Problem 20 - Calculus: Early Transcendental Functions

Question

Involve exponential decay. The radioactive element cesium- 137 has a decay constant of -0.023 year \(^{-1} .\) Find its half-life.

Step-by-Step Solution

Answer

The half-life of the radioactive element Cesium-137 is approximately 30.1 years.

Step 1: Define given

We are given that the decay constant \(k\) for Cesium 137 is -0.023 year \(^{-1}\)

Step 2: Determine the formula to use

The formula of half-life for a decreasing exponential function is given by, \(T_{1/2} = \frac{\ln(2)}{|k|}\), where \(T_{1/2}\) is the half-life, \(k\) is the decay constant, and \(|k|\) ensures we are using a positive value for the decay constant.

Step 3: Substitute the value

Substituting the given decay constant \(k=-0.023 year^{-1}\) into the formula, we get \(T_{1/2} = \frac{\ln(2)}{|-0.023 year^{-1}|}\)

Step 4: Solve for the Half-Life Period

Calculate the value of \(T_{1/2}\) to find the half-life period. In this case, the calculation results in \(T_{1/2} \approx 30.1 years \)

Key Concepts

Exponential Decay

When we talk about exponential decay, we're referring to the process where a quantity decreases at a rate that is proportional to its current value. This situation is commonly modeled with the formula:

\[N(t) = N_0 e^{kt}\]
where \(N(t)\) represents the quantity at time \(t\), \(N_0\) is the initial quantity, \(k\) is the decay constant, and \(e\) is the base of the natural logarithm.

The nature of exponential decay is such that the quantity diminishes rapidly at first, then more slowly, but it never fully reaches zero. Radioactive elements are a classic example, as they lose their radioactivity over time in a manner described by an exponential decay function.

In the context of the given exercise, cesium-137's radioactivity and hence the quantity would diminish according to such an exponential decay curve. Understanding this concept helps grasp why the half-life is constant regardless of the initial amount of the substance.

Decay Constant

The decay constant, denoted as \(k\), is a fundamental parameter in the formula for exponential decay. It provides the rate at which a substance decays over time and is specific to each radioactive element. This constant can be positive or negative, where a negative value indicates a decrease in quantity, typical for decay problems.

In any decay situation, knowing the decay constant allows us to predict how quickly a substance will decay. The absolute value of the decay constant is used when computing half-life to ensure a positive duration, as time cannot be negative.

In the exercise example, Cesium-137's decay constant is \(-0.023 year^{-1}\), which tells us that each year, approximately 2.3% of the cesium-137 will have decayed. This information is crucial when determining the half-life, as seen in the provided solution.

Radioactive Decay Mathematics

The branch of mathematics that deals with radioactive decay involves using logarithms and exponentials to model the behavior of decaying substances. It's important to understand this to solve radioactive decay problems accurately.

The half-life formula used in the exercise,
\[T_{1/2} = \frac{\ln(2)}{|k|}\]
illustrates how we use the natural logarithm of 2, symbolizing the time it takes for the substance to reduce to half its original amount. This mathematical relationship comes from setting the exponential decay formula equal to one half of the initial amount and solving for time.

The exercise's step-by-step solution showcases the use of this formula with a specified decay constant to find the half-life of cesium-137. Understanding the math behind radioactive decay enables students to tackle a multitude of problems related to half-life calculations and beyond.