Problem 30 - Calculus for Biology and Medicine

Question

An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction $$ \mathrm{A}+\mathrm{X} \rightarrow \mathrm{X} $$ If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by $$ R(x)=k x(a-x) $$ for \(0 \leq x \leq a\), where \(a\) is the initial concentration of \(A\) and \(x\) is the concentration of \(X\). (a) Show that \(R(x)\) is a polynomial and determine its degree. (b) Graph \(R(x)\) for \(k=2\) and \(a=6\). Find the value of \(x\) at which the reaction rate is maximal.

Step-by-Step Solution

Answer

(a) \( R(x) \) is a quadratic polynomial of degree 2. (b) The reaction rate is maximal at \( x = 3 \).

Step 1: Analyze the Reaction Rate Expression

Consider the reaction rate expression \( R(x) = kx(a-x) \). Expand it to check if it's a polynomial: \[ R(x) = k(ax - x^2) = kax - kx^2. \] This shows \( R(x) \) is a polynomial in \( x \), specifically a quadratic polynomial because the highest power of \( x \) is 2.

Step 2: Determine the Degree of the Polynomial

Since the expression for \( R(x) = kax - kx^2 \) has the highest power of \( x \) being 2, the degree of the polynomial is 2.

Step 3: Set up the Graph with Given Values

Substitute \( k = 2 \) and \( a = 6 \) into the equation: \[ R(x) = 2x(6-x) = 12x - 2x^2. \] This will help in graphing the polynomial function to observe its behavior and find the maximum.

Step 4: Find the Maximum Reaction Rate Using Calculus

The maximum value of \( R(x) \) can be found by taking the derivative: \( R'(x) = \frac{d}{dx}(12x - 2x^2) = 12 - 4x. \) Set \( R'(x) = 0 \) to find the critical points: \[ 12 - 4x = 0 \rightarrow x = 3. \] Check if it's a maximum by evaluating the second derivative, \( R''(x) = -4 \), which is negative, confirming a maximum at \( x = 3 \).

Step 5: Graph the Reaction Rate Function

Plot the function \( R(x) = 12x - 2x^2 \) using the range \( 0 \leq x \leq 6 \). Observe that the curve is a downward-opening parabola peaking at \( x = 3 \), where the maximum reaction rate occurs.

Key Concepts

Reaction Rate

The term **reaction rate** refers to how quickly a chemical reaction proceeds over time. In the case of the autocatalytic reaction described, it can be expressed by the equation: \[ R(x) = kx(a-x) \] where:

• \( R(x) \) is the reaction rate.
• \( k \) represents the rate constant.
• \( x \) is the concentration of product \( X \).
• \( a \) is the initial concentration of reactant \( A \).

The reaction rate in this autocatalytic process is unique because it depends on the concentration of both \( A \) and \( X \). As \( X \) increases, it facilitates the production of more \( X \), creating a feedback loop. The equation \( kx(a-x) \) allows us to understand how the concentration of substances influences the rate of reaction over time. Each factor is essential for modeling the changing nature of the reaction as more product forms.

Polynomial Degree

In mathematics, the **degree of a polynomial** indicates the highest exponent of its variable. For the reaction rate equation \( R(x) = kx(a-x) \), we can expand it to its polynomial form: \[ R(x) = k(ax - x^2) = kax - kx^2 \] The polynomial is expressed in terms of \( x \), and its highest power is 2. This reveals that the polynomial degree is 2, classifying it as a quadratic polynomial. Identifying the degree of a polynomial is crucial as it informs us about the graph's shape—to be more specific, the number of roots and the general behavior of the curve.

Quadratic Polynomial

A **quadratic polynomial** is a second-degree polynomial, typically represented as \( ax^2 + bx + c \). In the context of the reaction, after expanding \( R(x) = kx(a-x) \), we obtain: \[ R(x) = kax - kx^2 \] This is a quadratic with the following components:

• Leading term \( -kx^2 \): Determines the parabola's opening direction.
• Linear term \( kax \): Affects the slope and incline.
• Constant term: In this case, it is zero, simplifying the graph to intersect the origin.

The fact that our polynomial is quadratic implies it manifests as a parabolic curve. This curve opens downward due to the negative coefficient of \( x^2 \). Quadratic polynomials often serve various applications in physics and engineering, modeling phenomena like projectile motion and kinetics.

Maximum Reaction Rate

To find the **maximum reaction rate**, we use calculus to determine where the reaction rate peaks. For the equation \( R(x) = 12x - 2x^2 \), we calculate its derivative, \( R'(x) \), to find the rate of change of \( R(x) \): \[ R'(x) = \frac{d}{dx} (12x - 2x^2) = 12 - 4x \] Set \( R'(x) = 0 \) to find critical points, yielding \( x = 3 \). This value indicates where \( R(x) \) might have a maximum or minimum. We confirm a maximum via the second derivative, \( R''(x) = -4 \), which is negative, signifying a peak. This peak suggests the system reaches its highest rate of reaction at \( x = 3 \). Recognizing this maximum is crucial for optimizing reaction conditions in industrial and laboratory settings, as it informs scientists and engineers about the most efficient operational parameters.