Problem 25 - Applied Calculus

Question

The half-life of radioactive strontium-90 is 29 years. In 1960, radioactive strontium-90 was released into the atmosphere during testing of nuclear weapons, and was absorbed into people's bones. How many years does it take until only \(10 \%\) of the original amount absorbed remains?

Step-by-Step Solution

Answer

It takes approximately 97 years for strontium-90 to decay to 10% of its original amount.

Step 1: Understanding the Half-Life Formula

The half-life formula is used to determine how much of a substance remains after a certain period of time. It is given by the equation: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]where: - \(N(t)\) is the quantity of the substance that remains after time \(t\),- \(N_0\) is the initial quantity of the substance,- \(T_{1/2}\) is the half-life of the substance,- \(t\) is the time that has elapsed.

Step 2: Setting Up the Equation

We are given that the half-life \(T_{1/2}\) of strontium-90 is 29 years. We want to find out when only \(10\%\) of the original amount remains, so:\[ N(t) = 0.1 N_0 \]Substituting these values into the half-life formula gives:\[ 0.1 N_0 = N_0 \left( \frac{1}{2} \right)^{\frac{t}{29}} \]

Step 3: Solving for t

To solve for \(t\), we cancel \(N_0\) from both sides and take the natural logarithm:\[ 0.1 = \left( \frac{1}{2} \right)^{\frac{t}{29}} \]Taking the natural log on both sides to remove the exponent:\[ \ln(0.1) = \frac{t}{29} \ln\left(\frac{1}{2}\right) \]Solving for \(t\) gives:\[ t = \frac{29 \cdot \ln(0.1)}{\ln(0.5)} \]

Step 4: Calculating t

We perform the calculations using the natural logarithms:\[ t = \frac{29 \cdot \ln(0.1)}{\ln(0.5)} \]First, find \(\ln(0.1) \approx -2.302\) and \(\ln(0.5) \approx -0.693\).Now substitute these into the equation:\[ t = \frac{29 \cdot (-2.302)}{-0.693} \]Performing the division:\[ t \approx 96.8 \]

Step 5: Interpreting the Result

The computed value of \(t\) is approximately 96.8 years. This is the time it will take for the amount of strontium-90 to decay to \(10\%\) of its original quantity that was released in 1960.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This process changes the original element into a different one or reduces its concentration. The rate at which a radioactive substance decays is predictable, as it follows a characteristic pattern known as exponential decay. This decay continues until the substance reaches a stable state. In nature, different radioactive isotopes decay at different rates, which can be measured by their half-lives.

• The decay leads to the emission of alpha, beta, or gamma radiation.
• It can change the chemical element by altering the number of protons in the nucleus.
• It is a continuous process, meaning it does not stop until the substance becomes stable.

Understanding the concept of radioactive decay is crucial as it helps in applications like dating archaeological findings and managing nuclear waste. It's particularly important in the case of substances like strontium-90, which can be hazardous if released into the environment.

Exponential Decay Formula

The exponential decay formula is essential in understanding how quickly a radioactive substance decreases in quantity over time. It is defined mathematically, allowing for precise calculations of how much of a substance remains after a given time period. For any substance experiencing exponential decay, the formula is:\[N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\]This formula facilitates the calculation by breaking down key components:

• \(N(t)\): the amount of substance remaining after time \(t\).
• \(N_0\): the initial amount of the substance.
• \(T_{1/2}\): the half-life, the time it takes for the substance to reduce to half its original quantity.
• \(t\): the time elapsed.

This formula is particularly helpful in calculating the decay over long periods, like determining how many years are needed for radioactive strontium-90 to decay to a fraction of its original concentration.

Strontium-90

Strontium-90 is a man-made radioactive isotope that emerged as a byproduct in nuclear fission, especially during the testing of nuclear weapons. Upon release into the environment, strontium-90 becomes a serious health concern due to its chemical similarity to calcium.

• It can replace calcium in bones and lead to various health effects, including bone cancer.
• Its half-life of 29 years means it remains in the environment for many decades.

Because it mimics calcium, strontium-90 can easily be absorbed by living organisms, being stored in the bones and teeth. Its long half-life of 29 years indicates that it takes decades for the substance to diminish through radioactive decay, necessitating careful monitoring and management to minimize its impact on human and environmental health. Understanding its decay rate is essential in assessing long-term radioactivity and implementing safety measures.

Natural Logarithm

The natural logarithm is a mathematical function often denoted as \(\ln\), used throughout the sciences for calculations involving growth and decay. It is based on the mathematical constant \(e\), approximately equal to 2.71828. In the context of radioactive decay, the natural logarithm is crucial in determining time-related aspects of decay processes.

• It simplifies complex exponentiation calculations into manageable algebraic operations.
• Logarithms convert multiplicative processes into additive ones, which is useful for solving exponential decay equations.

Using the natural logarithm helps to resolve the exponential decay equation by allowing us to isolate the variable \(t\) (time). As in the example calculation for determining the decay of strontium-90 until 10% remains, taking the logarithm of both sides of the equation allows the exponent to be transformed into a linear form, making it easier to solve for \(t\). Such methods illustrate the power of logarithms in converting difficult exponential problems into simpler algebraic ones.