Problem 108 - Chemistry The Molecular Science

Question

In this chapter Boyle's, Charles's, and Avogadro's laws were presented as word statements and mathematical relationships. Express each of these laws graphically.

Step-by-Step Solution

Answer

Boyle's Law: Inverse curve; Charles's Law: Straight line; Avogadro's Law: Straight line.

Step 1: Boyle's Law

**Boyle's Law** states that at constant temperature, the pressure of a gas is inversely proportional to its volume. Mathematically, it's expressed as: \( PV = k \), where \( P \) is the pressure, \( V \) is the volume, and \( k \) is a constant. Graphically, this is represented as a hyperbola on a graph with Pressure \( P \) on the y-axis and Volume \( V \) on the x-axis.

Step 2: Charles's Law

**Charles's Law** states that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. It is mathematically given by \( \frac{V}{T} = k \), where \( V \) is the volume, \( T \) is the temperature, and \( k \) is a constant. Graphically, this is represented as a straight line on a graph with Volume \( V \) on the y-axis and Temperature \( T \) on the x-axis, indicating a linear relationship.

Step 3: Avogadro's Law

**Avogadro's Law** states that at constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas. It's expressed as \( V = kn \), where \( V \) is the volume, \( n \) is the number of moles, and \( k \) is a constant. Graphically, it's represented as a straight line on a graph with Volume \( V \) on the y-axis and Moles \( n \) on the x-axis.

Key Concepts

Boyle's Law

Boyle's Law forms a fundamental part of understanding gas behavior under varying pressure and volume. Imagine you have a balloon. When you squeeze the balloon, you decrease its volume, and you’ll notice that the pressure inside the balloon increases. This is a perfect demonstration of Boyle's Law at work. Simply put, if you keep the temperature the same, increasing the pressure will decrease the volume of the gas, and vice versa.
This idea can be represented mathematically by the equation \( PV = k \), where \( P \) represents pressure, \( V \) is volume, and \( k \) is a constant. In terms of graphing, this equation creates a hyperbolic curve when you plot pressure against volume. As pressure goes up, the volume reduces, showing the inverse relationship between them. This law is particularly useful in understanding the behavior of gases in closed systems, such as in the workings of a piston in an engine.

Charles's Law

Charles's Law is all about the comforting and ever-predictable relationship between volume and temperature in gases. If you've ever watched a hot air balloon rise into the sky, you've seen Charles's Law in action. When the air inside the balloon is heated, it expands, making the balloon grow larger and lighter than the cooler air outside.
The mathematical expression for Charles's Law is \( \frac{V}{T} = k \), where \( V \) is the volume, \( T \) is the temperature in Kelvins, and \( k \) is a constant. This means that as the temperature of a gas increases, so does its volume, as long as the pressure remains constant. When graphing this law, you'll see a straight line, reflecting the direct proportionality between temperature and volume. Understanding Charles's Law helps explain everyday phenomena like why car tires may seem deflated in cold weather compared to their condition on a hot day.

Avogadro's Law

Avogadro's Law highlights the simple yet fascinating relationship between the quantity of gas and its volume. Picture a bicycle pump. The more strokes you add, infusing the pump with more air molecules, the greater the pressure and therefore the volume you can pump into the bicycle tire.
According to Avogadro's Law, at a constant temperature and pressure, the volume of a gas is directly proportional to the number of gas molecules (or moles). The mathematical form of the law is \( V = kn \), where \( V \) is volume, \( n \) is the number of moles, and \( k \) is a constant. On a graph, plotting volume against the number of moles will yield a straight line, demonstrating their direct relationship. Avogadro's Law is essential for understanding chemical reactions involving gases, such as the way gases are measured and used in car airbags.