Question

Is the differential equation that describes a stirred tank reaction (as developed in this section) linear or nonlinear? What is its order?

Short answer

Answer: The stirred tank reaction equation is linear if the reaction order (n) is equal to 1 and nonlinear otherwise. The order of the differential equation is 1.

Step-by-step solution

Understand the stirred tank reaction

A stirred tank reaction is a chemical reaction that takes place in a well-mixed, closed vessel or tank. The concentration of the reactants and products change with time as the reaction proceeds, and the rate of this change depends on the reaction kinetics.

Analyze the differential equation of a stirred tank reaction

The differential equation that describes the stirred tank reaction is given by: dC/dt = (F/V) * (C_in - C) - k * C^n where: - C(t) is the concentration of the reactant in the tank at time t. - C_in is the concentration of the reactant in the incoming stream. - F is the flow rate of the incoming stream. - V is the volume of the tank. - k is the rate constant of the reaction. - n is the reaction order.

Determine if the differential equation is linear or nonlinear

To determine if the differential equation is linear or nonlinear, let's first recall the definition of a linear differential equation. A differential equation is linear if it can be written in the form: P(t) * dC/dt + Q(t) * C = R(t) where P(t), Q(t), and R(t) are continuous functions of t. Looking at the stirred tank reaction equation, we can see that we can not write it in this form if n ≠ 1. Hence, the differential equation is linear if n=1 and nonlinear otherwise.

Determine the order of the differential equation

An equation's order is the highest order of the derivative present in it. In this case, the highest-order derivative present in the stirred tank reaction equation is the first derivative dC/dt. Therefore, the order of the differential equation is 1. In conclusion, the differential equation that describes a stirred tank reaction is linear if the reaction order (n) is equal to 1 and nonlinear otherwise. The order of the differential equation is 1.

Key concepts

Linear Differential Equations

Linear differential equations are those that can be written in a specific form where the dependent variable and its derivatives appear linearly. In simpler terms, this means that you won't see any terms like \(C^2\) or \(e^C\) in the equation. Instead, the equation should follow the structure:

• \(P(t) \cdot \frac{dC}{dt} + Q(t) \cdot C = R(t)\)

Here, \(P(t)\), \(Q(t)\), and \(R(t)\) are functions of the independent variable \(t\), and only the first power of \(C\) and \(dC/dt\) are present.
An example of a linear differential equation is the simple growth or decay process, such as the cooling of an object or interest rates over time. If the equation for our stirred tank reaction fits this mold, specifically when \(n = 1\), it behaves like a linear system. However, in many chemical reactions, \(n\) is not equal to 1, rendering the equation nonlinear.

Nonlinear Differential Equations

Nonlinear differential equations differ because they contain higher powers or products of the dependent variable or its derivatives. This makes them more complex to solve and analyze because their solutions can't generally be written as a simple combination of basic functions. The equation \(\frac{dC}{dt} = \frac{F}{V} \cdot (C_{in} - C) - k \cdot C^n\) for a stirred tank reactor becomes nonlinear if \(n\) is anything except 1. In such a case, the term \(k \cdot C^n\) introduces the nonlinearity.
These types of differential equations appear in various real-world scenarios such as population growth models, climate change predictions, or chemical reactions with complex kinetics. Solving nonlinear differential equations often requires numerical methods or approximations because closed-form solutions are rare.

Order of Differential Equation

The order of a differential equation is determined by the highest derivative present in the equation. It is a crucial aspect because it dictates both the nature of the solutions and the methods required for solving the equation. In the context of our stirred tank reaction, the equation involves \(\frac{dC}{dt}\), which is the first derivative of concentration \(C\) with respect to time \(t\). Thus, the order of this equation is 1.
A first-order differential equation typically describes systems where the rate of change of a variable is directly related to the variable itself, such as in the case of radioactive decay or temperature change. Understanding the order helps us determine the behavior of the solution over time and choose the right mathematical tools for the analysis.

Chemical Reaction Kinetics

Chemical reaction kinetics deals with the rates at which chemical reactions occur and how various conditions affect these rates. It's essential in predicting how quickly reactants turn into products over time. In the stirred tank reaction, the rate at which the concentration \(C\) changes is driven by the differential equation:

• \(\frac{dC}{dt} = \frac{F}{V} \cdot (C_{in} - C) - k \cdot C^n\)

Here, the kinetics are described by parameters like the flow rate \(F\), tank volume \(V\), incoming concentration \(C_{in}\), and the rate constant \(k\). The exponent \(n\) indicates the reaction order, showing how the concentration affects the reaction rate.
Understanding this equation helps in both the design and the control of chemical processes, ensuring the reaction occurs as efficiently and safely as possible.