A rock contains 270 mmol of 238 U (t1/2= 4.5*10^9 yr) and

110 mmol of 206 Pb. Assuming that all the 206 Pb comes from decay
of the 238 U, estimate the rock’s age.

mmols U at finish = 270

mmols Pb at finish = 110
mmol U at start = 380

ln(380/270) = kt
Solve for t
You need k and k = 0.693/t1/2 = ?

To estimate the rock's age, we can use the concept of half-life.

The half-life of 238U is given as 4.5 * 10^9 years. This means that after every half-life, the amount of 238U reduces to half of its initial value.

Given that the rock contains 270 mmol of 238U, we need to determine the number of half-lives it has undergone to reach 110 mmol of 238U (which has decayed into 206Pb).

Let's calculate the number of half-lives:

Number of half-lives = ln (N_final / N_initial) / ln(0.5)

Where N_initial is the initial amount of 238U (270 mmol), and N_final is the final amount of 238U (110 mmol).

Number of half-lives = ln(110 / 270) / ln(0.5)
≈ -0.6086 / -0.6931
≈ 0.878

Since we cannot have a fraction of a half-life, we can round this up to 1 whole half-life.

Now, each half-life corresponds to 4.5 * 10^9 years. Therefore, the rock's age can be estimated as:

Rock's age ≈ Number of half-lives * Half-life duration
≈ 1 * 4.5 * 10^9 years
≈ 4.5 * 10^9 years

Therefore, based on the given information, the estimated age of the rock is approximately 4.5 * 10^9 years.

To estimate the rock's age based on the given information, we can make use of the concept of radioactive decay and the decay equation for Uranium-238.

The decay equation for Uranium-238 is:
N(t) = N(0) * (1/2)^(t / t1/2)

Where:
N(t) is the amount of Uranium-238 at time t
N(0) is the initial amount of Uranium-238
t is the time passed since the rock was formed
t1/2 is the half-life of Uranium-238

Given:
Initial amount of Uranium-238 (N(0)) = 270 mmol
Amount of Lead-206 (Pb-206) = 110 mmol

Since all the Pb-206 in the rock is a result of the decay of U-238, we can equate the amount of Pb-206 to the amount of U-238 decayed.

Using the decay equation, we can express the amount of Uranium-238 at the rock's age (N(t)) as:
N(t) = N(0) * (1/2)^(t / t1/2)

Now, since we know that the amount of Pb-206 is 110 mmol and all of it comes from the decay of U-238, we can equate it as:
110 mmol = 270 mmol * (1/2)^(t / t1/2)

To solve for t, we need to rearrange the equation and take the logarithm of both sides to isolate t:
(1/2)^(t / t1/2) = 110 mmol / 270 mmol
t / t1/2 = log(110/270) / log(1/2)
t = (t1/2) * log(110/270) / log(1/2)

Now we can substitute the values:
t = (4.5 * 10^9 yr) * log(110/270) / log(1/2)

Calculating this expression will give us the estimated age of the rock.