Problem: Nuclear decay is a first-order kinetic process. Therefore, If N0 nuclei are present at time t = 0, the number remaining at time t is given by the equation:N = N0e-ktThe decay of radioactive nuclei with known half-lives enables geochemists to measure the age of rocks from their compositions. For example, it is known that 238U decays to 206Pb with a half-life of 4.51 x 109 years.Suppose that a uranium-bearing rock was deposited 1.8 billion (1.80 x 109) years ago and remained geologically unaltered to the present time.What ratio of 206Pb to 238U would you expect to find in the 1.8 billion year old rock? Please circle your answer.

We’re being asked to calculate the ratio of ²⁰⁶Pb to ²³⁸U would you expect to find in the 1.8 billion-year-old rock

Recall that half-life is the time needed for the amount of a reactant to decrease by 50% or one-half. 

The half-life of a first-order reaction is given by:

$\boxed{{\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{ln}\mathbf{2}}{\mathbf{k}}}$

where:

t_1/2 = half-life

k = decay constant

We don’t know the decay constant but we can calculate it from the given values. 

Recall that radioactive/nuclear decay of isotopes follows first-order kinetics, and the rate law for first-order reactions is:

$\boxed{{\mathbf{N}}_{\mathbf{t}}\mathbf{ }\mathbf{=}\mathbf{ }{\mathbf{N}}_{\mathbf{0}}{\mathbf{e}}^{\mathbf{-}\mathbf{kt}}}$

where:

N_t = amount at time t

k = decay constant

t = time

N₀ = initial amount 

We use the following steps to solve the problem:

Step 1. Calculate the decay constant

Step 2. Calculate the fraction remaining

Step 3. Calculate the ratio of ²⁰⁶Pb to ²³⁸U