Problem 20 - Calculus for Biology and Medicine

Question

Chemicals \(\mathrm{A}\) and \(\mathrm{X}\) react through an autocatalytic reaction: $$ \mathrm{A}+\mathrm{X} \longrightarrow 2 \mathrm{X} $$ (a) If the rate constant for the reaction is \(k\), and the amounts of A and \(X\) present are denoted by \([\mathrm{A}]\) and \([\mathrm{X}]\), then explain why $$ \frac{d[\mathrm{~A}]}{d t}=-k[\mathrm{~A}][\mathrm{X}] $$ (b) Find a differential equation that describes the rate of change of the amount of \([\mathrm{X}]\).

Step-by-Step Solution

Answer

(a) \( \frac{d[\mathrm{A}]}{dt} = -k[\mathrm{A}][\mathrm{X}] \) because it's a bimolecular reaction; (b) \( \frac{d[\mathrm{X}]}{dt} = k[\mathrm{A}][\mathrm{X}] \).

Step 1: Analyze the Given Reaction

We start by examining the reaction \( \mathrm{A}+\mathrm{X} \rightarrow 2\mathrm{X} \). This indicates that one molecule of \( \mathrm{A} \) reacts with one molecule of \( \mathrm{X} \) to produce two molecules of \( \mathrm{X} \). This type of reaction is known as an autocatalytic reaction because \( \mathrm{X} \), one of the products, also acts as a catalyst.

Step 2: Equation for Rate of Change of \([\mathrm{A}]\)

The rate of reaction is typically dependent on the concentrations of the reactants. Since \( \mathrm{A} \) and \( \mathrm{X} \) are involved in the reaction, the rate of disappearance of \( [\mathrm{A}] \) can be described by the rate equation: \[ \frac{d[\mathrm{A}]}{dt} = -k[\mathrm{A}][\mathrm{X}] \]Here, \( k \) is the rate constant, and the negative sign indicates that \( [\mathrm{A}] \) is decreasing over time.

Step 3: Relation to \([\mathrm{X}]\)

From the stoichiometry of the reaction, each mole of \( \mathrm{A} \) that reacts produces an additional mole of \( \mathrm{X} \). Therefore, the rate of production of \( [\mathrm{X}] \) due to the reaction is: \[ \frac{d[\mathrm{X}]}{dt} = k[\mathrm{A}][\mathrm{X}] \]As the reaction progresses, the concentration of \( \mathrm{X} \) increases by the same rate as the consumption of \( \mathrm{A} \).

Key Concepts

Differential Equations

In the realm of chemistry, differential equations are used to describe how the concentration of reactants and products changes over time. These equations are crafted based on rate laws, which indicate how fast a reaction proceeds. In our case, we're looking at an autocatalytic reaction where the rate of disappearance of species \([A]\) is described using a first-order differential equation:

• \(\frac{d[\mathrm{A}]}{dt} = -k[\mathrm{A}][\mathrm{X}]\)

The negative sign tells us that the concentration of \([A]\) decreases with time. The equation represents the fact that the change in \([A]\)'s concentration depends on both its current concentration and the concentration of \([X]\).
The essential part of using a differential equation lies in its ability to project future behavior of the system. Solving such equations can show us how \([A]\) and \([X]\) will evolve as time passes. Solving differential equations often involves techniques that integrate or approximate solutions based on initial conditions.

Rate of Reaction

The rate of reaction is a measure of how quickly reactants are converted into products in a chemical process. It's influenced by factors like concentration, presence of catalysts, and temperature. In the autocatalytic reaction \(A + X \rightarrow 2X\), the rate is critical because the formation of X also increases the rate at which the reaction proceeds, due to its role as a catalyst. This autocatalytic nature can lead to surprisingly rapid changes once the reaction reaches a certain point.
The rate of reaction specific to the disappearance of A is provided by the equation:

• \(\frac{d[\mathrm{A}]}{dt} = -k[\mathrm{A}][\mathrm{X}]\)

This equation implies that the reaction rate is proportional to the product of the concentrations of \([A]\) and \([X]\), illustrating how crucial both reactants are. Higher concentrations of either substance increase the rate significantly.
A distinctive aspect of autocatalytic reactions is that the presence of product \([X]\) actually speeds up the rate of reaction, which is different from typical reactions where products do not influence the rate.

Stoichiometry

Stoichiometry provides us with a method of using balanced equations to determine the relative amounts of reactants and products involved in a reaction. For the reaction \[A + X \rightarrow 2X\], the stoichiometry is particularly interesting because it shows that \([X]\) is both a reactant and a product.

• For every molecule of \([A]\) that disappears, \([X]\) contributes to the reaction and also results in two molecules of \[X\] being formed.

Using stoichiometry, one can understand how the rate of change in concentration of \([X]\) is connected to \([A]\). As derived from the step-by-step solution:

• \(\frac{d[\mathrm{X}]}{dt} = k[\mathrm{A}][\mathrm{X}]\)

This shows the formation of \([X]\) as positively affecting the rate of creation, while the consumption of \([A]\) matches the rate of \([X]\)'s formation.
Ultimately, stoichiometry in an autocatalytic reaction like this is essential to grasp not only the quantitative outcome of products but also the dynamics that affect the reaction rate.