Question

The half-life for the process 238U→206Pb is 4.5×109yr . A mineral sample contains 72.0 mg of 238U and 18.0 mg of 206Pb .

What is the age of the material?

Answer 1

The age of the mineral sample can be determined using the half-life of U-238 and the current amounts of U-238 and Pb-206. By comparing the remaining U-238 to the amount of Pb-206, and applying the formula for radioactive decay, we can estimate the time elapsed since the rock formed.

Step-by-step explanation:

To determine the age of the material using radioactive dating and U-238's half-life, we compare the amount of uranium-238 (238U) left in the sample to the amount of lead-206 (206Pb), which is the end product of its decay.

Knowing that the half-life of U-238 is 4.5 × 109 years, we use the following formula that relates the original amount of U-238 (No), the current amount of U-238 (N1), and time (t):

N1 = No(1/2)(t/half-life)

We are given that the sample currently contains 72.0 mg of U-238 and 18.0 mg of Pb-206. Since one mole of U-238 decays into one mole of Pb-206, this means that originally there were (72.0 mg + 18.0 mg) of U-238 in the sample. Using (72 mg + 18 mg) as No and the current 72 mg of U-238 as N1, we substitute these values into the formula:

72 = (72 + 18)(1/2)(t/4.5 × 109)

Solving for t, the age of the rock, we get:

t = (4.5 × 109) × log2((72+18)/72)

Calculating this gives us an age estimate for the rock sample.

Answer 2

The age of the material can be determined using the half-life of the decay process between 238U and 206Pb.

Step-by-step explanation:

To determine the age of the material, we can use the half-life of the decay process between 238U and 206Pb. The half-life of 238U is 4.5 x 10^9 years.

First, we need to determine the ratio of 238U to 206Pb in the sample.

Let's assume the sample originally had N0 mg of 238U and 0 mg of 206Pb. After one half-life, we would have N0/2 mg of 238U and N0/2 mg of 206Pb.

Given that the sample contains 72.0 mg of 238U and 18.0 mg of 206Pb, we can set up the following equation:

72.0 mg / (N0/2 mg) = (4.5 x 10^9 years) / t

Where t is the age of the material in years.

Solving for t, we find that the age of the material is approximately 2.25 x 10^9 years.