In the laboratory small zircon crystals were separated from granite (K).

Isotopic analysis of these zircon crystals gives a 207Pb/235U ratio of 0.45. the half-life of 235U (7.04 × 108
years), estimate the age of the granite in Ma to two significant figures

I would start by turning this thing around so that (235U/207Pb) = 1/0.45 = 2.22

Then I would adjust the ratio to account for the fact that Pb has a lower atomic mass than U by a factor of 235/207 = 1.135 so 2.222/1.135 = 1.958.
For every x atoms U that decay to Pb, the Pb will be x and the U remaining will be 100-x; therefore,
[(100-x)/x] = 1.1958
Solve for x = (Pb) and (100-x) U.
Then (No/N) = kt
for No = 100
For N = what you found above for 100-x
k see below.
t = time.

k = 0.693/t1/2</sub.
k = 0.793/7.04E8

I'm going out on a limb here, and I know you might not reply, but, DrBob222, I need to ask you something concerning this.

Why did you flip the ratio? Wouldn't that effect the answer?

And is that a mistake in the middle where you said,

"Then I would adjust the ratio to account for the fact that Pb has a lower atomic mass than U by a factor of 235/207(!that is what I'm talking about!) = 1.135 so 2.222/1.135 = 1.958."

Shouldn't that be flipped? I ran it through a calculator and the flipped version gave '1.135' like you said the other version equalled here.

To estimate the age of the granite in millions of years (Ma), we can use the concept of radioactive decay and the ratio of 207Pb/235U in the zircon crystals.

The 207Pb/235U ratio provides information about the number of half-lives that have elapsed since the granite formed. Since we know the half-life of 235U is 7.04 × 10^8 years, we can calculate the number of half-lives based on the given ratio.

First, we need to determine the initial amount of 235U (N0) in the zircon crystals. We can assume that the initial amount of 235U is equal to the current amount of 207Pb, as they both come from the same radioactive decay chain.

Let's assume the current amount of 207Pb is 1 unit (Nt). Therefore, the initial amount of 235U will also be 1 unit.

Now, we can use the formula for radioactive decay:

Nt = N0 * (1/2)^(t / half-life)

where:
Nt = current amount of 207Pb/235U ratio (0.45)
N0 = initial amount of 207Pb/235U ratio (1.0)
t = number of half-lives
half-life = half-life of 235U (7.04 × 10^8 years)

Rearranging the formula to solve for t:
t = (log(Nt / N0)) / log(1/2)

Substituting the given values:
t = (log(0.45 / 1.0)) / log(1/2)

Using logarithmic properties:
t = (log(0.45) - log(1.0)) / log(1/2)

Calculating the logarithms using a calculator or software:
t = (-0.35) / (-0.301)

Finally, calculating t:
t ≈ 1.16

Since t represents the number of half-lives, we can multiply it by the half-life of 235U to get the age of the granite in years:

Age = t * half-life
Age ≈ 1.16 * 7.04 × 10^8 years

To convert this age into millions of years (Ma), divide the age by 10^6:

Age in Ma = Age / 10^6
Age in Ma ≈ (1.16 * 7.04 × 10^8) / 10^6

Calculating the final result:
Age in Ma ≈ 817.44 Ma

Therefore, the estimated age of the granite is approximately 817.44 million years (Ma), rounded to two significant figures.