Mixture Problems

Definition

Mixture problems model the change in the amount of a substance $y(t)$ in a tank as fluid flows in and out. The fundamental relationship is the rate equation: $\frac{dy}{dt} = (\text{rate in}) - (\text{rate out})$

• How to read: “The derivative of y with respect to time t is equal to the rate in minus the rate out.”
• Meaning: The net rate of change of solute mass equals what flows in minus what flows out—conservation of mass.

where the rate is typically (flow rate) $\times$ (concentration).

Why It Matters

In chemical engineering and pharmacology, failing to model mixtures correctly can lead to toxic concentrations or ineffective dosages. These problems are the real-world application of rates of change, and errors here can result in environmental disasters or medical fatalities.

Core Concepts

• Uniform Mixing: The model assumes the substance is perfectly stirred, so the concentration in the tank is $\frac{y(t)}{V(t)}$ at all times.

  • How to read: “The concentration is equal to the amount y at time t divided by the volume V at time t.”
  • Meaning: Total solute divided by current tank volume gives the uniform concentration everywhere in the tank.

• Variable Volume: If the inflow and outflow rates are different, the volume $V(t)$ changes over time, affecting the concentration.

• Linear Equation: These problems almost always result in a first-order linear differential equation that can be solved using an integrating factor.

Connected Concepts

• Classification of Differential Equations
• Superposition Principle (Differential Equations)
• Autonomous Differential Equations