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The half-life \(\left(t_{1 / 2}\right)\) of a radioactive species is the time it takes for half of the species to emit radiation and decay (turn into a different species). If a quantity \(N_{0}\) of the species is present at time \(t=0,\) the amount present at a later time \(t\) is given by the expression \(N=N_{0}\left(\frac{1}{2}\right)^{t / t_{1 / 2}}\) Each decay event involves the emission of radiation. A unit of the intensity of radioactivity is a curie (Ci), defined as \(3.7 \times 10^{10}\) decay events per second. A 300,000 -gallon tank has been storing aqueous radioactive waste since \(1945 .\) The waste contains the radioactive isotope cesium-137 \(\left(^{137} \mathrm{Cs}\right)\), which has a half-life of 30.1 years and a specific radioactivity of 86.58 Cilg. The isotope undergoes beta decay to radioactive barium- 137 , which in turn emits gamma rays and decays to stable (nonradioactive) barium with a half-life of 2.5 minutes. The concentration of \(^{137} \mathrm{Cs}\) in 2013 was \(2.50 \times 10^{-3} \mathrm{g} / \mathrm{L}\) (a) What fraction of the \(^{137} \mathrm{Cs}\) would have to decay for the level of cesium-related radioactivity of the contents to be \(1.00 \times 10^{-3} \mathrm{Ci} / \mathrm{L} ?\) What total mass of cesium ( \(\mathrm{kg}\) ) would that loss represent? In what year would that level be reached? (b) What was the concentration of \(^{137} \mathrm{Cs}\) in the tank (g/L) when the waste was first stored? (c) Explain why the radioactive cesium in the tank poses a significant environmental threat while the radioactive barium does not.

Step by step solution

01

Determine fraction of Cs-137 that must decay

We are given the specific radioactivity of Cs-137 as 86.58 Ci/g. This implies that 86.58 Ci of radiation is emitted per gram of Cs-137. We need to find the amount of Cs-137 that need to decay in order to reduce the radioactivity to \(1.00 \times 10^{-3} \) Ci/L. So, we can write an equation for specific activity, i.e, \[1.00 \times 10^{-3} \, \text{Ci/L} = x \, \text{g/L} \times 86.58 \, \text{Ci/g}\] Solving for x (representing mass per volume or concentration), we find the concentration of Cs-137 that needs to remain.

02

Determine total mass of Cs-137 corresponds to the loss

After computing the concentration of Cs-137 that needs to remain in the tank, we subtract this amount from the initial concentration (which is 2.50 \( \times 10^{-3} \) g/L) to find out how much Cs-137 needs to decay. Then, we multiply this concentration by the volume of the tank to find out the total mass in Kg.

03

Determine the year level will be reached

Now we use the decay equation \(N = N_{0}(\frac{1}{2})^{t / t_{1 / 2}}\) where \(N_0\) represents the initial quantity, \(N\) is the quantity after time \(t\), \(t_{1 / 2}\) is the half-life. We know the values \(N\), \(N_0\) and \(t_{1 / 2}\), and we need to solve this equation for \(t\). This will give us the time elapsed since 1945, and we can calculate the current year by adding this time to 1945.

04

Find the initial concentration of Cs-137

To find initial concentration, we must consider the time that has passed since the initial moment when the tank was filled up. We use the decay equation inversely, setting \(N_0\) as the variable we seek, while \(N\), \(t\), and \(t_{1 / 2}\) are known values. We know the time that passed since the tank was filled (\(t = 2013 - 1945\)) and we apply it to find \(N_0\).

05

Explain environmental implications of Cs-137 and Ba-137 decay

This doesn't involve a numerical solution but requires understanding of radioactive decay. While \(^{137}Cs\) and \(^{137}Ba\) are both radioactive, only Cs poses an environmental threat, because the half-life of Ba is very short (2.5 minutes), therefore it decays rapidly to stable, non-radioactive Barium shortly after its formation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

half-life

The concept of half-life is a fundamental aspect of radioactive decay. It is the time required for half of the radioactive nuclei in a sample to decay. In simpler terms, it's how long it takes for half of the radioactive material to lose its radioactivity.
Imagine you have a substance with 100 radioactive atoms. After one half-life, only 50 of those atoms would remain radioactive. After two half-lives, only 25 would remain, and so on.
Half-life varies greatly among different substances. For example, cesium-137 has a relatively long half-life of 30.1 years. This means it remains radioactive for many years, increasing its potential environmental impact.
Mathematically, the decay of a radioactive substance over time can be described by the equation:

• \( N = N_{0} \left(\frac{1}{2}\right)^{t/t_{1/2}} \)

Where:

• \( N \) is the remaining quantity of the substance.
• \( N_{0} \) is the initial quantity.
• \( t \) represents the time elapsed, and
• \( t_{1/2} \) is the half-life.

cesium-137

Cesium-137, often denoted as \( ^{137} \text{Cs} \), is a radioactive isotope of cesium. It is a byproduct of nuclear fission processes involving uranium and plutonium, such as those occurring in nuclear reactors and during nuclear weapon explosions.
This isotope is of significant concern due to its properties and environmental persistence.

• **Half-Life**: One of the key characteristics of cesium-137 is its half-life of 30.1 years. This means that it takes over three decades for half of any given amount to decay.
• **Decay Products**: It primarily undergoes beta decay, transforming into barium-137m. This decay process involves the emission of beta particles and gamma radiation, making cesium-137 a potent source of radioactivity.
• **Environmental Impact**: Due to its long half-life and the penetrating nature of its gamma radiation, cesium-137 is a significant environmental hazard. It can spread readily in the environment, contaminating soil and water.
• **Usage and Risks**: It is often found in medical radioisotope therapy and is used in industrial gauges. However, improper handling or accidental release can lead to widespread contamination and pose health risks.

specific radioactivity

Specific radioactivity refers to the activity per unit mass of a radioactive material. It provides a measure of how active a particular amount of a substance is, which is crucial for assessing potential exposure and risks.
For cesium-137, the specific activity is a measure of how many decay events occur per second per gram of this substance, often expressed in curies per gram (Ci/g). In this context:

• Cesium-137 has a specific radioactivity of 86.58 Ci/g. This means that each gram of cesium-137 emits a significant amount of radiation, amounting to 86.58 curies worth of decay events per second.
• Specific radioactivity helps determine how long a particular substance will remain hazardous. High specific radioactivity implies a high decay rate, leading to intense emissions of radiation over a short period. For cesium-137, even small quantities can be highly radioactive.
• Understanding specific radioactivity is important for safe handling and storage of radioactive materials, and for the calculation of how much material can present a danger to living organisms or the environment.

The concept is crucial when calculating allowable concentrations of radioactive substances in environmental and safety standards, ensuring that exposure remains within manageable limits.