Problem 55 - Chemistry and Chemical Reactivity

Question

The isotope of polonium that was most likely isolated by Marie Curie in her pioneering studies is polonium-210. A sample of this element was prepared in a nuclear reaction. Initially, its activity ( \(\alpha\) emission) was \(7840 \mathrm{dpm}\). Measuring radioactivity over time produced the data below. Determine the half-life of polonium-210. $$\begin{array}{cc}\text { Activity (dpm) } & \text { Time (days) } \\\\\hline 7840 & 0 \\\7570 & 7 \\\7300 & 14 \\ 5920 & 56 \\\5470 & 72 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The half-life of polonium-210 is approximately 138 days.

Step 1: Understand the Given Data

We are provided with initial activity data of a polonium-210 sample measured over different days. The activity decreases as time progresses from 0 days to 72 days.

Step 2: Identify the Decay Rate

The activity at time \(t\) is governed by the equation \(A(t) = A_0 e^{-\lambda t}\), where \(A_0\) is initial activity, \(\lambda\) is the decay constant, and \(t\) is time. We need to find \(\lambda\) using the provided data.

Step 3: Calculate Decay Constant \(\lambda\)

Calculate the decay constant using two points from the data. For simplicity, let's use 0 days and 14 days: \(7840 = 7300 e^{-14 \lambda}\). Solving for \(\lambda\) gives \(-\ln(7300/7840) / 14\).

Step 4: Compute Half-Life

The relationship between the decay constant \(\lambda\) and the half-life \(T_{1/2}\) is given by \(T_{1/2} = \frac{\ln(2)}{\lambda}\). Substitute the calculated \(\lambda\) from Step 3 to find the half-life of polonium-210.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay can happen in several ways, but most commonly it involves the emission of alpha particles, beta particles, or gamma rays. In the case of polonium-210, it primarily undergoes alpha decay, releasing alpha particles and transforming into a more stable element. The rate at which a radioactive isotope decays is characterized as an exponential function. This means that over equal intervals of time, the substance decays by a consistent fraction of its remaining amount. Some key points about radioactive decay:

• The decay process is spontaneous and cannot be influenced by external factors such as temperature or pressure.
• Each radioactive isotope decays at its own constant rate, determined by its decay constant.
• The half-life of any isotope is a crucial factor that describes the time it takes for half of the sample to decay.

Understanding these concepts is key when studying radioactive materials and predicting how they will behave over time. When dealing with polonium-210, its specific decay characteristics will inform us about its stability and longevity.

Decay Constant Calculation

To understand the decay constant, we look at the formula that dictates the decay of radioactive substances: \[ A(t) = A_0 e^{-\lambda t} \]In this formula, \( A(t) \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.The decay constant \( \lambda \) is a crucial value that tells us how quickly a substance will decay. Calculating \( \lambda \) involves using observed decay data over time:

• Identify initial and later activity levels from your data set. For instance, in the provided exercise, activities at time 0 days and 14 days were used.
• Insert these values into the decay formula to form an equation.
• Rearrange the equation to solve for \( \lambda \). This typically involves taking the natural logarithm of both sides of the equation.

Once \( \lambda \) is known, it greatly simplifies predicting future activity levels and calculating specific properties like half-life.

Isotope Activity

Activity in the context of radioactive isotopes refers to the measure of decay events occurring per unit of time. Commonly expressed in disintegrations per minute (dpm) or Becquerels (Bq), it is a direct indicator of how "active" a radioactive isotope is. Key aspects of isotope activity:

• Activity decreases over time as the isotope decays. Initially high activity indicates a large number of radioactive nuclei undergoing decay.
• Activity is proportional to the number of undecayed nuclei still present. As more nuclei decay, activity diminishes.
• Isotope activity can help us monitor and measure the rate of decay, providing practical ways to calculate other essential properties like the decay constant and half-life.

By using the decay equation and understanding changes in activity over time, we can derive valuable insights into the behavior of isotopes like polonium-210, making these mathematical relationships vital in predicting isotope longevity and behavior.