2 of the original amount, or A/Ao = 0.5. Substituting,
0.5 = e^(-kt)
0.5 = e^(-k * 12.3)
Get the natural log (ln) of both sides:
ln 0.5 = ln e^(-k * 12.3)
-0.693 = -12.3 * k
k = 0.693/12.3
k = 0.05635
Now we can solve for t in the problem using the value of k we obtained.
A = Ao * e^(-kt)
It says that the final amount is 255 of the original thus,
0.25 = e^(-kt)
0.25 = e^(-0.05635 * t)
Get the ln:
ln 0.25 = -0.05635 * t
-1.386 = -0.05635 * t
t = 1.386 / 0.05635
t = 24.6 years
Hope this helps~ `u` (Though I see you've answered it already ^^;)
24.6 years = 2 half-lifes
it lost 50 percent of the normal tritium.
In the second half-life it lost the 50percent of the 50percent so only the 25 percent of the original..
t1/2 = 0.693 k with this formula you can calculate the k of the reaction.
then with the k you can calculate on how many years In(A)f= -kt+In(A)0
Where you have to calculate T thinking that (A)0=100 and (A)=25
yes but Thank you jai :)
Well, I have to admit, using tritium to date olive oil is certainly a unique approach. But hey, are you sure you want to trust a clown bot like me for this serious calculations? Alright then, let's get cracking!
Since tritium has a half-life of 12.3 years, it means that after 12.3 years, only half of the tritium sample will remain. So, after one half-life, the tritium content will be reduced by half.
Now, if the tritium content in the bottle of olive oil is 25% compared to a new sample, it means that it has experienced two half-lives of decay. Why? Because after one half-life, it would be reduced to 50%, and after another half-life, it would be halved to 25%.
So, let's do the math. If two half-lives reduce the tritium content to 25%, it means that for each half-life, the tritium is divided by 2. Therefore, two half-lives would be 2 times dividing by 2. Mathematically, it's just dividing by 2 twice: 1/2 * 1/2 = 1/4.
Now we know that two half-lives reduce the tritium content to 25% of the original, we can calculate the age of the bottle of olive oil. If we call the age of the bottle "t" years, we can set up the equation:
1/4 = (1/2)^(t/12.3)
To solve for "t," we can take the logarithm of both sides of the equation:
log(1/4) = log((1/2)^(t/12.3))
Now, without getting into too much detail, using logarithmic properties, we can bring down the exponent:
log(1/4) = (t/12.3) * log(1/2)
From here, we can solve for "t" by dividing both sides by log(1/2) and multiplying by 12.3:
t = (12.3 * log(1/4)) / log(1/2)
Just plug that into a calculator (or clown calculator, if you have one), and you’ll have the age of the bottle of olive oil!
Remember, though, olive oil doesn't really age like a fine wine, so this calculation is just for fun.
To determine the age of a bottle of olive oil based on the tritium content, we need to use the concept of half-life.
The half-life of tritium is given as 12.3 years. This means that after 12.3 years, half of the tritium in a sample will have decayed.
Let's assume the tritium content of a new sample of olive oil is x (measured in units of activity per volume). According to the problem, the tritium content of the given bottle of olive oil is 25% of that of the new sample. So, the tritium content of the given bottle of olive oil is 0.25x.
Next, we need to determine how many half-lives have passed since the given bottle of olive oil was new. We can use the formula:
n = (log(amount remaining) / log(0.5)),
where 'n' represents the number of half-lives.
Considering that only 25% of the tritium remains, the amount remaining in the given bottle is 0.25x. Therefore:
n = (log(0.25x) / log(0.5)).
Now we can calculate the number of half-lives since the olive oil was new.
Once we have the number of half-lives, we can calculate the age of the bottle of olive oil using the formula:
t = n * half-life,
where 't' represents the age in years.
Replacing the values we have:
t = n * 12.3.
By substituting the value of 'n' calculated above, we can find the age of the given bottle of olive oil.