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 just worked this for Martin. If you're Martin, here it is again.

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To estimate the rock's age based on the given information, we can use the concept of radioactive decay and the ratio of parent and daughter isotopes. Here's a step-by-step explanation:

Step 1: Calculate the ratio of the parent (238U) to daughter (206Pb) isotopes in the rock.
- Since all the 206Pb comes from the decay of 238U, the ratio of 238U to 206Pb is equal to the ratio of their initial amounts.
- The initial amount of 238U is given as 200 µmol, and the amount of 206Pb is given as 116 µmol.
- Therefore, the ratio of 238U to 206Pb is 200 µmol / 116 µmol.

Step 2: Determine the decay constant (λ) for the parent isotope (238U).
- The decay constant (λ) is defined as the probability of a nucleus decaying per unit time.
- The half-life (t1/2) of 238U is given as 4.5 × 10^9 years.
- The decay constant can be calculated using the relation: λ = ln(2) / t1/2, where ln denotes the natural logarithm.
- Substituting the value of t1/2, we get: λ = ln(2) / (4.5 × 10^9 years).

Step 3: Calculate the age of the rock using the ratio and decay constant.
- The relationship between the ratio (R) of parent to daughter isotopes and the time (t) elapsed since their initial amounts is given by the equation: R = e^(-λt), where e denotes the base of natural logarithm.
- Rearranging the equation, we get: t = -ln(R) / λ.
- Substituting the calculated value of λ and the ratio of 238U to 206Pb, we can find the age of the rock.

Let's calculate the age of the rock using the given values:

Step 1: Ratio of 238U to 206Pb = 200 µmol / 116 µmol.

Step 2: Decay constant (λ) = ln(2) / (4.5 × 10^9 years).

Step 3: Age of the rock (t) = -ln(Ratio) / λ.
- Substituting the values: t = -ln(200/116) / (ln(2) / (4.5 × 10^9 years)).

Now, you can use a calculator to perform the calculations and find the age of the rock.

To estimate the rock's age, we need to use the concept of radioactive decay and utilize the information given about the quantities of 238U and 206Pb isotopes in the rock.

Step 1: Understand the concept:
The radioactive decay of 238U to produce 206Pb is a known process with a half-life of 4.5 x 10^9 years. This means that after every half-life, half of the 238U atoms decay into 206Pb atoms. By measuring the ratio of 238U to 206Pb in a rock, we can determine how many half-lives have occurred and estimate the rock's age.

Step 2: Calculate the ratio of 238U to 206Pb:
Since we know the amount of 238U in the rock is 200 µmol and all the 206Pb is derived from the decay of 238U and is present in the rock as 116 µmol, we can calculate the ratio of 238U to 206Pb as follows:

238U / 206Pb = 200 µmol / 116 µmol

Step 3: Calculate the number of half-lives:
To find out how many half-lives have occurred, we need to take the natural logarithm (ln) of the above ratio and divide it by the natural logarithm of 0.5 (since it represents the fraction of 238U remaining after one half-life). The formula for this calculation is:

Number of half-lives (n) = ln(238U / 206Pb) / ln(0.5)

Step 4: Calculate the rock's age:
Now that we know the number of half-lives, we can estimate the rock's age by multiplying the number of half-lives by the half-life duration of 238U, which is 4.5 x 10^9 years.

Rock's age ≈ Number of half-lives x Half-life duration for 238U

By following these steps, you should be able to estimate the rock's age. Just plug in the values for 238U (200 µmol) and 206Pb (116 µmol) into the calculations and solve for the rock's age.