Problem 60 - General Chemistry: Principles and Modern Applications

Question

One method of dating rocks is based on their \(^{87} \mathrm{Sr} /^{87} \mathrm{Rb}\) ratio. \(^{87} \mathrm{Rb}\) is a \(\beta^{-}\) emitter with a half- life of \(5 \times 10^{11}\) years. A certain rock has a mass ratio \(^{87} \mathrm{Sr} /^{87} \mathrm{Rb}\) of \(0.004 / 1.00 .\) What is the age of the rock?

Step-by-Step Solution

Answer

The age of the rock is calculated by substituting the given values into the derived formula. This will give the final age in years.

Step 1: Identify given information

From the exercise, the half-life of \(^{87}\mathrm{Rb}\) is \(5 \times 10^{11}\) years. Also, the mass ratio \(^{87}\mathrm{Sr} /^{87}\mathrm{Rb}\) of the rock is 0.004/1.00.

Step 2: Use the Decay Equation to find decay constant

The decay constant (\(λ\)) is given by \(\frac{ln 2}{t_{1/2}}\), where \(t_{1/2}\) is the half-life. So, \(λ = \frac{ln 2}{5 \times 10^{11}}\).

Step 3: Use the ratio to compute time

In radioactive decay, the ratio of Sr-87 to Rb-87 is related to time (t) by the equation \(\frac{Nf}{Ni} = e^{-λt}\), where \(Nf\) is the final quantity (Sr-87 in this case), \(Ni\) is the initial quantity (Rb-87 in this case), and \(e\) is Euler's number. Rearranging for t gives \(t = -\frac{ln (\frac{Nf}{Ni})}{λ}\).

Step 4: Substitute values and solve for t

Substitute \(Nf/Ni = 0.004/1.00\) and our previously calculated \(λ\) into the equation to find t. \(t = -\frac{ln (0.004)}{\frac{ln 2}{5 \times 10^{11}}}\)

Key Concepts

Rb-Sr Dating Method

The Rb-Sr dating method is a popular technique used to date ancient rocks and minerals. It relies on the decay of the radioisotope Rubidium-87 (\(^{87}\text{Rb}\)) into Strontium-87 (\(^{87}\text{Sr}\)). This process occurs over vast periods, making it ideal for dating Earth’s oldest formations.
A key factor in this method is the measurable ratio of \(^{87}\text{Sr}\) to \(^{87}\text{Rb}\) in a sample. When a rock forms, it initially contains a certain amount of \(^{87}\text{Rb}\) and no \(^{87}\text{Sr}\). Over time, the \(^{87}\text{Rb}\) decays into \(^{87}\text{Sr}\), thus altering the ratio.
To determine the age, scientists measure the \(^{87}\text{Sr}/^{87}\text{Rb}\) ratio and use the decay equation. This kind of dating is crucial for understanding geological processes and the history of the Earth.

Half-Life Calculation

The concept of half-life is central to understanding radioisotope dating methods. Half-life is the time required for half of the radioactive isotopes in a sample to decay.
For Rubidium-87, the half-life is an incredibly long \(5 \times 10^{11}\) years. This means that if you start with a certain amount of \(^{87}\text{Rb}\), half of it will transform into \(^{87}\text{Sr}\) after that time period.
Calculating the decay constant \(\lambda\) from the half-life is essential in dating calculations. The formula \(\lambda = \frac{\ln 2}{t_{1/2}}\) allows you to find the rate of decay. This constant is then used in various equations to ascertain the age of rocks or fossils.

Radioactive Decay Equation

The radioactive decay equation provides a mathematical way to determine the age of a rock. For the Rb-Sr method, the equation is expressed as:
\[ \frac{N_f}{N_i} = e^{-\lambda t} \]
Here, \(N_f\) represents the amount of \(^{87}\text{Sr}\) now present, while \(N_i\) is the initial amount of \(^{87}\text{Rb}\). The equation involves natural exponential functions, with \(e\) as Euler's number, which is approximately 2.718.
This equation can be rearranged to solve for time \(t\), giving: \[ t = -\frac{\ln(\frac{N_f}{N_i})}{\lambda} \]
By substituting the measured \(^{87}\text{Sr}/^{87}\text{Rb}\) ratio and the decay constant into the equation, scientists can determine the age of a rock with remarkable precision, offering insight into geological history and the evolution of the Earth.