23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
b. Solve the initial value problem.


A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

Verified step by step guidance

1. Identify the variable to describe the system. Let \(Q(t)\) represent the amount of copper sulfate (in grams) in the tank at time \(t\) (in minutes).
2. Set up the differential equation based on the rate of change of \(Q(t)\). The rate of change equals the rate in minus the rate out:
3. \[ \frac{dQ}{dt} = \text{(rate in)} - \text{(rate out)} \]
4. Calculate the rate in: The inflow concentration is 20 g/L and the inflow rate is 4 L/min, so the rate in is \$4 \times 20 = 80$ g/min.
5. Calculate the rate out: The outflow rate is 4 L/min, and the concentration in the tank at time \(t\) is \(\frac{Q(t)}{500}\) g/L (since the tank volume is constant at 500 L). Thus, the rate out is \$4 \times \frac{Q(t)}{500} = \frac{4}{500} Q(t)$ g/min.
6. Write the initial value problem (IVP) combining these expressions:
7. \[ \frac{dQ}{dt} = 80 - \frac{4}{500} Q(t), \quad Q(0) = 0 \]
8. This is a first-order linear ordinary differential equation. To solve it, use an integrating factor or recognize it as a separable equation.

Key Concepts

Formulating the Initial Value Problem (IVP)

An initial value problem involves a differential equation along with a specified initial condition. In this context, it models the rate of change of copper sulfate concentration in the tank over time, starting from an initial state of pure water (zero concentration). Setting up the IVP requires expressing the inflow, outflow, and accumulation of the substance mathematically.

Mass Balance and Rate of Change

Mass balance principles state that the rate of change of the substance in the tank equals the rate in minus the rate out. Here, the inflow concentration and flow rate add copper sulfate, while the outflow removes it at the current tank concentration. This balance leads to a first-order linear differential equation describing the system.

Solving First-Order Linear Differential Equations

The differential equation derived is typically linear and separable or solvable using integrating factors. Solving it yields the concentration as a function of time, satisfying the initial condition. Understanding solution methods for such equations is essential to find the explicit concentration profile in the tank.