Problem 23 - Calculus: Early Transcendental Functions

Question

Involve exponential decay. Strontium-90 is a dangerous radioactive isotope. Because of its similarity to calcium, it is easily absorbed into human bones. The half-life of strontium- 90 is 28 years. If a certain amount is absorbed into the bones due to exposure to a nuclear explosion, what percentage will remain after 50 years?

Step-by-Step Solution

Answer

Around 39.34% of the strontium-90 will remain in the bones after 50 years.

Step 1: Identify all known variables

In this case, the known variables are the half-life, \(T = 28\) years, and the time that has passed after the nuclear explosion, \(t = 50\) years.

Step 2: Use the formula for exponential decay

To find the percentage remaining, use the exponential decay formula \(N_t = N_0 * 0.5^{(t/T)}\). In this case, you're trying to find \(N_t/N_0\), so the formula becomes \(N_t/N_0 = 0.5^{(t/T)}\) and you can simply substitute the known values into this equation.

Step 3: Calculation

Substitute \(T = 28\) years and \(t = 50\) years into the formula: \(N_t/N_0 = 0.5^{(50/28)}\). Calculate this to find the percentage remaining of Strontium-90. The result is approximately 0.3934.

Step 4: Convert to percentage

To convert the calculated 0.3934 to a percentage, multiply it by 100. So the remaining percentage of strontium-90 is 0.3934 * 100 = 39.34%.

Key Concepts

Radioactive Isotope

A radioactive isotope is a variant of an element that has an unstable nucleus and emits radiation as it decays to a stable form. This process, known as radioactive decay, involves the transformation of the isotope into another element or a different isotope of the same element. Radioactive isotopes are often used in various scientific, medical, and industrial applications, but they can also pose significant health risks if not handled properly.
What makes an isotope radioactive is its instability. The nuclei of these isotopes have an excess of energy or mass or both, making them seek stability through decay.

• This decay can produce different types of particles, such as alpha, beta, and gamma rays.
• Exposure to these particles can be harmful to living organisms.

Understanding radioactive isotopes is crucial, particularly in contexts like nuclear medicine and environmental science.

Half-Life

The half-life of a radioactive isotope is the time it takes for half of the original quantity of the isotope to decay. This measure is crucial for understanding how quickly a radioactive substance loses its radioactivity and becomes stable. Half-life is specific to each isotope and remains constant regardless of the initial amount of the substance.
Here's why half-life is important:

• It allows scientists to calculate the rate of decay and predict how long a substance will remain radioactive.
• This information helps in managing radioactive materials and understanding their long-term impact.

To calculate how much of a substance remains after a certain time period, the half-life can be used in decay equations, which facilitate these predictions.

Strontium-90

Strontium-90 is a radioactive isotope that behaves chemically similar to calcium, leading to its incorporation in bones when ingested or inhaled. This is particularly concerning because it can replace calcium in bone tissues, posing serious health risks.
Strontium-90 has a half-life of 28 years, meaning it remains active in the environment or living organisms for a long time.

• This sustained presence increases the chance of radiation-induced damage, such as bone cancer.
• Understanding Strontium-90's behavior is critical for evaluating the risks of exposure, especially in areas impacted by nuclear fallout.

Its persistence and bioaccumulative potential make it especially dangerous.

Exponential Decay Formula

The exponential decay formula is used to calculate the remaining amount of a substance after a period of time, given its decay rate. It is particularly helpful in determining how much of a radioactive isotope remains after several half-lives. The formula is expressed as:
\[ N_t = N_0 \times 0.5^{(t/T)} \]
Where:

• \( N_t \) is the remaining quantity of the substance after time \( t \).
• \( N_0 \) is the initial quantity of the substance.
• \( t \) is the time elapsed.
• \( T \) is the half-life of the substance.

This formula is crucial for understanding how quickly a radioactive material decreases in quantity and can help predict future exposure levels.
In the case of Strontium-90, by substituting its specific half-life and the elapsed time into this formula, one can easily determine how much of the isotope remains after 50 years.