Question

The half-life of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that, for any radioactive material that decays according to the equation \(Q^{\prime}=-r Q,\) the half-life \(\tau\) and the decay rater satisfy the equation \(r \tau=\ln 2 .\)

Short answer

Question: Derive the relationship between the decay rate constant (r), half-life (τ), and the natural logarithm of 2 (ln(2)) for a radioactive decay. Answer: The relationship between the decay rate constant (r), half-life (τ), and the natural logarithm of 2 (ln(2)) can be derived from the radioactive decay equation and is given by: $$r\tau = \ln{2}$$

Step-by-step solution

Identify the variables in the radioactive decay equation and set up the equation

We are given the differential equation for radioactive decay: $$\frac{dQ}{dt}=-rQ$$ Where Q(t) represents the quantity of the radioactive material at a given time, t, r is the decay rate constant, and -rQ represents the rate of decay.

Integrate the differential equation to find Q(t)

Now, we integrate the differential equation. First, separate variables by dividing by Q and multiplying by dt: $$\frac{dQ}{Q}=-r\,dt$$ Next, integrate both sides: $$\int\frac{dQ}{Q}=-r\int dt$$ After integration, we get: $$\ln(Q)=-rt+C$$ Where C is the integration constant. To find the equation for Q(t), we exponentiate both sides: $$Q(t)=Q_0e^{-rt}$$ Where Q_0 = Q(t=0) represents the initial quantity of the radioactive material.

Use the definition of half-life to relate τ and r

Now, we determine the relationship between r and τ. A half-life is the time needed for half of the radioactive material to decay. We can write the equation for this by setting Q(t) = 1/2 Q_0, where Q_0 is the initial quantity of the radioactive material and τ is the time taken to reach half of the initial quantity. $$\frac{1}{2}Q_0=Q_0e^{-r\tau}$$ Divide by Q_0: $$\frac{1}{2}=e^{-r\tau}$$

Derive the relationship between τ, r, and ln(2)

Taking the natural logarithm of both sides of the equation from Step 3, we get: $$\ln{\frac{1}{2}}=-r\tau$$ Now, we know that \(\ln{\frac{1}{2}}=\ln{2^{-1}}=-\ln{2}\), so we can write: $$-\ln{2}=-r\tau$$ Finally, we have the relationship between τ, r, and ln(2): $$r\tau = \ln{2}$$

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe the relationship between variables and their rates of change. They are the backbone of many scientific and engineering disciplines as they provide a means to model dynamic systems like populations, temperature changes, and yes, even radioactive decay. In our context, the radioactive decay equation, \( \frac{dQ}{dt} = -rQ \), is a first-order linear differential equation, where the rate of change of the quantity \( Q \) with respect to time \( t \) is proportional to the quantity itself. This—quite vitally—lets us predict how quickly a substance will decay over time.

Half-Life

Half-life is a concept that signifies the duration required for half the atoms in a sample of radioactive material to decay. It's a constant that provides an intuitive understanding of the decay process, as it is independent of the initial amount of the substance. In our equation, we use half-life, denoted \( \tau \) to establish a clear link between how long it takes for the quantity to reach half its value and the decay rate, \( r \). Though different radioactive materials have different half-lives, ranging from milliseconds to billions of years, the mathematical relationship remains consistent, enabling us to predict long-term behavior from a single measurement.

Exponential Decay

Exponential decay describes a process where the amount of substance decreases at a rate proportional to its current value, which is exactly what happens in radioactive decay. This leads to a rapid decline initially that slows down over time, creating a decaying curve when graphed. The formula \( Q(t) = Q_0e^{-rt} \) represents this phenomenon quantitatively, portraying how the quantity diminishes over time. Here, \( Q_0 \) is the initial quantity and \( e \) is the base of the natural logarithm, which is crucial to the continuous growth and decay in natural processes.

Integration of Equations

Integration can be seen as a way to 'sum up' continuous change, and it's the opposite process of differentiation. It's used to find quantities like areas, volumes, and, in our case, the total quantity of radioactive material remaining over time. In the step-by-step solution, we separated variables and integrated both sides to transition from \( \frac{dQ}{Q} = -rdt \) to \( \ln(Q) = -rt + C \), effectively 'adding up' the effects of decay over each moment in time—and thereby getting the formula for \( Q(t) \) reflecting the exponential decay model.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It plays a pivotal role in our understanding of growth and decay because many natural processes can be explained using exponential functions, which are inherently tied to \( e \) and its properties. When we encounter the equation \( \frac{1}{2} = e^{-r\tau} \) during the solution process, taking the natural logarithm of both sides lets us solve for the decay rate \( r \) in relation to the half-life \( \tau \) since \( \ln(e^x) = x \) and \( \ln(\frac{1}{2}) = -\ln(2) \)—a clean and elegant connection only made possible because of the properties of the natural logarithm.