Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In the beginning of a chemical reaction there are 800 moles of substance A and none of substance \(\mathrm{B}\). Over the course of the reaction, the \(800 \mathrm{moles}\) of substance \(\mathrm{A}\) are converted to 800 moles of substance B. (Each molecule of A is converted to a molecule of \(\mathrm{B}\) via the reaction.) Suppose the rate at which \(\mathrm{A}\) is turning into \(\mathrm{B}\) is proportional to the product of the number of moles of A and the number of moles of \(\mathrm{B}\). (a) Let \(N=N(t)\) be the number of moles of substance \(\mathrm{A}\) at time \(t .\) Translate the statement above into mathematical language. (Note: The number of moles of substance \(\mathrm{B}\) should be expressed in terms of the number of moles of substance A.) (b) \(N(t)\) is a decreasing function. The rate at which \(N\) is changing is a function of \(N\), the number of moles of substance A. When the rate at which A is being converted to \(\mathrm{B}\) is highest, how many moles are there of substance \(\mathrm{A}\) ?

Step-by-Step Solution

Answer

The conversion rate of substance A to B is at its highest in the beginning of the reaction when there are 800 moles of substance A.

Step 1: Translating the statement

The given states that the rate of conversion of A to B is proportional to the product of the number of moles of A and B. Since the number of moles of A is denoted as \(N(t)\) and the total moles (A and B combined) remains constant at 800, it follows that the moles of B at any point in time is \(800 - N(t)\). We can translate the given statement into the following differential equation: \[\frac{dN}{dt} = -kN(800 - N)\]where \(k\) is the constant of proportionality and the negative sign is because \(N(t)\) is decreasing as A is being converted into B.

Step 2: Finding the maximum conversion rate value

To find the maximum conversion rate, we need to solve the equation from step 1 for when \(\frac{dN}{dt} = 0\) (the function's maximum or minimum). Solving \(-kN(800 - N)=0\), we obtain the values for \(N(t)\) when the derivative equals zero. The solutions are \(N(t) = 0\) and \(N(t) = 800\). However, since \(N(t)\) is a decreasing function according to the problem, the function's maximum occurs when the number of A moles is at 800.

Step 3: Analyzing the results

Given that the rate function is a decreasing function and has a maximum when the number of A moles is at 800, the conversion rate is at its highest in the beginning of the reaction, when there are 800 moles of substance A.

Key Concepts

Chemical Reactions

Chemical reactions are processes where substances transform into new substances through breaking and forming chemical bonds. In this scenario, substance A is fully converted into substance B. The total amount of substance does not change, but only its form. This follows the law of conservation of mass, where the initial 800 moles of A turn into 800 moles of B. As the reaction progresses, A is consumed, and B is formed until all of A is converted. Understanding this concept helps in analyzing how reactions progress over time.

Reaction Rate

The reaction rate measures how quickly a chemical reaction occurs. It is influenced by various factors, including the concentration of reactants, temperature, and presence of catalysts. In the exercise, the rate of reaction is directly related to the number of moles of substance A and B. This means that as the reaction proceeds and substance A converts to B, the rate at which A disappears changes over time. The maximum rate of conversion initially occurs because the concentrations of A and potential B are greatest then. When considering physical or chemical transformations, knowing how fast they happen is crucial in practical applications.

Proportionality

Proportionality refers to a constant relation between two quantities. In differential equations, this often translates into one quantity changing at a rate proportional to another. For this exercise, the rate at which substance A converts to B is proportional to both the moles of A and B. Mathematically, this is depicted as \(-kN(800 - N)\), where \(k\) is a proportionality constant. This setup indicates that the change in A is driven by both the remaining amount of A and the accumulating B, supporting how reactions naturally slow down as one reactant diminishes.

Decreasing Functions

A decreasing function is one where, as the input (often time) increases, the output decreases. In the context of this reaction, \(N(t)\), representing moles of substance A, is decreasing over time. This matches the reaction's nature as A gets converted to B. The mathematical expression \(-kN(800 - N)\) indicates that as time progresses, and more A turns into B, fewer moles of A remain, decreasing the reaction's driving force and thus reducing the rate of change over time. Understanding decreasing functions helps in predicting how rapidly resources or substances are consumed in various processes.