Problem 74 - Chemistry: The Central Science

Question

An experiment was designed to determine whether an aquatic plant absorbed iodide ion from water. lodine\(131\left(t_{1 / 2}=8.02\right.\) 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.

Step-by-Step Solution

Answer

Yes, the plants absorbed iodide from the water. The calculated activity ratio after 30 days (≈ 0.0733) is higher than the expected activity ratio (≈ 0.0578) if no iodide was absorbed, indicating that the plants absorbed iodide ions from the water.

Step 1: Calculate the decay constant

We need to first calculate the decay constant, λ, using the half-life formula: \(t_{1/2} = \frac{ln(2)}{λ}\). Here \(t_{1/2}\) is 8.02 days. Rearranging the formula to find λ, we get: \(λ = \frac{ln(2)}{t_{1/2}}\)

Step 2: Plug in the half-life value

Now plug in the half-life value of 8.02 days into the equation: \(λ = \frac{ln(2)}{8.02}\) _lambda =\(\frac{ln(2)}{8.02}\approx 0.086\space day^{-1} So, the decay constant (λ) is approximately 0.086 per day.

Step 3: Calculate the activity ratio

Next, we calculate the activity ratio after 30 days using the equation \(\frac{A_t}{A_0} = e^{-λt}\), where A_t is the activity after t days and \(A_0\) is the initial activity. We have \(A_0 = 214\) counts/min and \(A_t = 15.7\) counts/min. \(\frac{A_t}{A_0}\) = \(\frac{15.7}{214}\) ≈ 0.0733

Step 4: Calculate the expected activity ratio

Now, we need to calculate the expected activity ratio if no iodide was absorbed by the plants. Use the equation we found earlier: \(\frac{A_t}{A_0} = e^{-λt}\) Plug in λ ≈ 0.086 per day and t = 30 days: \(\frac{A_t}{A_0}\) = \(e^{-(0.086)(30)}\) ≈ 0.0578

Step 5: Compare the activity ratio

Now, compare the calculated activity ratio with the expected activity ratio. We have: Calculated activity ratio: ≈ 0.0733; Expected activity ratio: ≈ 0.0578. Since the calculated activity ratio (≈ 0.0733) is higher than the expected activity ratio (≈ 0.0578), it indicates that the plants absorbed iodide ions from the water, which contribute to the higher activity ratio than expected for the 30 days period.

Key Concepts

Half-Life

The half-life of a radioactive element is the time it takes for half of the radioactive atoms in a sample to decay. This concept is crucial because it allows us to predict how long it will take for a substance to decrease to a certain level of activity. The formula for calculating half-life is given by:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
Where \( t_{1/2} \) represents the half-life, and \( \lambda \) is the decay constant. In the case of iodine-131, its half-life is 8.02 days. This means that every 8.02 days, the amount of iodine-131 in a sample will reduce to half of its original amount.

• This concept helps us understand how quickly a radioactive substance becomes less active.
• Knowing the half-life allows scientists to calculate the age of an artifact or predict the time it takes for a hazardous element to become less dangerous.

Decay Constant

The decay constant (\( \lambda \)) is a measure of the probability of decay of a radioactive atom per unit time. It provides important information about how quickly a substance will decay. This probability is constant for any given substance and does not change over time. The decay constant can be determined from the half-life of a substance using the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
For iodine-131, using a half-life of 8.02 days, we find that:
\[ \lambda \approx 0.086\, \text{day}^{-1} \]

• This means that approximately 8.6% of the iodine-131 nuclei in a sample decay per day.
• The decay constant helps predict the activity levels of radioactive substances over time.

Iodine-131

Iodine-131 is a radioactive isotope commonly used as a tracer in scientific experiments. Its properties are particularly useful for tracking the distribution and movement of substances, such as minerals and ions, in ecosystems and organisms.

• With a half-life of 8.02 days, iodine-131 is suitable for studies over short to medium periods, allowing for detailed tracking before it decays significantly.
• In this experiment, iodine-131 is used to trace the absorption of iodide ions by aquatic plants, helping determine if the plants uptake these ions from the water.

Activity Measurements

Activity measurements involve determining the rate at which a radioactive sample emits radiation. This rate, typically counted as disintegrations per second or counts per minute, helps assess how radioactive a sample is at any given time.
In our exercise, the activity measurement started with 214 counts per minute and decreased to 15.7 counts per minute after 30 days.

• By measuring the activity over time, we can understand how fast the radioactive material is decaying.
• Calculating the ratios of activity over time also allows us to determine whether external factors, like plant absorption, affect the decay rate differently from what would be expected naturally.

Plant Absorption

Plant absorption refers to the uptake of substances, such as nutrients and ions, from their environment. In the context of radioactive tracers like iodine-131, it is used to determine how much of the tracer substance the plant absorbs from its surroundings.
In this experiment, the comparison between the calculated and expected activity ratios indicated higher-than-expected iodine activity after 30 days, suggesting plants absorbed some of the iodide ions from the water.

• This conclusion was drawn because the measured activity was higher than what decay alone would predict, thus implying additional iodine-131 remained in the water due to plant uptake.
• Understanding plant absorption is crucial in ecological studies as it helps explain the movement and utilization of nutrients and tracers in environments.