Question

An experiment was designed to determine whether an aquatic plant absorbed iodide ion from water. Iodine\(131\left(t_{1 / 2}=8.02\right.\) days) was added as a tracer, in the form of iodide ion, to a tank containing the plants. The initial activity of a \(1.00-\mu \mathrm{L}\) sample of the water was 214 counts per minute. After 30 days the level of activity in a \(1.00-\mu \mathrm{L}\) sample was \(15.7\) counts per minute. Did the plants absorb iodide from the water? Explain.

Short answer

The observed final activity (15.7 counts per minute) is slightly higher than the expected activity (15.06 counts per minute) calculated through radioactive decay of Iodine-131. Therefore, we can conclude that the plants did not absorb any significant amount of iodide from the water.

Step-by-step solution

Identify given information

Given information: Initial activity (A₀) = 214 counts per minute Final activity (Aₓ) = 15.7 counts per minute Half-life of Iodine-131 (t₁₂) = 8.02 days Time elapsed (t) = 30 days

Calculate the decay constant

To calculate the decay constant (λ), use the half-life formula: \( t_{1 / 2} = \frac{\ln{2}}{\lambda} \) Rearranging the formula gives us: \( \lambda = \frac{\ln{2}}{t_{1 / 2}} \) Substitute the given half-life value into the formula: \( \lambda = \frac{\ln{2}}{8.02} \) \( \lambda \approx 0.08658 \) per day

Calculate the expected final activity

Now, we calculate the expected final activity (A') after 30 days using the decay formula: \( A' = A_0 \times e^{-\lambda t} \) Substitute the values of A₀, λ, and t into the formula: \( A' = 214 \times e^{-0.08658 \times 30} \) \( A' \approx 15.06 \) counts per minute

Compare observed and expected activity

Now we compare the expected activity A' with the observed activity (Aₓ = 15.7 counts per minute). Expected final activity (A') = 15.06 counts per minute Observed final activity (Aₓ) = 15.7 counts per minute Since the observed final activity (15.7 counts per minute) is slightly higher than the expected activity (15.06 counts per minute), we can conclude that the plants did not absorb any significant amount of iodide from the water.

Key concepts

Half-Life Calculation

Understanding half-life is crucial when studying radioactive decay in chemistry. Half-life, denoted as \( t_{1/2} \), is the time required for half of the radioactive nuclei in a sample to undergo radioactive decay. The concept of half-life makes it possible to predict how much radioactive material will remain after a certain period.

To calculate half-life, we can utilize the decay constant, \( \lambda \), and the relationship \( t_{1/2} = \frac{\ln{2}}{\lambda} \). This equation helps us understand the inherent property of the nucleus and is independent of the initial amount. In the context of practical applications, half-life calculations allow scientists to date archaeological finds, manage medical treatments, and handle radioactive waste.

For example, if a scientist wants to determine how long it will take for a sample with a decay constant \( \lambda \) to decrease to half its initial quantity, they would simply calculate the half-life using the aforementioned equation. Knowing the half-life also aids in addressing safety concerns by helping to determine when a radioactive sample will decay to a level that is safe for the environment or public exposure.

Decay Constant

The decay constant, symbolized by \( \lambda \), is a fundamental parameter in the study of radioactive decay. It represents the probability per unit time that a nucleus will decay and is a measure of the rate at which a radioactive substance undergoes decay. The decay constant is related to the half-life of a substance, as shown in the equation \( \lambda = \frac{\ln{2}}{t_{1/2}} \).

In practical scenarios, by knowing the decay constant, scientists and engineers can predict the stability of radioactive isotopes and assess the potential risks they pose. A larger decay constant means the radioactive material is more unstable and decays more quickly. Conversely, a smaller decay constant indicates that the material is more stable and decays more slowly. Thus, the decay constant provides crucial information for handling and storing radioactive materials, as well as for medical applications where radioactive isotopes are used for diagnosis or treatment.

Exponential Decay

Exponential decay is a key concept that describes the manner in which radioactive substances diminish in quantity over time. The decay of a radioactive element is a random process at the level of individual atoms, and it is described by an exponential function of the form \( A' = A_0 \times e^{-\lambda t} \), where \( A_0 \) is the initial activity, \( A' \) is the activity after time \( t \), \( \lambda \) is the decay constant, and \( e \) is the base of the natural logarithm. This equation reveals that the rate of decay is proportional to the current quantity of the substance.

What’s remarkable about exponential decay is that no matter the amount of substance you start with, it always takes the same amount of time to decay by half – a characteristic time defined by its half-life. This leads to the distinctive ‘half-life curve’ associated with exponential decay. In real-world scenarios, understanding exponential decay is essential for predicting the behavior of nuclear reactions, managing nuclear power plants, and developing radiopharmaceuticals for medical imaging.