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.