A rock containing 238U and 206Pb was examined to determine its appropriate age. The molar ratio of 206Pb to 238U was 0.1462. Assuming no lead was originally present and that all lead formed remained in the rock. What is the age of the rock in years? The half life of 238U is 4.500*10^9yr

amount remaining= original*e^(-693t/halflife)

.1462=1 e^( )
take the ln of each side

ln (.1462)=-.693 t/4.5E9

t= -4.5E9 /693 * ln (.1462) That is a long time.

I tried this but was told the answer was wrong.

To determine the age of the rock, we can use the concept of radioactive decay and the relationship between the isotopes 238U and 206Pb. The half-life of 238U is given as 4.500*10^9 years.

The molar ratio of 206Pb to 238U in the rock can be used to determine how many half-lives have passed for the uranium decay to lead. Since no lead was originally present and all the lead formed remained in the rock, we can assume that all the lead was produced through the decay of uranium.

Let's start by finding the number of half-lives:

1. Calculate the current ratio of 206Pb to 238U:
Ratio = [206Pb] / [238U] = 0.1462

2. Take the natural logarithm of both sides of the equation to convert the ratio to a time-based relationship:
ln(Ratio) = ln(0.1462)

3. Using the property of logarithms, we know that ln(a/b) = ln(a) - ln(b):
ln(0.1462) = ln([206Pb] / [238U]) = ln([206Pb]) - ln([238U])

4. Calculate the natural logarithm of the ratio:
ln([206Pb]) - ln([238U]) = ln(0.1462)

5. Since no lead was originally present, the initial concentration of [206Pb] was 0, so ln([206Pb]) = -∞. Therefore, we can ignore it in the equation:
-ln([238U]) = ln(0.1462)

6. Multiply both sides by -1 to isolate ln([238U]):
ln([238U]) = -ln(0.1462)

7. Use the relationship between natural logarithm and exponential functions to solve for [238U]:
[238U] = e^(-ln(0.1462))

8. Calculate the number of half-lives by dividing the ratio of [238U] to its initial concentration (which is 1) by 0.5 (representing one half-life):
Number of half-lives = ln([238U]) / ln(0.5)

9. Finally, calculate the age of the rock by multiplying the number of half-lives by the half-life of 238U:
Age of the rock = Number of half-lives * half-life of 238U

By following these steps, you should be able to determine the age of the rock in years based on the given molar ratio of 206Pb to 238U and the half-life of 238U.