A crystal of zircon separated from an igneous rock, was analyzed, and found to contain 7930 atoms of 207Pb, and 12070 atoms of 235U. Knowing the half life of 235U is 713 million years, how old is the zircon crystal? Please show your calculations so that I can figure out how to do this type of problem.
The method you need to use is explained at
http://geology.about.com/od/geotime_dating/a/uraniumlead.htm
To determine the age of the zircon crystal, we need to use the concept of radioactive decay and the ratio of isotopes within the crystal. Here's how you can calculate it step by step:
1. Determine the ratio of parent to daughter atoms:
The ratio of 207Pb (parent) to 235U (daughter) can be calculated by dividing the number of 207Pb atoms by the number of 235U atoms:
Ratio = (number of 207Pb atoms) / (number of 235U atoms)
= 7930 / 12070
2. Calculate the age of the crystal using the ratio:
The ratio of parent to daughter atoms can be expressed using the radioactive decay equation:
Ratio = e^(-λt)
Where:
- λ is the decay constant, calculated as ln(2)/T (T is the half-life of the isotope)
- t is the age of the crystal in years
Substituting the values into the equation:
7930 / 12070 = e^(-λt)
3. Solve for the age of the crystal:
To solve for t, we need to isolate the variable t. Take the natural logarithm (ln) of both sides of the equation:
ln(7930 / 12070) = -λt
Now, rearrange the equation to solve for t:
t = (ln(7930 / 12070)) / -λ
4. Calculate the decay constant (λ):
The decay constant (λ) can be calculated using the formula: λ = ln(2) / T, where T is the half-life of 235U:
λ = ln(2) / (713 million years)
5. Substitute the value of λ into the equation from step 3 and solve for t:
t = (ln(7930 / 12070)) / (-ln(2) / (713 million years))
6. Perform the calculations:
Divide the natural logarithm of 7930 divided by 12070:
ln(7930 / 12070) ≈ -0.372
Substitute the values into the equation:
t = (-0.372) / (-ln(2) / (713 million years))
Simplify the equation further:
t = (-0.372) / (-0.693 / (713 million years))
7. Calculate the final result:
Now, simply perform the calculation:
t ≈ (0.372) * (713 million years) / 0.693
The final result will give you the age of the zircon crystal.