Problem 46 - Chemistry and Chemical Reactivity

Question

Strontium-90 is a hazardous radioactive isotope that resulted from atmospheric testing of nuclear weapons. A sample of strontium carbonate containing \(^{90}\) Sr is found to have an activity of \(1.0 \times 10^{3} \mathrm{dpm} .\) One year later, the activity of this sample is \(975 \mathrm{dpm} .\) (a) Calculate the half-life of strontium-90 from this information. (b) How long will it take for the activity of this sample to drop to \(1.0 \%\) of the initial value?

Step-by-Step Solution

Answer

(a) The half-life of strontium-90 is approximately 27.39 years. (b) It takes about 183.83 years for the activity to drop to 1% of the initial value.

Step 1: Understand the concept of decay

Radioactive decay is an exponential process. The activity of a radioactive sample decreases over time according to the formula: \[ N(t) = N_0 \times e^{-\lambda t} \] where \( N(t) \) is the activity at time \( t \), \( N_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is the time.

Step 2: Calculate the decay constant (λ)

Given: - Initial activity \( N_0 = 1.0 \times 10^{3} \) dpm - Activity after 1 year \( N(1) = 975 \) dpmWe can use the formula:\[ N(t) = N_0 \times e^{-\lambda t} \]Substituting values:\[ 975 = 1.0 \times 10^{3} \times e^{-\lambda \times 1} \]\[ \frac{975}{1000} = e^{-\lambda} \]\[ e^{-\lambda} = 0.975 \]Now, take the natural logarithm on both sides:\[ -\lambda = \ln(0.975) \]\[ \lambda = -\ln(0.975) \approx 0.0253 \text{ year}^{-1} \]

Step 3: Calculate the half-life using the decay constant

The half-life \( t_{1/2} \) is related to the decay constant \( \lambda \) by the formula:\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]Substitute the decay constant found in Step 2:\[ t_{1/2} = \frac{\ln(2)}{0.0253} \approx 27.39 \text{ years} \]

Step 4: Calculate time to reduce activity to 1%

We are asked to determine the time it takes for the activity to drop to 1% of the initial activity. If the initial activity is \( N_0 = 1.0 \times 10^{3} \) dpm, then 1% of this is \( 10 \) dpm.Use the formula:\[ N(t) = N_0 \times e^{-\lambda t} \]Substitute values:\[ 10 = 1.0 \times 10^{3} \times e^{-0.0253 t} \]\[ e^{-0.0253 t} = \frac{10}{1000} \]\[ e^{-0.0253 t} = 0.01 \]Take the natural logarithm on both sides:\[ -0.0253 t = \ln(0.01) \]\[ t \approx \frac{\ln(0.01)}{-0.0253} \approx 183.83 \text{ years} \]

Key Concepts

Strontium-90

Strontium-90 is a significant byproduct of nuclear fallout resulting from the atmospheric testing of nuclear weapons. It is a radioactive isotope, which means it has an unstable nuclear configuration and releases energy as it decays over time. This energy is emitted in the form of radiation. Strontium-90 poses environmental and health risks mainly because it can replace calcium in biological systems, leading to its accumulation in bones and disrupting biological processes. This accumulation can potentially expose living tissues to harmful radiation. The understanding of Strontium-90 and its decay process is crucial not only for scientific purposes but also for mitigating its impacts on health and the environment. Monitoring and managing its levels are critical for ensuring safety and minimizing exposure risks to humans and wildlife.

Half-Life Calculation

Half-life is a key concept in radioactive decay, referring to the time it takes for half of a sample of a radioactive isotope to decay. In simple terms, it's the time needed for a 50% reduction in the sample's radioactivity. Understanding half-life helps predict how long a radioactive sample will remain active or hazardous.To calculate the half-life of a radioactive isotope like Strontium-90, we use a mathematical equation that connects the decay constant, \( \lambda \), to the half-life, \( t_{1/2} \). The formula used is:

• \( t_{1/2} = \frac{\ln(2)}{\lambda} \)

By determining the decay constant through experiments or calculations, such as measuring the reduction in radioactivity of a sample over time, you can easily find its half-life. For instance, following the decay constant calculation, the half-life of Strontium-90 is approximately 27.39 years.

Decay Constant

The decay constant, denoted \( \lambda \), is a crucial parameter in understanding radioactive decay. It represents the probability per unit time that a particular radioactive nucleus will decay. Think of it as the rate at which a sample loses its radioactivity.The decay constant can be extracted from the exponential decay formula:

• \( N(t) = N_0 \times e^{-\lambda t} \)

where \( N(t) \) is the remaining activity at time \( t \), and \( N_0 \) is the initial activity.Through this formula, it becomes apparent that the value of \( \lambda \) impacts how quickly the radioactivity of a substance decreases. A higher decay constant means the substance decays more quickly, while a lower decay constant results in slower decay. By solving the equation for \( \lambda \), you can determine the specific decay rate for a substance, such as Strontium-90, in any given environment.