The half-life for the process 238 U --> 206 Pb is 4.5*10^9 yr. A mineral sample contains 66.0 mg of 238 U and 16.0 mg of 206Pb. What is the age of the mineral?
Well, let's do some calculations here. Since the half-life of 238 U is 4.5*10^9 years, that means after 4.5*10^9 years, half of the original 238 U will have decayed into 206 Pb.
Now, let's see how much 238 U is left in the mineral sample. We start with 66.0 mg of 238 U, and after 4.5*10^9 years, half of it would have decayed, so we're left with 33.0 mg of 238 U.
Since the amount of 206 Pb in the sample is 16.0 mg, that means it was produced as a result of the decay of the 238 U. Therefore, the 16.0 mg of 206 Pb was originally part of the 33.0 mg of 238 U.
Now, we need to figure out how long it takes for half of 33.0 mg (which is 16.5 mg) to decay into 206 Pb. It's like a never-ending game of divide and conquer!
But, here's where the joke comes in... The age of the mineral is so old that it's practically prehistoric! We're talking about billions of years! So, brace yourself for the punchline: The age of the mineral is "ancient"!
To find the age of the mineral, we need to use the concept of half-life and the ratio of parent to daughter isotopes.
Let's start by calculating the number of half-lives that have passed for the decay of 238 U to 206 Pb.
The half-life of 238 U is given as 4.5 * 10^9 years.
Formula to calculate the number of half-lives (n) is n = t / T½,
where t is the age of the mineral and T½ is the half-life of the radioactive substance.
So, using the given half-life, we have:
n = t / (4.5 * 10^9)
Now, let's find the ratio of 238 U to 206 Pb in the mineral sample.
The initial amount of 238 U is 66.0 mg, and the amount of 206 Pb is 16.0 mg.
The ratio of 238 U to 206 Pb is:
238 U / 206 Pb = 66.0 mg / 16.0 mg
Let's simplify this ratio:
238 U / 206 Pb ≈ 4.125
For each half-life that passes, the ratio of 238 U to 206 Pb is halved because half of the 238 U decays to 206 Pb. So, after "n" half-lives, the ratio is given by:
(238 U / 206 Pb)^n = 4.125^n
We know that the ratio at the time of the sample is 4.125. So, we can set up an equation:
4.125^n = 238 U / 206 Pb ≈ 4.125
Solving for "n":
n ≈ log base 4.125 of 4.125
n ≈ 1
This means one half-life has passed.
Therefore, the age of the mineral sample is equal to the half-life:
t = (1 * 4.5 * 10^9) years
t ≈ 4.5 * 10^9 years
Hence, the age of the mineral is approximately 4.5 billion years.
To find the age of the mineral, we can use the concept of half-life and the ratio of the amounts of parent isotope (238 U) to the daughter isotope (206 Pb).
Here's how you can do it step by step:
1. Determine the number of half-lives that have passed by comparing the ratio of 238 U to 206 Pb in the mineral sample.
- Start by finding the ratio of 238 U to 206 Pb:
238 U / 206 Pb = 66 mg / 16 mg
- Simplify the ratio by dividing both sides by 16:
238 U / 206 Pb = 4.125 mg / 1 mg
So, the ratio of 238 U to 206 Pb is 4.125:1.
2. Calculate the number of half-lives (n) that have passed using the ratio:
- The formula to calculate the number of half-lives is: n = ln(ratio) / ln(1/2), where ln represents the natural logarithm.
- Plug in the values into the formula:
n = ln(4.125) / ln(1/2)
n ≈ 3.0275
3. Calculate the age of the mineral:
- The age of the mineral (t) can be obtained by multiplying the number of half-lives (n) by the half-life time (t1/2):
t = n * t1/2
- Plug in the values:
t = 3.0275 * (4.5 * 10^9 yr)
t ≈ 13.72 billion years (rounded to two decimal places)
So, the age of the mineral is approximately 13.72 billion years.
mass U-238 originally is mass Pb x (238/206) = 16.0 x (206/238) = about 13.8 but you need to get a more accurate answer.
mass U-238 initially = 66 mg + 13.8 = about 80.
k = 0.693/t1/2 - 0.693/4.5E9 = 1.5E-10(approx)
Substitute these values into the first order equation below and solve for t (in years).
ln(No/N) = kt
No = about 80
N = 66
k from above.
t = unknown.
All of the numbers above are estimates. You should go through and confirm everything.