Geologists can estimate the age of rocks by their uranium-238 content. The uranium is incorporated in the rock as it hardens and then decays with first-order kinetics and a half-life of 4.5 billion years. A rock contains 65.7% of the amount of uranium- 238 that it contained when it was formed. (The amount that the rock contained when it was formed can be deduced from the presence of the decay products of U-238.) How old is the rock?

k = 0.693/t1/2

ln(No/N) = kt
No = 100
N = 65.7
k from above.
solve for t in years.

To find the age of the rock, we can use the equation for first-order decay:

N(t) = N₀ * e^(-kt)

Where:
N(t) = the remaining amount of uranium-238 at time t
N₀ = the initial amount of uranium-238
k = decay constant
t = time

We are given that the rock contains 65.7% of the original amount of uranium-238. Therefore, N(t) = 0.657 * N₀. We also know that the half-life of uranium-238 is 4.5 billion years.

To find the decay constant (k), we can use the half-life equation:

t₁/₂ = ln(2) / k

Substituting the given value for the half-life:

4.5 billion years = ln(2) / k

Now, let's solve for k:
k = ln(2) / 4.5 billion years

Substituting the values into the first-order decay equation:

0.657 * N₀ = N₀ * e^(-(ln(2) / 4.5 billion years) * t)

Simplifying the equation by canceling out N₀:

0.657 = e^(-(ln(2) / 4.5 billion years) * t)

Taking the natural logarithm of both sides to isolate t:

ln(0.657) = -(ln(2) / 4.5 billion years) * t

Now, let's solve for t:

t = ln(0.657) / (-(ln(2) / 4.5 billion years)

Using a calculator:

t ≈ 1.091 billion years

Therefore, the rock is approximately 1.091 billion years old.

To determine the age of the rock, we can use the equation for radioactive decay:

N(t) = N(0) * (1/2)^(t / T)

Where:
N(t) is the amount of uranium-238 at time t.
N(0) is the initial amount of uranium-238 when the rock was formed.
t is the age of the rock.
T is the half-life of uranium-238, which is 4.5 billion years.

Given that the rock currently contains 65.7% of the initial amount of uranium-238, we can write:

0.657 = (1/2)^(t / T)

Taking the logarithm of both sides of the equation:

log(0.657) = log[(1/2)^(t / T)]

Using the logarithmic property, we can bring down the exponent:

log(0.657) = (t / T) * log(1/2)

Rearranging the equation to solve for t:

t / T = (log(0.657) / log(1/2))

Substituting the values:

t / 4.5 billion years = (log(0.657) / log(1/2))

Now we can solve for t by rearranging the equation:

t = (log(0.657) / log(1/2)) * 4.5 billion years

Calculating this using a calculator:

t ≈ (0.183 / -0.693) * 4.5 billion years

t ≈ (-0.26388) * 4.5 billion years

t ≈ -1.18746 billion years

Since time cannot be negative, we can ignore the negative sign:

t ≈ 1.19 billion years

Therefore, the rock is approximately 1.19 billion years old.