Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

Verified step by step guidance

1. Identify the decay model for Uranium-238, which follows exponential decay. The amount of Uranium-238 remaining after time \(t\) can be modeled by the equation \(N(t) = N_0 e^{-kt}\), where \(N_0\) is the original amount, \(N(t)\) is the amount remaining at time \(t\), and \(k\) is the decay constant.
2. Use the half-life information to find the decay constant \(k\). The half-life \(T_{1/2}\) is related to \(k\) by the formula \(T_{1/2} = \frac{\ln(2)}{k}\). Rearranging gives \(k = \frac{\ln(2)}{T_{1/2}}\).
3. Substitute the given half-life of 4.5 billion years into the formula to calculate \(k\): \(k = \frac{\ln(2)}{4.5 \times 10^9}\) years\(^{-1}\).
4. Use the given information that 85% of the original Uranium-238 remains, so \(\frac{N(t)}{N_0} = 0.85\). Substitute this into the decay model: \$0.85 = e^{-kt}$.
5. Solve for the age \(t\) of the rock by taking the natural logarithm of both sides: \(\ln(0.85) = -kt\), then rearrange to find \(t = -\frac{\ln(0.85)}{k}\). Substitute the value of \(k\) from step 3 to express \(t\) in terms of known quantities.

Key Concepts

Radioactive Decay and Half-Life

Radioactive decay is the process by which unstable nuclei lose energy by emitting radiation. The half-life is the time required for half of a given amount of a radioactive substance to decay. It is a constant property for each isotope and is used to determine the age of materials by measuring remaining radioactive atoms.

Exponential Decay Model

Radioactive decay follows an exponential decay model, where the quantity of a substance decreases at a rate proportional to its current amount. The formula N(t) = N0 * (1/2)^(t/T) relates the remaining amount N(t) to the initial amount N0, time t, and half-life T, allowing calculation of elapsed time from remaining material.

Solving for Time in Decay Problems

To find the age of a sample, we solve the exponential decay equation for time t. Given the fraction of remaining substance and the half-life, logarithms are used to isolate t, enabling calculation of how long the decay has been occurring, which corresponds to the sample's age.