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? Answer this question for strontium- 90 .

Step-by-Step Solution

Answer

Cesium-137 reaches 1% in 2187 and Strontium-90 in 2177.

Step 1: Understand the Problem

We need to find the year when the amount of each substance (cesium-137 and strontium-90) is equal to 1% of its original amount released in 1986. This involves calculating the number of half-lives required for the substance to decay to 1% of its original amount.

Step 2: Determine the Formula for Exponential Decay

The formula to use for radioactive decay is \( A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{HL}} \), where \( A \) is the remaining amount, \( A_0 \) is the initial amount, \( t \) is the time, and \( HL \) is the half-life. We set \( A = 0.01 \times A_0 \) to find when the substance decays to 1% of \( A_0 \).

Step 3: Solve for Cesium-137 Time

Substitute \( HL = 30.22 \) and \( A = 0.01 \times A_0 \) into the formula: \( 0.01 = \left(\frac{1}{2}\right)^{\frac{t}{30.22}} \). Taking log base 10 on both sides gives \( \log_{10}(0.01) = \frac{t}{30.22} \times \log_{10}(0.5) \). Solve for \( t \): \( t = 30.22 \times \frac{\log_{10}(0.01)}{\log_{10}(0.5)} \).

Step 4: Calculate Cesium-137 Time

Calculate \( t \) using a calculator. The values are \( \log_{10}(0.01) = -2 \) and \( \log_{10}(0.5) \approx -0.3010 \). Thus, \( t \approx 30.22 \times \frac{-2}{-0.3010} = 200.8 \). Round to the nearest year, so \( t \approx 201 \) years after 1986.

Step 5: Determine Year for Cesium-137

Starting from 1986, add the 201 years calculated to find the year: 1986 + 201 = 2187.

Step 6: Solve for Strontium-90 Time

Use the same method as steps 3 and 4 but with \( HL = 28.8 \). Substitute and solve: \( t = 28.8 \times \frac{\log_{10}(0.01)}{\log_{10}(0.5)} \), which gives \( t \approx 28.8 \times 6.644 = 191.3 \). Round to the nearest year, so \( t \approx 191 \) years after 1986.

Step 7: Determine Year for Strontium-90

Starting from 1986, add the 191 years calculated: 1986 + 191 = 2177.

Key Concepts

Half-Life

Radioactive decay is a process where an unstable atomic nucleus loses energy by emitting radiation. The term "half-life" is fundamental because it tells us how long it takes for half of a radioactive substance to decay. For example, if you start with 100 atoms of a radionuclide, after one half-life, you’d expect only 50 atoms to remain. This quantified approach makes it manageable to predict when a substance will decrease to a certain level.
In the case of Cesium-137, the half-life is 30.22 years, while Strontium-90 has a half-life of 28.8 years. This means over approximately 30 and 29 years, respectively, half of the radioactive material will have decayed into more stable forms. Half-life is a critical property used in calculating how long it will take for the radioactive materials to reduce to safer levels.

Exponential Decay Formula

The decay of radioactive substances is not linear, but exponential. To describe how the amount of a radioactive substance decreases over time, we use the exponential decay formula: \[ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{HL}} \]where:

• \( A \) is the remaining amount of the substance after time \( t \).
• \( A_0 \) is the initial amount of the substance.
• \( HL \) is the half-life of the substance.

This formula shows us how, over successive half-lives, the remaining quantity of a radioactive substance decreases by half. As seen in the steps for the initial problem, solving for \( t \) helps us find out when the substance will be at only 1% of its initial amount.

Chernobyl

Chernobyl was the site of a catastrophic nuclear disaster that occurred in April 1986 in the Soviet Union. It is one of the most infamous nuclear accidents, primarily because of the massive release of radioactive materials into the environment.
The disaster caused a large area to be contaminated with radioactive substances like Cesium-137 and Strontium-90, which have long half-lives, thus impacting the area for many years.
The Chernobyl disaster serves as a stark reminder of the potential dangers associated with nuclear power and the importance of stringent safety protocols to mitigate risks.

Cesium-137

Cesium-137 is a byproduct of nuclear fission, a process used in nuclear reactors and atomic bombs. It is one of the most harmful radioactive isotopes released during the Chernobyl disaster due to its relatively long half-life of 30.22 years.
The element is absorbed by plants and enters the food chain, posing health risks to humans and animals. Over time, as its levels decrease, the risks reduce, but initially, the contamination is significant, necessitating evacuation and restriction of land use.
Roadmaps to predicting the decontamination period often use Cesium-137’s half-life to determine when the levels will drop to a safer threshold.

Strontium-90

Strontium-90, much like Cesium-137, is another significant isotope released during nuclear fission. With a half-life of 28.8 years, Strontium-90 decays relatively quickly compared to other isotopes, but still presents a long-term hazard.
Strontium-90 is particularly dangerous because it mimics calcium and accumulates in bones and teeth, increasing the risk of bone cancer and leukemia. This incorporation means that it can persist in organisms long after the initial exposure.
Countries affected by nuclear fallout, such as Ukraine post-Chernobyl, monitor Strontium-90 levels closely to manage and reduce exposure over the decades following any nuclear event.