Problem 39 - Calculus

Question

Many organ pipes in old European churches are made of tin. In cold climates such pipes can be affected with tin pest, when the tin becomes brittle and crumbles into a gray powder. This transformation can appear to take place very suddenly because the presence of the gray powder encourages the reaction to proceed. The rate of the reaction is proportional to the product of the amount of tin left and the quantity of gray powder, \(p,\) present at time \(t .\) Assume that when metallic tin is converted to gray powder, its mass does not change. (a) Write a differential equation for \(p .\) Let the total quantity of metallic tin present originally be \(B\) (b) Sketch a graph of the solution \(p=f(t)\) if there is a small quantity of powder initially. How much metallic tin has crumbled when it is crumbling fastest? (c) Suppose there is no gray powder initially. (For example, suppose the tin is completely new.) What does this model predict will happen? How do you reconcile this with the fact that many organ pipes do get tin pest?

Step-by-Step Solution

Answer

The differential equation is \( \frac{dp}{dt} = k(A-p)p \). Fastest crumbling occurs when \( p = \frac{A}{2} \). Without any gray powder initially, no tin pest develops.

Step 1: Understand the Relationship

The problem describes a scenario where the reaction, or crumbling of tin, increases with the presence of gray powder. It states that the rate of reaction is proportional to the product of the amount of tin left and the quantity of gray powder present.

Step 2: Setup the Differential Equation

Since the rate of reaction is proportional to both tin and gray powder, we denote the rate as \( \frac{dp}{dt} \) where \( p \) is the quantity of gray powder. Then, \( \frac{dp}{dt} = k(A-p)p \), where \( A \) is the initial amount of metallic tin and \( k \) is a proportional constant.

Step 3: Graph the Solution

To sketch the graph of \( p = f(t) \), note that the function will start slow if \( p \) is initially small and escalate as \( p \) increases, reaching its peak crumbling activity when the derivative \( \frac{dp}{dt} \) is maximized.

Step 4: Find Maximum Crumbling

To find when the tin crumbles fastest, find the maximum of \( \frac{dp}{dt} = k(A-p)p \). This happens when \( \frac{d}{dp} (k(A-p)p) = 0 \). Solving \( A-2p = 0 \), we get \( p = \frac{A}{2} \).

Step 5: Examine Initial Condition of No Powder

If there is no initial gray powder (\( p = 0 \) initially), the differential equation \( \frac{dp}{dt} = k(A-p)p \) indicates no change, \( \frac{dp}{dt} = 0 \). Thus, no outbreak of tin pest occurs in absence of initial powder, explaining the need for a catalyst.

Key Concepts

Reaction Rate

Understanding reaction rates is crucial, especially in the context of physical chemistry, where it helps predict how fast a chemical reaction proceeds. In this exercise involving tin and its transformation to gray powder, the rate of reaction is described as the pace at which tin turns into powder. The reaction rate increases with the presence of the product itself—the gray powder. This is because the reaction is autocatalytic, meaning that the product of the reaction actually promotes more of the reaction.
This is a typical scenario in chemistry where the presence of a reaction product increases the rate of the overall reaction. The reaction rate here is represented by a differential equation, which mathematically describes how fast this transformation occurs over time. So, in simple terms, the more powder you have, the faster it increases, which can make the transformation appear sudden.

Proportional Relationships

Proportional relationships are a fundamental concept in understanding the dynamics of differential equations. In this problem, the rate at which tin turns into gray powder is proportional to the product of the remaining amount of tin and the current amount of gray powder.
Expressing this in the form of a differential equation looks like: \[ \frac{dp}{dt} = k(A-p)p \]Here:

• \(k\) is a constant of proportionality that measures how forcefully the reaction progresses.
• \(A\) is the initial quantity of tin, and \(p\) is the amount of gray powder at time \(t\).

This equation states that the change in quantity of powder over time is influenced by both how much tin is left as well as how much has already turned into powder. Without these proportional relationships, predicting the speed and extent of the reaction wouldn't be possible.

Graphing Solutions

Graphing the solution of a differential equation helps visualize how a reaction progresses over time. In this exercise, the goal is to graph \( p = f(t) \), which represents the amount of gray powder over time. Starting with a small amount of powder means the reaction is initially slow. As time progresses, and as more gray powder forms, the rate increases sharply.
When graphing this scenario, your plot begins near the origin, slowly at first. It ramps up steeply as the amount of powder increases, creating an upswung curve that peaks where the reaction is fastest. Finding this peak mathematically is done by finding where the derivative of the rate equation is maximized, \( \frac{dp}{dt} = k(A-p)p \), occurring at \( p = \frac{A}{2} \). This point signifies the greatest rate of tin transformation into powder.

Physical Chemistry

Physical chemistry delves into how chemistry fundamentally works, often through studying reactions like metals turning brittle. In this exercise, we explore a real-world problem involving tin pest, where physical conditions (temperature and initial presence of powder) define the reaction's behavior.
From a physical chemistry perspective, the absence of gray powder, to begin with, means that no reaction occurs since there's no catalysis to trigger the transformation. The differential equation confirms this: starting from zero means \( \frac{dp}{dt} = 0 \) without that initial powder. This underlines the importance of catalysts in physical chemistry, which facilitate and expedite otherwise dormant reactions.
Even though new tin might seem safe, in reality, external environmental factors can introduce the catalyst (gray powder), thereby igniting the reaction over time.