Problem 25 - Calculus for Biology and Medicine

Question

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes \(V_{1}\) and \(V_{2}\). $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{\infty}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ where \(C_{1}(t)\) is the concentration in the first tank and \(C_{2}(t)\) is the concentration in the second tank, and \(q\) is the volume of water flowing between the two tanks in one unit of time. When we analyzed \((8.93)\) and \((8.94)\) in the main text we assumed that \(V_{1} \neq V_{2} .\) Now consider how the analysis must be modified if \(V_{1}=V_{2}\), and \(C_{1}(0)=C_{2}(0)=0 .\) (a) Show that \(C_{1}(t)=C_{\infty}\left(1-e^{-q t / V_{1}}\right)\) and \(C_{2}(t)=\) \(C_{\infty}\left(1-\left(1+\frac{q t}{V_{1}}\right) e^{-q t / V_{1}}\right)\) (b) Show that \(\lim _{t \rightarrow \infty} C_{1}(t)=C_{\infty}\) and \(\lim _{t \rightarrow \infty} C_{2}(t)=C_{\infty}\).

Step-by-Step Solution

Answer

Both concentrations reach \(C_{\infty}\) as time goes to infinity.

Step 1: Setting the Initial Conditions

We know that initially, both tanks have zero concentration, i.e., \(C_{1}(0)=C_{2}(0)=0\). This provides our initial conditions to solve the differential equations for \(C_{1}(t)\) and \(C_{2}(t)\).

Step 2: Solving the Differential Equation for C1(t)

The equation \(\frac{d C_{1}}{d t}=\frac{q}{V_{1}}(C_{\infty}-C_{1})\) is a standard first-order linear differential equation. Solving this, we find the general solution for \(C_{1}(t)\): \(C_{1}(t)=C_{\infty}\left(1-e^{-qt/V_{1}}\right)\). This matches the form provided, confirming part (a) for \(C_{1}(t)\).

Step 3: Solving the Differential Equation for C2(t)

The equation \(\frac{d C_{2}}{d t}=\frac{q}{V_{2}}(C_{1}-C_{2})\) is similar to a standard form. We substitute \(C_{1}(t)=C_{\infty}(1-e^{-qt/V_{1}})\), and given \(V_1 = V_2\), integrate to find \(C_{2}(t)=C_{\infty}\left(1-\left(1+\frac{qt}{V_{1}}\right)e^{-qt/V_{1}}\right)\). This confirms part (a) for \(C_{2}(t)\).

Step 4: Checking the Limits for C1(t) as t Approaches Infinity

To find \(\lim _{t \to \infty} C_{1}(t)\), we evaluate the limit. As \(t \rightarrow \infty\), \(e^{-qt/V_{1}} \rightarrow 0\), resulting in \(C_{1}(t)\) approaching \(C_{\infty}\).

Step 5: Checking the Limits for C2(t) as t Approaches Infinity

To find \(\lim _{t \to \infty} C_{2}(t)\), we evaluate the limit of the expression \(C_{\infty}\left(1-\left(1+\frac{qt}{V_{1}}\right)e^{-qt/V_{1}}\right)\). As \(t \rightarrow \infty\), the term \(\left(1+\frac{qt}{V_{1}}\right)e^{-qt/V_{1}} \rightarrow 0\), leading to \(C_{2}(t)\) approaching \(C_{\infty}\). This confirms both required limits.

Key Concepts

Differential Equations in the Two-Compartment Model

In the world of mathematics, differential equations have a special place because they allow us to describe the change of quantities over time. In this two-compartment model, we have two tanks that interact through the flow of liquid. This is described by two connected differential equations: - The first equation, \(\frac{d C_{1}}{d t}=\frac{q}{V_{1}}(C_{\infty}-C_{1})\), describes how the concentration \(C_1(t)\) in the first tank changes. It considers the difference between the steady-state concentration \(C_\infty\) and the current concentration \(C_1\). - The second equation, \(\frac{d C_{2}}{d t}=\frac{q}{V_{2}}(C_{1}-C_{2})\), expresses how the concentration \(C_2(t)\) in the second tank evolves. This depends on the difference between the concentration in the first and second tanks.These equations are **first-order linear differential equations** and are fundamental because they predict how the concentration in both tanks changes when water flows between them. The beauty of these equations is in their ability to model dynamic processes in a simple format by using the flow rate \(q\) and the volumes \(V_1\) and \(V_2\) as parameters.

Understanding Concentration in Water Tanks

Concentration is the amount of a substance in a given volume, and it's a crucial concept in the two-compartment model. In this scenario, concentration \(C_1(t)\) and \(C_2(t)\) refer to how much of a substance is present in tanks 1 and 2, respectively, at any given time \(t\). It's important to note that concentrations change over time due to the dynamic nature of water exchange, governed by the flow of water (\(q\) in this model). The concentrations start at zero, which helps us understand their time evolution from a baseline. We focus on the ultimate goal: reaching a steady state where the concentrations \(C_1(t)\) and \(C_2(t)\) in both tanks eventually reach the steady-state concentration \(C_\infty\).- In practical terms, understanding concentration dynamics helps predict how substances dissolve or distribute within reservoirs, affecting fields like environmental engineering and chemical processing.

Initial Conditions and Their Role

Initial conditions set the stage for any differential equation problem. They determine the specific solution among many possibilities. For our problem involving two tanks, the initial conditions are given as \(C_1(0) = C_2(0) = 0\), meaning both tanks start with zero concentration.Here's why initial conditions are important:

• Initial conditions allow us to capture the unique actual progress of a system under study. Without them, we could theoretically have infinite solutions.
• They also anchor us for modeling real-world problems where measurements need a starting point.

In our model, the zero initial concentrations mean that no substance is present in either tank at the start. Over time, as the tanks interact, these values evolve based on the differential equations. Understanding initial conditions ensures we grasp how a system transitions from its start to eventual equilibrium.

Evaluating Limits in Mathematical Models

A limit in mathematics describes the behavior of a function as its input approaches a certain value. In our exercise, we're particularly interested in the limits of \(C_1(t)\) and \(C_2(t)\) as time \(t\) grows infinitely large.Evaluating the limits of the concentrations tells us their steady-state values. For \(C_1(t)\), as \(t\) approaches infinity, the exponential term \(e^{-qt/V_1}\) becomes negligible, resulting in \(C_1(t)\) reaching \(C_{\infty}\). Similarly, for \(C_2(t)\), the term \(\left(1+\frac{qt}{V_{1}}\right)e^{-qt/V_{1}}\) approaches zero, making \(C_2(t)\) also reach \(C_{\infty}\).This shows that both tanks ultimately equilibrate at the steady-state concentration. Exploring limits gives insight into the long-term behavior of the system. By confirming these limits, we verify that no matter the starting condition, the system will stabilize over time, value relevant for predicting steady-state conditions in engineering models.