A rock contains 200. µmol 238U (t1/2 = 4.5 109 yr) and 116 µmol 206Pb. Assuming that all the 206Pb comes from decay of the 238U, estimate the rock's age.
Please explain step by step.
I would do this.
Take the ratio of moles U/moles Pb which is 200/116 = 1.72
Suppose we start with 100 atoms U, then X of them will decay to form Pb and 100-X will remain.
U/Pb = (100-X/X) = 1.72 and solve for X. I get something like 63 but that's an estimate and you need to recalculate it.
Then k = 0.693/t1/2 and substitute below.
ln(No/N) = kt
Use 100 for No
Use 63 (the more accurate number you get from above).
k from above
Solve for t in years.
I don't understand the part when it says 100-X/X. How do you solve for X?
Cross multiply.
100-X = 1.72X and go from there.
where are you getting that k value from? if you look at the half life of U i calculated it to be 1.5e-10
To estimate the rock's age based on the given information, we can use the concept of radioactive decay and the decay equation.
The decay equation for 238U is: 238U → 206Pb + 8He
Here are the steps to estimate the rock's age:
Step 1: Calculate the number of moles of 206Pb:
To convert the given mass of 206Pb into moles, we need to know the molar mass of 206Pb.
Molar mass of 206Pb = 206 g/mol
Number of moles of 206Pb = mass of 206Pb / molar mass of 206Pb
Number of moles of 206Pb = 116 µmol / (206 g/mol × 10^-6 mol/µmol)
Number of moles of 206Pb = 116 × 10^-6 mol / (206 × 10^-6)
Number of moles of 206Pb = 0.563 mol
Step 2: Calculate the number of moles of 238U:
Given that the number of moles of 206Pb is equal to the number of moles produced by the decay of 238U, we can conclude that the number of moles of 238U is also 0.563 mol.
Step 3: Determine the ratio of the number of moles of 238U to the number of moles remaining of 238U:
The decay equation for 238U shows that one atom of 238U produces one atom of 206Pb.
Since the number of moles is a measure of the number of atoms, the ratio of the number of moles of 238U to the number of moles remaining of 238U is always 1:1.
Step 4: Calculate the number of 238U atoms in the rock:
The Avogadro's number (6.022 × 10^23) is used to convert moles to atoms.
Number of 238U atoms = number of moles of 238U × Avogadro's number
Number of 238U atoms = 0.563 mol × 6.022 × 10^23 atoms/mol
Step 5: Calculate the number of 238U atoms that have decayed:
Since the number of moles remaining of 238U is 0.563 mol, the number of moles that have decayed is the initial number of moles minus the remaining number of moles, which is also 0.563 mol.
Step 6: Calculate the number of 238U atoms that have decayed over time:
Given that the half-life of 238U is 4.5 × 10^9 years, we can determine the number of decays that have occurred over time using the formula:
Number of decays = initial number of atoms × (1/2)^(time elapsed / half-life)
Step 7: Calculate the rock's age:
Since we have the number of decays and we need the time, we rearrange the decay equation to solve for time:
Number of decays = initial number of atoms × (1/2)^(time / half-life)
time = half-life × log(base 1/2) (Number of decays / initial number of atoms)
Now we can plug in the values:
time = (4.5 × 10^9 years) × log(base 1/2) (0.563 mol / (0.563 mol × 6.022 × 10^23 atoms/mol))
Simplifying this equation should give us the estimated age of the rock in years.