Based on mass spectrometer analysis of Uranium-235 and Lead-207, I have determined the age of a zircon to be 89.125 million years old. How many half-lives passed? What are the percentages of Uranium-235 and Lead-207?
To determine the number of half-lives that have passed, we can use the radioactive decay equation:
N = N₀ * (1/2)^(t/T)
where:
N = the current amount of the radioactive isotope
N₀ = the initial amount of the radioactive isotope
t = the time that has passed
T = the half-life of the radioactive isotope
In this case, we know that the zircon is 89.125 million years old. Let's assume the initial amount of Uranium-235 (N₀) was equal to 100. We can substitute the given values into the equation:
N = 100 * (1/2)^(89.125 / T)
Now, we need to solve for T, the half-life of Uranium-235. We can do this by rearranging the equation:
(1/2)^(89.125 / T) = N / 100
Take the logarithm of both sides to isolate T:
log[(1/2)^(89.125 / T)] = log(N / 100)
Using logarithmic rules, we can rewrite the equation as:
(89.125 / T) * log(1/2) = log(N / 100)
Finally, solve for T:
T = (89.125 / (log(N / 100) / log(1/2)))
Now that we have the value of T (the half-life), we can calculate the number of half-lives that have passed by dividing the age of the zircon by the half-life:
Number of half-lives = Age of the zircon / T
To determine the percentages of Uranium-235 and Lead-207, we need to know the ratio of the amounts of each isotope in the zircon. Since we have the number of half-lives that have passed, we can calculate the remaining percentage of Uranium-235 by using the equation:
Percentage of Uranium-235 = (1/2)^(Number of half-lives) * 100
And the percentage of Lead-207 is simply the remaining percentage after subtracting the Uranium-235 percentage:
Percentage of Lead-207 = 100 - Percentage of Uranium-235
Using these calculations, we can determine the number of half-lives that have passed and the percentages of Uranium-235 and Lead-207 in the zircon.