Historical Context & Motivation
Humans have been mixing substances for millennia — from blending medicinal tonics to purifying drinking water. However, predicting exactly how a concentration changes over time required the language of calculus. Mixing problems are one of the earliest and most practical applications of first-order ordinary differential equations (ODEs). They model situations where a substance — like salt, a pollutant, or a chemical — enters and leaves a container at known rates, and we want to know the amount of that substance at any moment.
The central question a mixing problem answers is deceptively simple: If a solution flows into and out of a tank at known rates and concentrations, how much of the dissolved substance is in the tank at time t? Answering this question with algebra alone is impossible because the amount inside the tank is constantly changing, which in turn changes the concentration of what flows out. A differential equation captures this feedback loop perfectly.
Core Principles & Definitions
Every mixing problem revolves around the same fundamental idea: the rate at which the amount of substance in a tank changes equals the rate it comes in minus the rate it goes out. Before we write any equations, let's nail down the key concepts you'll use again and again.
Rate In
Rate Out
Well-Stirred Assumption
Volume Function V(t)
Visual Explanation — The Tank Model
rᵢₙ with incoming concentration cᵢₙ. The well-stirred mixture leaves through the outflow pipe (pink arrow) at rate rₒᵤₜ with a concentration equal to Q(t) ÷ V(t). The governing ODE is shown in the amber bar at the bottom.The diagram above captures everything you need to set up a mixing problem. Notice that the inflow rate is entirely given information — you always know how fast liquid enters and what concentration it carries. The outflow concentration, however, depends on Q(t) and V(t), both of which can change with time. This dependency is precisely why we need a differential equation rather than a simple algebraic formula.
Mathematical Framework
Let's build the governing equation step by step. We define Q(t) as the amount (in grams, kilograms, etc.) of the dissolved substance in the tank at time t. The key principle is a balance law: the rate of change of Q equals what comes in minus what goes out.
Solving with the Integrating Factor
The standard approach to solving mixing problems uses the integrating factor method. For the constant-volume case where rᵢₙ = rₒᵤₜ = r, we write the ODE in standard linear form and then multiply both sides by a carefully chosen function that makes the left side collapse into a single derivative.
The final formula tells us something powerful: as t → ∞, the exponential term e^(−rt/V) decays to zero, so Q(t) approaches V × cᵢₙ. In plain language, the tank eventually reaches the same concentration as the incoming solution — no matter what the starting concentration was. This is known as the steady-state or equilibrium value of the system.
Worked Example — Salt in a Tank
A tank initially holds 100 liters of water with 50 grams of salt dissolved in it. Brine (salt water) containing 2 g/L of salt flows into the tank at 5 L/min, and the well-stirred mixture flows out at 5 L/min. Find the amount of salt Q(t) in the tank at any time t, and determine how much salt is in the tank after 20 minutes.
Notice that the equilibrium amount is V × cᵢₙ = 100 × 2 = 200 g. After 20 minutes the tank already holds about 145 g, well on its way toward 200 g. As time goes on, the exponential term shrinks, and Q(t) gets closer and closer to 200 g but never quite reaches it in finite time.
Strengths, Limitations & Variations
| Feature | Strengths | Limitations |
|---|---|---|
| Well-stirred assumption | Greatly simplifies the model; yields a single ODE rather than a partial differential equation. | Real tanks may have dead zones or layering, making the assumption inaccurate for large or poorly mixed vessels. |
| Constant flow rates | Keeps the ODE linear with constant coefficients, making it solvable by hand with an integrating factor. | In practice, flow rates often vary with time, requiring numerical methods or piecewise solutions. |
| Single tank | One ODE captures the full dynamics; the solution is a single explicit formula. | Many real systems involve cascades of tanks (e.g., wastewater treatment), requiring coupled ODEs. |
| No reactions | Ideal for inert substances like salt or dye where the substance doesn't degrade. | Fails for substances that react, decay, or are consumed (e.g., medications metabolizing in the body). |
Connection to Advanced Topics
Mixing problems are a gateway to many powerful ideas in mathematics and science. Once you're comfortable with the single-tank model, you can extend the framework in several directions. The table below compares the introductory model you've learned to its more advanced counterparts.
| Basic Model | Advanced Extension | What Changes |
|---|---|---|
| Single tank, constant volume | Single tank, variable volume | V(t) = V₀ + (rᵢₙ − rₒᵤₜ)t replaces V; the integrating factor involves a power function instead of an exponential. |
| Single tank | Cascading tanks (systems of ODEs) | The outflow of Tank 1 becomes the inflow of Tank 2. You solve a system of coupled first-order ODEs, often using matrices. |
| Inert substance (no reaction) | Substance with decay or reaction | An additional term −kQ is added to model radioactive decay or chemical consumption, leading to a modified linear ODE. |
| Constant inflow concentration | Time-varying inflow (e.g., periodic dosing) | cᵢₙ becomes a function of t, so the non-homogeneous term changes and may require Laplace transforms or numerical integration. |
If you continue studying differential equations, you'll encounter systems of ODEs and Laplace transforms — both of which build directly on the skills you develop here. The mixing problem is also structurally identical to models used in pharmacokinetics (how drugs move through your body) and compartmental epidemiology (how diseases spread through populations), so mastering this framework opens the door to a wide range of applications.
Practice Problems
Summary — Mixing Problems
Mixing problems model how the amount of a dissolved substance changes inside a tank when solution flows in and out continuously. The governing equation is built from a simple balance law: dQ/dt = (Rate In) − (Rate Out), where Rate In = rᵢₙ × cᵢₙ and Rate Out = rₒᵤₜ × Q(t)/V(t). The well-stirred assumption ensures the outflow concentration equals the uniform concentration inside the tank. When inflow and outflow rates are equal, the volume stays constant and the ODE is a standard first-order linear ODE solvable with an integrating factor.
The general solution for a constant-volume tank is Q(t) = V × cᵢₙ + (Q₀ − V × cᵢₙ) × e^(−rt/V), revealing that Q(t) always approaches the steady-state value V × cᵢₙ as time increases. When flow rates differ, V(t) changes linearly, producing a variable-coefficient ODE. The mixing-problem framework extends naturally to cascading tanks, pharmacokinetics, and environmental modeling, making it one of the most versatile applications of first-order ODEs.