Question

(Estimating glacial ages) The following oxygen- 18 data were obtained from soil samples taken at different depths in Ontario. Canada. Assuming that all the \(^{18} \mathrm{O}\) was laid down during the last glacial age and that the transport of \(^{18} \mathrm{C}\) to the surface takes place by molecular diffusion. estimate the number of years since the last glacial age from the following data. Independent measure. ments give the diffusivity of \(^{18} \mathrm{O}\) in soil as \(2.64 \times 10^{-10} \mathrm{m}^{2} / \mathrm{s}\) $$\begin{array}{l|cccccc} \text {Depth (m)} &{\text { (surface) }} \\ & 0 & 3 & 6 & 9 & 12 & 18 \\ \hline ^{18} \mathrm{O} \text {Conc. Ratio }\left(\mathrm{C} / \mathrm{C}_{0}\right) & 0 & 0.35 & 0.65 & 0.83 & 0.94 & 1.0 \end{array}$$

Short answer

It is estimated that approximately 4040 years have passed since the last glacial age, using the given oxygen-18 data and the Gaussian distribution function as a solution to Fick's second law for molecular diffusion.

Step-by-step solution

Choose the correct diffusion model

Since the transport of oxygen-18 to the surface takes place by molecular diffusion, we can use Fick's second law to model the diffusion: \[ \frac{\partial C}{\partial t} = D \cdot \frac{\partial^2 C}{\partial x^2} \] where \(C\) is the concentration of oxygen-18 at position \(x\) and time \(t\), and \(D\) is the diffusivity of oxygen-18 in the soil.

Solve the diffusion model for concentration as a function of time and position

Due to the initial condition of all the oxygen-18 laid down during the last glacial age and the given concentration ratios (\(C/C_0\)), we take the Gaussian distribution function as the solution for the diffusion model (Fick's second law) problem: \[ C(x, t) = C_0 \cdot \exp \left[ -\frac{x^2}{4 \cdot D \cdot t} \right] \]

Calculate the time elapsed

Rearrange the Gaussian distribution function to solve for time: \[t = \frac{x^2}{4 \cdot D \cdot \ln{\frac{C_0}{C}}} \] Now our task is to estimate the time elapsed since the last glacial age. To do this, we can choose any data point from the given data. For instance let's use the data at depth of 6 m, where the concentration ratio \(C/C_0 = 0.65\): \[t = \frac{(6 \,\mathrm{m})^2}{4 \cdot (2.64 \times 10^{-10}\, \mathrm{m}^2 / \mathrm{s}) \cdot \ln{\frac{1}{0.65}}}\]

Convert the time to years

Calculate the time in seconds and then convert it to years: \[t = \frac{(6 \,\mathrm{m})^2}{4 \cdot (2.64 \times 10^{-10}\, \mathrm{m}^2 / \mathrm{s}) \cdot \ln{\frac{1}{0.65}}} = 1.274 \times 10^{11} \, \mathrm{s} \] Since there are \(3.1536 \times 10^7\) seconds in a year: \[t = \frac{1.274 \times 10^{11}\, \mathrm{s}}{3.1536 \times 10^7 \, \mathrm{s/yr}} = 4.04 \times 10^3 \, \mathrm{years} \] So, it is estimated that approximately 4040 years have passed since the last glacial age.

Key concepts

Molecular Diffusion

Molecular diffusion is a process where molecules spread from an area of high concentration to an area of low concentration. It is driven by the random motion of molecules, also known as Brownian motion. This phenomenon is essential in many natural and industrial processes. In the scenario of estimating glacial ages from soil samples, we assume the transport of oxygen-18 isotopes to the surface occurs via molecular diffusion.

Fick's second law describes how diffusion causes the concentration of a substance to change over time in a given space. The law is represented by the equation:\[\frac{\partial C}{\partial t} = D \cdot \frac{\partial^2 C}{\partial x^2}\]Here, \(C\) is the concentration level, \(t\) is time, \(x\) represents the position, and \(D\) is the diffusion coefficient. Understanding molecular diffusion is crucial to accurately modeling and predicting how substances like oxygen-18 move through soil or other materials.

Oxygen-18 Isotopes

Oxygen-18 is a stable isotope of oxygen, often used in paleoenvironmental and paleoclimate studies. It naturally occurs in the Earth's environment and can be found in water, ice, carbonates, and silicates. By studying the concentration of oxygen-18 in historical layers, researchers can gather valuable information about past climates and environmental conditions.

When soil samples are analyzed for estimating glacial ages, the concentration ratio of oxygen-18 is investigated. This ratio can indicate the period when certain amounts of oxygen-18 were embedded in soil layers. The concentration of this isotope is integral to understanding how different climatic events — such as glaciations — have occurred over time. Because oxygen-18 can tell us about rainfall, temperature, and other climate factors in the past, it plays a vital role in geological and climatological research.

Glacial Age Estimation

Estimating glacial ages involves determining when the last major ice ages occurred by analyzing geological and isotopic evidence. In the given exercise, this is done using oxygen-18 isotopes found in soil samples. By applying principles of molecular diffusion and isotopic concentrations, scientists can backtrack the changes in these concentrations to estimate the age of past glacial events.

The process requires an understanding of both the initial conditions (i.e., when the oxygen-18 was last deposited during a glaciation event) and the molecular diffusion that spreads these isotopes over time. By solving the diffusion model — specifically using Fick's second law — and examining the isotopic concentration at certain depths, scientists can estimate the number of years since the last glacial period. This calculation provides insights into how Earth's climate has shifted over millennia.

Diffusion Coefficient Calculation

The diffusion coefficient \(D\) is a measure of how quickly molecules or particles spread out due to diffusion. For oxygen-18 in the soil, this coefficient is a known value that aids in predicting how the isotope moves through the Earth's layers. In the given exercise, the diffusion coefficient is provided as \(2.64 \times 10^{-10} \ \text{m}^2/\text{s}\).

Given this coefficient and the initial positioning of the isotopes, we use Fick's second law to model their diffusion over time. Specifically, the Gaussian distribution function helps solve the concentration over time:\[ C(x, t) = C_0 \cdot \exp\left(-\frac{x^2}{4 \cdot D \cdot t}\right) \]By rearranging this formula, the time \(t\) it took for the isotopes to diffuse can be calculated, utilizing the known values of \(D\) and initial concentrations. This approach is crucial in determining the relative period since an event, such as the end of a glacial age. Understanding and calculating diffusion coefficients provide a foundation for numerous practical applications beyond geology, including chemical engineering and environmental science.