Question

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?

Accepted Answer

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}$$, where N_0$ is the original amount, N(t$$)$$ is the amount remaining at time t$, and k$ is the decay constant. Use the half-life information to find the decay constant k$. The half-life T$$_{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}}$$. Substitute the given half-life of 4.5 billion years into the formula to calculate k$: k$$ = \frac{\ln(2)}{4.5 	imes $$10^9$$}$$ years$$^{-1}$$. Use the given information that 85% of the original Uranium-238 remains, so $$\frac{N(t)}{$$N_0$$} = 0.85$$. Substitute this into the decay model: 0.85$$ = $$e^{-kt}. $$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.

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 video answer for a similar problem:

Key Concepts

Radioactive Decay and Half-Life

Exponential Decay Model

Solving for Time in Decay Problems