Answer:
Step-by-step explanation:
To calculate the dates for a zircon, we will use the uranium-lead (U-Pb) dating method. The U-Pb dating method is based on the radioactive decay of uranium isotopes to lead isotopes. The decay of uranium to lead occurs at a known rate, so by measuring the amount of uranium and lead isotopes in a sample, we can calculate the age of the sample.
First, we need to calculate the ratios of the radiogenic (formed by radioactive decay) Pb isotopes to the non-radiogenic Pb isotope in the sample:
206Pb*/204Pb = (206Pb/204Pb) - (206Pb/204Pb)i
= 1960.8 - 14.2
= 1946.6
207Pb*/204Pb = (207Pb/204Pb) - (207Pb/204Pb)i
= 464.9 - 15.0
= 449.9
208Pb*/204Pb = (208Pb/204Pb) - (208Pb/204Pb)i
= 147.4
Next, we can calculate the age of the zircon using the following equation:
t = ln[(1 + (Pb*/U))/((Pb*/U) × e^(λt))] / λ
where t is the age of the zircon, Pb*/U is the ratio of radiogenic Pb isotopes to uranium, λ is the decay constant for uranium, and e is the mathematical constant approximately equal to 2.718.
We can assume the initial ratio of Pb*/U to be zero, as the non-radiogenic lead is inherited from the rock material where the zircon formed. The decay constant for U-238 is 1.55125×10^(-10) per year.
For the first date, we can calculate the age using the 206Pb*/238U ratio:
206Pb*/238U = (206Pb/238U) × (U/204Pb) × (204Pb/204Pb*)
= (962 × 10^(-6)) × (1/548) × (1/1946.6)
= 9.26×10^(-12)
t = ln[(1 + (206Pb*/238U))/((206Pb*/238U) × e^(1.55125×10^(-10) × t))] / (1.55125×10^(-10))
Solving for t, we get:
t = 1.15 billion years
For the second date, we can calculate the age using the 207Pb*/235U ratio:
207Pb*/235U = (207Pb/235U) × (U/204Pb) × (204Pb/204Pb*)
= (962 × 10^(-6)) × (1/548) × (1/449.9)
= 1.77×10^(-11)
t = ln[(1 + (207Pb*/235U))/((207Pb*/235U) × e^(9.8485×10^(-10) × t))] / (9.8485×10^(-10))
Solving for t, we get:
t = 1.21 billion years
For the third date, we can calculate the age using the 206Pb*/207Pb* ratio:
206Pb*/207Pb* = (206Pb/207Pb) - (206Pb/207Pb)i
= (1960.8/464.9) - (14.2