Problem 78 - Chemistry and Chemical Reactivity

Question

You might wonder how it is possible to determine the half-life of long-lived radioactive isotopes such as \(^{238}\) U. With a half-life of more than \(10^{9}\) years, the radioactivity of a sample of uranium will not measurably change in your lifetime. In fact, you can calculate the half-life using the mathematics governing first-order reactions. It can be shown that a 1.0 -mg sample of \(^{238}\) U decays at the rate of \(12 \alpha\) emissions per second. Set up a mathematical equation for the rate of decay, \(\Delta N / \Delta t=-k N,\) where \(N\) is the number of nuclei in the 1.0 -mg sample and \(\Delta N / \Delta t\) is 12 dps. Solve this equation for the rate constant for this process, and then relate the rate constant to the half-life of the reaction. Carry out this calculation, and compare your result with the literature value, \(4.5 \times 10^{9}\) years.

Step-by-Step Solution

Answer

The calculated half-life is approximately \(4.63 \times 10^{9}\) years, close to the literature value of \(4.5 \times 10^{9}\) years.

Step 1: Identify the Number of Nuclei

First, determine the number of nuclei \(N\) in a 1.0 mg sample of \(^{238}\)U. We use Avogadro's number, which is \(6.022 \times 10^{23}\) atoms/mol, and the molar mass of \(^{238}\)U, which is approximately 238 g/mol. Calculate \(N\) as follows:\[N = \frac{1.0 \times 10^{-3} \text{ g}}{238 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ atoms/mol}\approx 2.53 \times 10^{18} \text{ atoms}\]

Step 2: Set up the Decay Equation

Given the decay rate \(\Delta N / \Delta t = 12\) disintegrations per second (dps), plug the known values into the equation \(\Delta N / \Delta t = -k N\). This yields:\[12 = -k \times 2.53 \times 10^{18}\]

Step 3: Solve for the Rate Constant \(k\)

Rearrange the equation from Step 2 to solve for the rate constant \(k\):\[k = \frac{12}{2.53 \times 10^{18}} \approx 4.74 \times 10^{-18} \text{ s}^{-1}\]

Step 4: Relate the Rate Constant to the Half-Life

The relationship between the rate constant \(k\) and the half-life \(t_{1/2}\) of a first-order reaction is given by \(t_{1/2} = \frac{0.693}{k}\). Substitute the value of \(k\):\[t_{1/2} = \frac{0.693}{4.74 \times 10^{-18}} \approx 1.46 \times 10^{17} \text{ seconds}\]

Step 5: Convert Half-Life to Years

Convert the half-life from seconds to years, knowing that there are approximately \(3.1536 \times 10^{7}\) seconds in a year:\[t_{1/2} \approx \frac{1.46 \times 10^{17}}{3.1536 \times 10^{7}} \approx 4.63 \times 10^{9} \text{ years}\]

Step 6: Compare with Literature Value

The calculated half-life of \(4.63 \times 10^{9}\) years is close to the literature value of \(4.5 \times 10^{9}\) years. The slight difference could be due to rounding or measurement approximations.

Key Concepts

Half-life Calculation

Understanding how to calculate half-life is crucial for studying substances like uranium-238. The half-life is the time required for half of a radioactive material to decay. To calculate it, we use the formula for first-order reactions. In this exercise, we explored the half-life of a 1.0 mg sample of uranium-238, leading to a calculated half-life of approximately \(4.63 \times 10^{9}\) years. This is very close to the known half-life of \(4.5 \times 10^{9}\) years, showing the effectiveness of our calculations.

• First, we determined the number of uranium atoms using its mass and Avogadro's number.
• We then set up the decay equation \(\Delta N / \Delta t = -k N\), where \(k\) is the rate constant.
• Finally, we used the relationship \(t_{1/2} = \frac{0.693}{k}\) to find the half-life.

This approach allows scientists to predict how long it will take for a significant portion of a radioactive sample to decay.

First-order Reactions

Radioactive decay often follows first-order kinetics, which means the rate of decay is directly proportional to the number of nuclei present. This proportionality can be expressed mathematically as:\[\Delta N / \Delta t = -k N\]

• The negative sign indicates the decrease in quantity over time.
• The rate constant \(k\) governs how quickly the reaction proceeds.

Because the reaction rate depends on the current quantity, no fixed amount of material will always decay over a fixed time, but rather a fixed fraction. This is why the concept of half-life is so important in first-order reactions—it provides a consistent measure of decay speed regardless of how much material you start with.

Rate Constant

The rate constant \(k\) is a critical factor in determining the speed of a radioactive decay process. It helps quantify how fast radioactive atoms are lost over time:

• In our uranium-238 example, we calculated \(k\) to be \(4.74 \times 10^{-18} \text{s}^{-1}\).
• The rate constant reflects the probability per unit time of a single atom decaying.

The rate constant directly influences the half-life through the formula \(t_{1/2} = \frac{0.693}{k}\), meaning a smaller \(k\) results in a longer half-life. Understanding \(k\) allows chemists and physicists to predict how quickly a substance will decay, which has significant applications in fields ranging from archaeology to nuclear medicine.