Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Cesium- 137 and strontium-90 are two radioactive chemicals that were released at the Chernobyl nuclear reactor in April \(1986 .\) The half-life of cesium- 137 is \(30.22\) years, and that of strontium-90 is \(28.8\) years. In what year will the amount of cesium- 137 be equal to \(1 \%\) of what was released?

Step-by-Step Solution

Answer

The cesium-137 will be 1% of its original amount by the year 2186.

Step 1: Determine the formula for radioactive decay

The amount of a radioactive substance remaining after a certain amount of time can be calculated using the formula:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \]where:- \( N(t) \) is the remaining amount of the substance at time \( t \).- \( N_0 \) is the initial amount of the substance.- \( T \) is the half-life of the substance.- \( t \) is the time that has passed.

Step 2: Set up the equation for 1% remaining

We want to find the time \( t \) when the amount of cesium-137 is 1% of the initial amount:\\[ N(t) = 0.01 \times N_0 \]Substitute this into the decay formula:\[ 0.01 \times N_0 = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{30.22}} \]Divide both sides by \( N_0 \):\[ 0.01 = \left( \frac{1}{2} \right)^{\frac{t}{30.22}} \]

Step 3: Solve for the time \( t \)

To solve the equation \[ 0.01 = \left( \frac{1}{2} \right)^{\frac{t}{30.22}} \], take the logarithm of both sides:\[ \log(0.01) = \log \left( \left( \frac{1}{2} \right)^{\frac{t}{30.22}} \right) \]Using logarithmic identities:\[ \log(0.01) = \frac{t}{30.22} \cdot \log \left( \frac{1}{2} \right) \]Solving for \( t \):\[ t = 30.22 \cdot \frac{\log(0.01)}{\log(0.5)} \]

Step 4: Calculate the value of \( t \)

Use a calculator to find the numerical value of \( t \):\[ t = 30.22 \cdot \frac{\log(0.01)}{\log(0.5)} \approx 199.899 \]This means it takes approximately 200 years for the amount of cesium-137 to decay to 1% of its initial value.

Step 5: Determine the future year

The release occurred in 1986, so to find the future year, add the time \( t \) to 1986:\[ \text{Future Year} = 1986 + 200 \approx 2186 \]

Step 6: Conclusion

The cesium-137 will decay to 1% of its original amount by approximately the year 2186.

Key Concepts

Half-Life

Half-life is a measure used in nuclear chemistry to determine the duration over which half of a radioisotope's atoms decay. This concept is central to understanding the behavior of radioactive elements, like cesium-137 and strontium-90.

To calculate the half-life with accuracy, one examines how the quantity of a particular substance decreases by half over specific repeated periods. For instance, cesium-137 has a half-life of 30.22 years, meaning every 30.22 years, half of its radioactive atoms will have decayed into another element. This predictable process helps scientists predict how long a radioactive substance will remain active and potent.

Knowing half-life is essential for various applications such as nuclear medicine, radioactive dating, and nuclear power. It allows scientists to gauge both the risks associated with a radioactive material and its potential benefits. Understanding these dynamics is crucial when handling substances released during nuclear incidents, as we saw with Chernobyl.

Logarithmic Functions

Logarithmic functions play a vital role in solving problems related to radioactive decay. They provide an effective means for dealing with exponential equations, often required in nuclear chemistry calculations.

When we consider the decay of a substance like cesium-137 to a specific remaining fraction, such as 1%, logarithmic functions allow us to reverse the exponential decay equation. For example, using the decay formula, we set up an equation and apply logarithms to solve for the time required for the cesium to reduce to this minimal amount.

In mathematical terms, logarithms convert products into sums, making complex multiplication and power-spanning equations more manageable. This mathematical property eases the manipulation of exponential decay equations when determining unknowns, such as time or remaining quantity.

Radioisotopes

Radioisotopes are atoms that possess excess nuclear energy, making them unstable. This instability leads these atoms to release this energy in the form of radiation, resulting in radioactive decay.

Both cesium-137 and strontium-90 are examples of radioisotopes that were dispersed during the Chernobyl disaster. Because of their radioactive nature, radioisotopes can pose significant health and environmental hazards.

Despite the risks, radioisotopes hold numerous applications:

• Medical imaging and treatments
• Food irradiation, improving preservation
• Scientific research for tracing and analysis

Understanding and measuring the behavior of radioisotopes, including their decay rates, enable these applications and help in disaster mitigation efforts.

Nuclear Chemistry

Nuclear chemistry interfaces with chemical and physical processes that involve changes in atomic nuclei. It encompasses the study of radioactive elements, nuclear reactions, and the harnessing of nuclear energy.

A key focus within nuclear chemistry is the monitoring and understanding of nuclear decay processes, essential for everything from energy production to medical applications. The chain of events after a nuclear release, like that at Chernobyl, illustrates the importance of this scientific field.

Within nuclear chemistry, scientists explore:

• Nuclear reactions, including fission and fusion
• Nuclear energy generation and its safe management
• Environmental impact assessments of radioactive substances

The comprehensive study in this branch helps not only in mitigating the hazards of nuclear materials but also in innovating technologies that benefit society.