Problem 49 - Calculus for Biology and Medicine

Question

Use the mass action law to translate each chemical reaction into a system of differential equations. \(\mathrm{A}+\mathrm{B} \stackrel{k}{\longrightarrow} \mathrm{C}\)

Step-by-Step Solution

Answer

The system of differential equations is: 1) \( \frac{d[A]}{dt} = -k[A][B] \), 2) \( \frac{d[B]}{dt} = -k[A][B] \), 3) \( \frac{d[C]}{dt} = k[A][B] \).

Step 1: Understand the Reaction

We have a reaction where one molecule of A reacts with one molecule of B to produce one molecule of C. This reaction is determined by the rate constant \( k \).

Step 2: Write the Rate of Reaction

According to the mass action law, the rate of reaction is proportional to the product of the concentrations of the reactants. Thus, the rate of the reaction \( A + B \rightarrow C \) is given by: \( r = k[A][B] \), where \([A]\) and \([B]\) are the concentrations of A and B, respectively.

Step 3: Formulate Differential Equations for Reactants

The concentration of reactants A and B decreases as the reaction proceeds. Therefore, the rate of change of the concentration of A is given by: \( \frac{d[A]}{dt} = -k[A][B] \). Similarly, for B: \( \frac{d[B]}{dt} = -k[A][B] \).

Step 4: Formulate Differential Equation for Product

As the reactants are consumed, the concentration of product C increases at a rate equal to the rate at which A and B are consumed. Therefore, the rate of change of the concentration of C is given by: \( \frac{d[C]}{dt} = k[A][B] \).

Step 5: Compile the System of Differential Equations

Combine the above equations to get the full system of differential equations for the reaction: 1. \( \frac{d[A]}{dt} = -k[A][B] \) 2. \( \frac{d[B]}{dt} = -k[A][B] \) 3. \( \frac{d[C]}{dt} = k[A][B] \).

Key Concepts

Mass Action Law

The Mass Action Law is a fundamental principle in chemistry used to describe the rate of a chemical reaction. It states that the rate of a reaction is directly proportional to the product of the concentrations of the reactants, each raised to the power of their respective coefficients in the balanced equation. In this context, for a simple reaction like \(A + B \rightarrow C\), the rate can be expressed as follows:

• Reaction rate \( r = k[A][B] \)

Here, \([A]\) and \([B]\) denote the molar concentrations of reactants \(A\) and \(B\), and \(k\) is the rate constant—a value that is determined experimentally and varies with temperature.
The mass action law provides a systematic way to predict how changes in concentration affect the reaction rate, making it invaluable in chemical kinetics.

Chemical Reactions

Chemical reactions are processes where substances—reactants—transform into different substances, known as products. These transformations involve the breaking and forming of bonds, ultimately rearranging atoms.
In the reaction \(A + B \rightarrow C\), a molecule of \(A\) reacts with a molecule of \(B\) to create a molecule of \(C\). Understanding reactions requires grasping:

• Reactants and Products: Starting and ending materials, such as \(A\) and \(B\) as reactants, and \(C\) as the product.
• Rate of Reaction: How quickly a reaction progresses, influenced by factors like temperature, pressure, and concentration.
• Conservation of Mass: In a chemical reaction, atoms are neither created nor destroyed, ensuring mass is conserved.

Changes in concentrations over time during reactions are tracked using differential equations, making it possible to predict system behavior under various conditions.

Reaction Kinetics

Reaction kinetics is the study of the rates of chemical reactions and the factors affecting them. It focuses on:

• Rate Laws: Expressions like \( r=k[A][B] \) derived from the mass action law, showing how reaction rates depend on reactant concentrations.
• Order of Reaction: Determined by the sum of the exponents in the rate law; for \( r=k[A][B] \), the overall order is 2, indicating both reactants influence the rate.
• Rate Constants: Constants that vary with temperature and are specific to each reaction, playing a crucial role in these expressions.

Understanding kinetics involves formulating differential equations like:

• \( \frac{d[A]}{dt} = -k[A][B] \)
• \( \frac{d[C]}{dt} = k[A][B] \)

These equations depict how concentrations of \(A\), \(B\), and \(C\) change with time, enabling predictions about reaction completion and finding conditions for optimal reaction rates.