Two radioactive sources X and Y have half-lives of 3.0 hours and 2.0 hours, respectively. The product of the decay is a stable isotope of the element Z. Six hours ago a mixture contained the same number of atoms of both X and Y, and no other atoms.
What fraction of the mixture is now made up of atoms of Z?
(the answer is 12/16)
So the issue is having X,Y equal to make Z. The amount of X after t hours is
A is original amount) X(t)=Ae^-kt/3
and amount of Y after t hours is
Y(t)=Ae^-kt/2
The slower of the two is X changing, so the number of Z= X(t)
Fraction unchanged = Z/(total excess Y+unchanged atoms)
total excess Y= Y(t)-X(t)
unchanged X= X0-X(t) = A-X(t)
unchanged y=A-y(t)
fraction mixture unchanged= Z/(total excess Y+unchanged atoms)
= X(t)/ (Y(t)-X(t)+A-X(t)+A-Y(t) )
= X(t)/ ( -2X(t)+2A )=1/2 (1/(1-e-kt/3)
= 1/2 (1-1/2)=1/4
fraction changed (ie, now Z)= 1-fraction unchanged= 3/4 or 12/16 if you prefer that.
To determine the fraction of the mixture made up of atoms of Z, we need to calculate how many half-life periods have passed for each radioactive source.
Let's start with source X, which has a half-life of 3.0 hours. Since 6 hours have passed, we can divide 6 by the half-life to find the number of half-life periods:
Number of half-life periods for X = 6 hours / 3.0 hours per half-life = 2
Therefore, source X has undergone 2 half-life periods.
Now let's do the same for source Y, which has a half-life of 2.0 hours. Similarly, we can divide 6 by the half-life:
Number of half-life periods for Y = 6 hours / 2.0 hours per half-life = 3
Therefore, source Y has undergone 3 half-life periods.
Next, we need to determine the fraction of each source that remains after the given number of half-life periods.
Each half-life period reduces the initial amount of a source by half (i.e., half of the atoms decay). So, after 2 half-life periods, the fraction remaining for source X is:
Fraction remaining for X = 1/2 * 1/2 = 1/4
Similarly, after 3 half-life periods, the fraction remaining for source Y is:
Fraction remaining for Y = 1/2 * 1/2 * 1/2 = 1/8
Finally, to find the fraction of the mixture that is made up of atoms of Z, we subtract the fractions remaining for X and Y from 1 (since the remaining fraction is the fraction of decayed atoms):
Fraction of Z = 1 - fraction remaining for X - fraction remaining for Y
= 1 - 1/4 - 1/8
= 8/8 - 2/8 - 1/8
= 5/8
However, the problem states that the answer is 12/16, which is equivalent to 3/4. Given the information provided, it seems the answer may be incorrect.
To determine the fraction of the mixture that is now made up of atoms of Z, we need to understand how much of X and Y remains after six hours and how much of the decay product Z is formed.
First, let's calculate the number of half-lives that have passed for each radioactive source. We can use the formula N = N₀ * (1/2)^(t / T), where N is the remaining quantity, N₀ is the initial quantity, t is the time passed, and T is the half-life.
For source X:
N = N₀ * (1/2)^(t / T)
N = N₀ * (1/2)^(6 / 3)
N = N₀ * (1/2)^2
N = N₀ / 4
For source Y:
N = N₀ * (1/2)^(t / T)
N = N₀ * (1/2)^(6 / 2)
N = N₀ * (1/2)^3
N = N₀ / 8
After six hours, both sources X and Y have decayed to 1/4 and 1/8 of their initial quantities, respectively.
Next, let's calculate the number of atoms of Z formed. Since Z is the product of the decay of both X and Y, the number of atoms of Z formed will be equal to the number of atoms that have decayed from X and Y.
For source X: (3/4) * N₀
For source Y: (7/8) * N₀
Calculating the total number of atoms of Z formed:
Total Z = (3/4) * N₀ + (7/8) * N₀
Total Z = (24/32) * N₀ + (28/32) * N₀
Total Z = (52/32) * N₀
Now, to find the fraction of the mixture that is made up of atoms of Z, we divide the total number of atoms of Z by the initial number of atoms in the mixture:
Fraction Z = Total Z / (N₀ + N₀)
Fraction Z = (52/32) * N₀ / (2 * N₀)
Fraction Z = (52/32) / 2
Fraction Z = 52/64
Fraction Z = 13/16
So, the fraction of the mixture now made up of atoms of Z is 13/16, not 12/16 as mentioned.