Problem 68 - Calculus for Biology and Medicine

Question

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the half-life of the material is 3 days. Find the differential equation for the radioactive decay function \(W(t)\).

Step-by-Step Solution

Answer

The differential equation is \( \frac{dW}{dt} = \frac{\ln 2}{3} W \).

Step 1: Understanding Half-life and Exponential Decay

The half-life of a radioactive material is the time it takes for half of the material to decay. Since the half-life is given as 3 days, this implies that after 3 days, the amount of the material becomes half of its initial amount.

Step 2: Model of Exponential Decay

The process of radioactive decay follows an exponential decay model, which can be expressed with the equation:\[ W(t) = W_0 e^{-kt} \]where \( W_0 \) is the initial amount of the material, \( k \) is the decay constant, and \( t \) is time.

Step 3: Relating Half-life to Decay Constant

To find the decay constant \( k \), use the half-life information. At half-life, the remaining material is half of \( W_0 \). Substitute into the exponential decay model:\[ \frac{W_0}{2} = W_0 e^{-3k} \]which simplifies to:\[ \frac{1}{2} = e^{-3k} \]

Step 4: Solving for the Decay Constant

Take the natural logarithm of both sides to solve for \( k \):\[ \ln\left( \frac{1}{2} \right) = -3k \]which gives:\[ k = -\frac{\ln 2}{3} \]

Step 5: Formulating the Differential Equation

The differential equation representing this decay is derived from the exponential model:\[ \frac{dW}{dt} = -kW \] Substituting the value of \( k \):\[ \frac{dW}{dt} = \frac{\ln 2}{3} W \] This represents the rate of change of the material over time and is the complete differential equation sought.

Key Concepts

Differential Equation in Radioactive Decay

A differential equation is a mathematical equation that relates a function with its derivatives. In the context of radioactive decay, it captures how the quantity of radioactive substance decreases over time. The differential equation used in radioactive decay helps us understand how the rate of decay is proportional to the current amount of substance. This is crucial because it allows us to describe the process systematically, predicting how much of a material will remain at any given time.

In the case of radioactive decay, the differential equation is derived from the principle that the rate of decay of a substance is directly proportional to its current amount. This relationship is expressed as:\[ \frac{dW}{dt} = -kW \]where

• \( \frac{dW}{dt} \) is the rate of decay of the substance over time,
• \( k \) is the decay constant specific to the material,
• \( W \) is the amount of substance remaining at time \( t \).

This equation provides the foundation for further analysis and is essential in scientific fields where understanding decay behavior is important.

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current value. This kind of decay occurs in many natural processes, especially in radioactive materials, which decays over time at a rate dependent on its current quantity.

The mathematical model for exponential decay is given by the formula:\[ W(t) = W_0 e^{-kt} \]Here:

• \( W(t) \) is the amount of substance remaining at time \( t \),
• \( W_0 \) is the initial amount of the substance,
• \( e \) is the base of natural logarithms, approximately equal to 2.718,
• \( k \) is the decay constant, illustrating how quickly the substance decays,
• \( t \) is the time elapsed.

This model shows that as time goes on, the remaining quantity decreases exponentially. It's crucial because it helps in estimating the half-life and understanding how quickly a substance will lose its radioactivity.

Understanding Half-Life

Half-life is the term used to describe the time required for half of the radioactive nuclei in a sample to decay. It's a critical measure in the study of radioactive substances because it gives insight into how long a material remains active.

The half-life is related to the exponential decay model. At half-life, the amount of material is exactly half of the initial quantity, meaning:\[ W(t_{1/2}) = \frac{W_0}{2} \]Using the exponential decay formula, substituting the half-life condition gives:\[ \frac{W_0}{2} = W_0 e^{-kt} \]which simplifies to:\[ \frac{1}{2} = e^{-kt} \]When simplified further, allows us to find the decay constant \( k \). For example, if the half-life is known, one can solve for \( k \) using the natural logarithm, as demonstrated in calculations within the step-by-step solution by:\[ k = -\frac{\ln 2}{t_{1/2}} \]Half-life is fundamental in determining how quickly a material becomes half as potent and helps predict long-term behavior of radioactive substances.