Problem 62 - Thomas Calculus

Question

Boyle's Law Boyle's Law says that the volume \(V\) of a gas at constant temperature increases whenever the pressure \(P\) decreases, so that \(V\) and \(P\) are inversely proportional. If \(P=14.7 \mathrm{b} / \mathrm{in}^{2}\) when \(V=1000 \mathrm{in}^{3},\) then what is \(V\) when \(P=23.4 \mathrm{lb} / \mathrm{in}^{2} ?\)

Step-by-Step Solution

Answer

The new volume \( V_2 \) is 628.21 in\(^3\).

Step 1: Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is always a constant. This can be expressed mathematically as \( P_1 V_1 = P_2 V_2 \), where \( P_1 \) and \( V_1 \) are initial conditions and \( P_2 \) and \( V_2 \) are final conditions.

Step 2: Identify Given Values

From the problem, we know the initial pressure \( P_1 = 14.7 \) lb/in\(^2\) and initial volume \( V_1 = 1000 \) in\(^3\). We are asked to find the new volume \( V_2 \) when the pressure \( P_2 = 23.4 \) lb/in\(^2\).

Step 3: Apply Boyle's Law Formula

Using Boyle's Law, \( P_1 V_1 = P_2 V_2 \), substitute the given values into the equation: \( 14.7 \times 1000 = 23.4 \times V_2 \).

Step 4: Solve for the New Volume \( V_2 \)

First, compute the left side of the equation, which is \( 14,700 \). Set up the equation as \( 14,700 = 23.4 \times V_2 \) and solve for \( V_2 \) by dividing both sides by 23.4: \( V_2 = \frac{14,700}{23.4} \).

Step 5: Calculate the Result

Perform the division to find \( V_2 \). The calculation is \( V_2 = 628.21 \) in\(^3\) (rounded to two decimal places).

Key Concepts

Inverse Proportionality

In everyday life, you might notice situations where two things are related such that as one quantity increases, the other decreases. This relationship is known as inverse proportionality. In mathematical terms, if two variables, say \( x \) and \( y \), are inversely proportional, we write this relationship as \( x \cdot y = k \), where \( k \) is a constant.
This means when \( x \) goes up, \( y \) must go down to keep the product \( x \cdot y \) the same, and vice versa.

An inverse proportionality is a fundamental concept in many areas, including physics, economics, and even biology. Understanding how changing one factor affects another inversely can help in figuring out how systems work.
In the context of gas laws and specifically Boyle's Law, understanding inverse proportionality helps us predict how pressure and volume of a gas relate to each other under constant temperature.

Gas Laws

Gas laws are a set of principles that describe the behavior of gases in response to various changes in conditions like temperature, pressure, and volume. These laws form the cornerstone of thermodynamics and physical chemistry, aiding in understanding how gases respond to changes in their environments.

Some of the key gas laws include:

• **Boyle's Law**: It shows the relationship between pressure and volume at constant temperature.
• **Charles's Law**: It explores the volume and temperature relationship at constant pressure.
• **Avogadro's Law**: It relates to the volume of gas and the number of particles at constant pressure and temperature.

Understanding these fundamental gas laws is crucial for a variety of applications, from filling up balloons to complicated industrial processes. They assist in understanding not just how gases behave but also in predicting outcomes of chemical reactions involving gases.

Pressure-Volume Relationship

The pressure-volume relationship of gases is vital in understanding how gases behave under different conditions. According to Boyle's Law, there is an inverse relationship between the pressure of a gas and its volume when the temperature is held constant. This means, as you increase the pressure on a gas, the volume will decrease proportionally, so long as temperature does not change.

Here’s how it works:

• When pressure increases, gas particles are forced closer together, reducing the space they occupy—hence decreasing volume.
• Conversely, when pressure decreases, particles have more room to spread out, leading to an increase in volume.

It's important to note that this relationship assumes ideal conditions (without significant interactions between gas particles), which might not always apply in real-world scenarios.
Despite its simplification, this pressure-volume relationship underpins many practical applications, including breathing, the operation of syringes, and even how car engines function.