Question

Uranium- 238 has a half-life of \(4.47 \times 10^{9}\) years and decays through a series of events to yield lead-206. Estimate the age of a rock that contains \(105 \mu \mathrm{mol}\) of \({ }^{238} \mathrm{U}\) and 33 \mumol of \({ }^{206} \mathrm{~Pb}\). Assume all the \({ }^{206} \mathrm{~Pb}\) is from the decay of \({ }^{238} \mathrm{U}\).

Short answer

The rock is approximately 1.79 billion years old.

Step-by-step solution

Writing the Decay Equation

The relationship between the initial amount of Uranium-238 and the amount of Uranium-238 and Lead-206 in the rock can be written using the decay equation:\[ N_t = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \(N_t\) is the amount of Uranium-238 remaining, \(N_0\) is the initial amount, \(t\) is the time, and \(T_{1/2}\) is the half-life.

Find Initial Uranium-238 Amount

Initially, the amount of Uranium was the sum of the current Uranium-238 and the formed Lead-206, because each Lead-206 originates from a Uranium-238 atom. Thus,\[ N_0 = 105 + 33 = 138 \, \mu\mathrm{mol} \]

Identify Amount Remaining

The amount of Uranium-238 remaining is given as:\[ N_t = 105 \, \mu\mathrm{mol} \]

Set Up Decay Equation

Substitute the known values into the decay equation:\[ 105 = 138 \left(\frac{1}{2}\right)^{\frac{t}{4.47 \times 10^9}} \]

Solve for Time t

Rearrange the equation to solve for \(t\):1. Divide both sides by 138: \[ \frac{105}{138} = \left(\frac{1}{2}\right)^{\frac{t}{4.47 \times 10^9}} \]2. Take the natural logarithm of both sides: \[ \ln \left(\frac{105}{138}\right) = \ln \left(\left(\frac{1}{2}\right)^{\frac{t}{4.47 \times 10^9}}\right) \]3. Simplify using logarithm properties: \[ \ln \left(\frac{105}{138}\right) = \frac{t}{4.47 \times 10^9} \ln \left(\frac{1}{2}\right) \]4. Solve for \(t\): \[ t = \frac{4.47 \times 10^9 \cdot \ln \left(\frac{105}{138}\right)}{\ln \left(\frac{1}{2}\right)} \]5. Calculate \(t\): \[ t \approx 1.79 \times 10^9 \text{ years} \]

Conclude the Estimate

The estimated age of the rock is approximately \(1.79 \times 10^9\) years.

Key concepts

Half-life

The concept of half-life is central to understanding radioactive decay. A half-life is defined as the amount of time required for half of the radioactive substance to decay into another element. For Uranium-238, the half-life is an incredibly long 4.47 billion years. This means that every 4.47 billion years, half of a given quantity of Uranium-238 will have transformed into different elements through a series of decay steps. Want to know exactly how this affects our calculations?

• It gives us a timeframe to estimate ages of rocks and minerals, making it a valuable tool in geology.
• It helps scientists understand the timeline of earth’s history through radioactive dating methods like Uranium-Lead dating.

Understanding the concept of half-life ensures a grasp on how long-lasting the materials being studied are, and how this impacts the entire dating process.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In our focus of Uranium-238, radioactive decay occurs in a stepwise fashion leading ultimately to Lead-206. This decay process involves a series of transformations over millions to billions of years. How does this decay occur in detail?

• It begins with Uranium-238 decaying by alpha emission.
• Each decay step reduces the atomic number, gradually transforming the atom into elements like Thorium, Radium, and ultimately Lead-206.

The radiation emitted during decay can be in the form of alpha particles, beta particles, or gamma rays, depending on the position of the atom in the decay series. Understanding radioactive decay is crucial as it lays down the principles of how elements transition over time.

Isotope Geochemistry

Isotope geochemistry is the study of the relative and absolute concentrations of elements and their isotopes in natural environments. When we talk about Uranium-Lead dating, isotope geochemistry allows scientists to use the precise measurement of isotopic ratios in minerals to estimate the ages of rocks.

• The isotopic composition of minerals gives clues about the conditions and processes they have undergone.
• This method relies on accurate measurements of parent and daughter isotopes, such as Uranium-238 and Lead-206.

By studying isotopic ratios, geochemists can unveil the history and evolution of Earth’s crust, providing invaluable insights into geological timescales and processes.

Uranium-238

Uranium-238 is a heavy, naturally occurring radioactive isotope of uranium. It is the most abundant uranium isotope, making up about 99.3% of natural uranium. Its significance in dating processes, especially concerning its decay to Lead-206, cannot be overemphasized.

• With its long half-life of 4.47 billion years, it highly suits the dating of geological events over large time scales.
• Uranium-238 decays through a series of 14 different steps before becoming Lead-206, offering insights along various points in the timeline.

This remarkable isotope allows geologists to unravel the age and evolution of the Earth's crust, enhancing the accuracy of age estimations of rocks. The presence, abundance, and nature of Uranium-238 play an instrumental role in Earth Science research.