Problem 53 - Basic Engineering Mathematics

Question

Boyle's law states that at constant temperature, the volume \(V\) of a fixed mass of gas is inversely proportional to its absolute pressure \(p .\) If a gas occupies a volume of \(0.08 \mathrm{~m}^{3}\) at a pressure of \(1.5 \times 10^{6}\) pascals determine (a) the coefficient of proportionality and (b) the volume if the pressure is changed to \(4 \times 10^{6}\) pascals..

Step-by-Step Solution

Answer

Coefficient of proportionality is \(1.2 \times 10^5\) m³·Pa and new volume is \(0.03\) m³.

Step 1: Understand Boyle's Law

Boyle's Law states that at constant temperature, the volume of a gas, \( V \), is inversely proportional to its pressure, \( p \). This can be expressed mathematically as \( V = \frac{K}{p} \), where \( K \) is the proportionality constant or coefficient.

Step 2: Calculate the Coefficient of Proportionality

Given \( V = 0.08 \text{ m}^3 \) and \( p = 1.5 \times 10^6 \text{ Pa} \), substitute these values into the equation \( V = \frac{K}{p} \) to find \( K \): \[ 0.08 = \frac{K}{1.5 \times 10^6} \] Solving for \( K \), we multiply both sides by \( 1.5 \times 10^6 \): \[ K = 0.08 \times 1.5 \times 10^6 = 1.2 \times 10^5 \text{ m}^3 \times \text{Pa} \] Thus, the coefficient of proportionality is \( 1.2 \times 10^5 \text{ m}^3 \times \text{Pa} \).

Step 3: Calculate the New Volume

Using the previously calculated \( K = 1.2 \times 10^5 \text{ m}^3 \times \text{Pa} \), and the new pressure \( p = 4 \times 10^6 \text{ Pa} \), find the new volume \( V \): \[ V = \frac{K}{p} = \frac{1.2 \times 10^5}{4 \times 10^6} \] \[ V = 0.03 \text{ m}^3 \] Therefore, the new volume of the gas at a pressure of \( 4 \times 10^6 \text{ Pa} \) is \( 0.03 \text{ m}^3 \).

Key Concepts

pressure-volume relationship

Boyle's Law provides a clear picture of the pressure-volume relationship for gases. According to this law, the volume occupied by a gas is inversely proportional to its pressure when temperature remains constant. This means that as the pressure on a gas increases, its volume decreases, and vice versa.

Mathematically, this can be represented by the equation: \[ V \propto \frac{1}{p} \] where \( V \) represents the volume and \( p \) is the pressure. The implication here is that if the pressure is doubled, the volume will be halved, assuming all other conditions remain constant.

Understanding this relationship is crucial in various applications such as in syringes, pneumatic controls, and even human lungs, where volume changes affect pressure and vice versa.

proportionality constant

In the context of Boyle's Law, the proportionality constant, often represented as \( K \), is a crucial element. It represents the product of pressure and volume for a given amount of gas at constant temperature.

The role of \( K \) is significant because it remains constant for a fixed amount of gas at a constant temperature, regardless of changes in pressure or volume. This can be mathematically expressed as: \[ V = \frac{K}{p} \] where \( K = V \times p \). This equation shows that the product of volume and pressure in any situation is always equivalent to \( K \).

To find \( K \), one simply multiplies the initial values of pressure and volume, as seen in the solution process: \[ K = 0.08 \, \text{m}^3 \times 1.5 \times 10^6 \, \text{Pa} = 1.2 \times 10^5 \, \text{m}^3 \times \text{Pa} \].

inverse proportionality

Inverse proportionality is a fundamental concept in understanding Boyle's Law. This particular relationship means that a variable's increase will result in a decrease of another variable and vice versa.

With gases, this translates to an increase in gas pressure resulting in a decrease in gas volume, as long as temperature remains constant. The practical application includes the compression of gases, where increased pressure results in decreased volume—helpful in industrial and scientific scenarios.

When plotted on a graph, this relationship yields a hyperbolic curve, where pressure multiplied by volume remains constant. For example, in the equation: \[ V = \frac{K}{p} \], it becomes evident that as \( p \) increases, \( V \) must decrease to maintain a constant \( K \). This is quintessential in the mathematical model of inverse relationships.

gas laws

The gas laws are a collection of a few simple principles and equations that describe how gases behave. Boyle's Law is one of these foundational gas laws, focusing specifically on the interplay between pressure and volume at a constant temperature.

Other gas laws include Charles's Law, which examines the direct relationship between volume and temperature, and Gay-Lussac's Law, which focuses on the direct relationship between pressure and temperature.

Together, these laws form the Ideal Gas Law, encapsulated by the equation: \[ PV = nRT \] where \( P \) stands for pressure, \( V \) for volume, \( n \) for amount of substance (in moles), \( R \) for the ideal gas constant, and \( T \) for temperature. Boyle's Law accounts for the combined behaviors of the other gas laws under different scenarios, providing essential insights into thermodynamics and fluid mechanics.