Problem 48 - Calculus for Biology and Medicine

Question

Consider the chemical reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x\) denotes the concentration of \(\mathrm{AB}\) at time \(t\), then the reaction rate \(R(x)\) is given by $$ R(x)=k(a-x)(b-x) $$ where \(k, a\), and \(b\) are positive constants. Differentiate \(R(x)\).

Step-by-Step Solution

Answer

The derivative of \( R(x) \) is \( R'(x) = k(2x - a - b) \).

Step 1: Understanding the Problem

We are given a chemical reaction rate equation \( R(x) = k(a-x)(b-x) \) that depends on the concentration of AB, denoted by \( x \). To solve the problem, we need to find the derivative of \( R(x) \) with respect to \( x \).

Step 2: Using the Product Rule

Since \( R(x) \) is the product of two functions of \( x \), i.e., \( (a-x) \) and \( (b-x) \), we will use the product rule for differentiation. The product rule states that if we have two functions \( u(x) \) and \( v(x) \), their derivative is \( u'(x)v(x) + u(x)v'(x) \).

Step 3: Applying the Product Rule Step-by-Step

Define \( u(x) = (a-x) \) and \( v(x) = (b-x) \). Therefore, \( R(x) = k \, u(x) \, v(x) \). We first find the derivatives: \( u'(x) = -1 \) and \( v'(x) = -1 \). The derivative \( R'(x) \) is given by:\[ R'(x) = k \left( u'(x)v(x) + u(x)v'(x) \right) \]\[ = k \left( (-1)(b-x) + (a-x)(-1) \right) \].

Step 4: Simplifying the Derivative

Substitute \( u'(x) = -1 \) and \( v'(x) = -1 \):\[ R'(x) = k \left( -(b-x) - (a-x) \right) \]Simplify the expression:\[ = k(-b + x - a + x) \]\[ = k(-a - b + 2x) \].

Step 5: Final Expression for the Derivative

Thus, the derivative of the reaction rate \( R(x) \) with respect to \( x \) is:\[ R'(x) = k(2x - a - b) \].

Key Concepts

Reaction Rate

In chemical kinetics, the reaction rate is a key concept. It refers to how fast reactants turn into products in a chemical reaction over time. For the reaction \( \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} \), the rate at which \( \mathrm{AB} \) is formed is represented by the function \( R(x) \), where \( x \) is the concentration of \( \mathrm{AB} \) at time \( t \). Understanding reaction rates is crucial to predict how quickly a reaction proceeds. For our specific equation, the reaction rate is given by:

• \( R(x) = k(a-x)(b-x) \)

Here, \( k \) is the rate constant, while \( a \) and \( b \) are initial concentrations of reactants \( \mathrm{A} \) and \( \mathrm{B} \), respectively. As \( x \) changes, so does \( R(x) \); thus, differentiating \( R(x) \) with respect to \( x \) tells us how the rate changes as the concentration of \( \mathrm{AB} \) varies.

Product Rule

The product rule is a fundamental principle in calculus used to differentiate functions that are products of two other functions. If we have two functions \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) is given by:

• \( u'(x)v(x) + u(x)v'(x) \)

This rule is particularly useful when faced with a problem like ours, where \( R(x) = k(a-x)(b-x) \). We treat \( (a-x) \) and \( (b-x) \) as separate functions:

• \( u(x) = (a-x) \)
• \( v(x) = (b-x) \)

Differentiating each function independently:

• \( u'(x) = -1 \)
• \( v'(x) = -1 \)

The product rule helps us systematically combine these derivatives to find \( R'(x) \), the rate at which the reaction rate changes with \( x \).

Derivative Calculation

Calculating derivatives is a method to find how a function changes as its input changes. In our case, we want to determine how \( R(x) \)—the reaction rate—evolves as \( x \), the concentration of \( \mathrm{AB} \), varies. Given the expression \( R(x) = k(a-x)(b-x) \), we apply the product rule because it is a product of \( (a-x) \) and \( (b-x) \). With the derivatives:

• \( u'(x) = -1 \)
• \( v'(x) = -1 \)

Substitute these into the product rule formula:

• \( R'(x) = k(-1)(b-x) + k(a-x)(-1) \)

This simplifies step-by-step to:

• \( R'(x) = k \left( -(b-x) - (a-x) \right) \)

Upon further simplification:

• \( R'(x) = k(-b + x - a + x) \)
• \( R'(x) = k(2x - a - b) \)

Thus, the derivative \( R'(x) = k(2x - a - b) \) indicates the rate at which the reaction rate changes, offering valuable insights into the dynamics of the reaction.