We’re being asked to** calculate the ratio of ^{206}Pb to ^{238}U 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:

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

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:

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

where:

**N _{t}** = amount at time t

**k** = decay constant

**t** = time

**N _{0}** = 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 ^{206}Pb to ^{238}U

Nuclear decay is a first-order kinetic process. Therefore, If N_{0} nuclei are present at time t = 0, the number remaining at time t is given by the equation:

N = N_{0}e^{-kt}

The 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 ^{238}U decays to ^{206}Pb with a half-life of 4.51 x 10^{9} years.

Suppose that a uranium-bearing rock was deposited 1.8 billion (1.80 x 10^{9}) years ago and remained geologically unaltered to the present time.

What ratio of ^{206}Pb to ^{238}U would you expect to find in the 1.8 billion year old rock? Please circle your answer.