Question

Sketch a graph of pressure versus inverse volume, assuming temperature is constant. Label the vertical axis \(P\) and the horizontal axis \(1 / \mathrm{V}\).

Short answer

The graph is a downward-curving line starting from the top, showing the inverse relationship between pressure and volume.

Step-by-step solution

Understand the Relationship

This problem is based on Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume when temperature is kept constant. Mathematically, it can be expressed as \( P \propto \frac{1}{V} \) or \( PV = k \) where \( k \) is a constant.

Identify Variables for Graph Axes

According to the problem, we need to graph pressure \( P \) on the vertical axis against inverse volume \( \frac{1}{V} \) on the horizontal axis. This means our x-axis will represent \( \frac{1}{V} \) and the y-axis will represent the pressure \( P \).

Choose Coordinates

To sketch the graph, select a few values for \( V \) to calculate corresponding \( P \) and \( \frac{1}{V} \). For instance, for \( V = 1, 2, 3 \), \( \frac{1}{V} \) becomes \( 1, 0.5, 0.33 \) respectively. Assume a constant \( k \) to find \( P = k \cdot \frac{1}{V} \).

Plot the Points

Using the selected values, plot the points on the graph: If \( V = 1 \) then \( P = k \cdot 1 = k \). If \( V = 2 \), then \( P = k \cdot 0.5 \). If \( V = 3 \), then \( P = k \cdot 0.33 \). Plot these calculated \( (\frac{1}{V}, P) \) values.

Draw the Graph

After plotting the points, draw a smooth curve that passes through them, starting from the top left and asymptotically approaching the axes as it moves rightward and downward. This graph should be a decreasing curve, representing an inverse relationship.

Label the Graph

Ensure your graph is properly labeled: mark the vertical axis as \( P \) (Pressure) and the horizontal axis as \( \frac{1}{V} \) (Inverse Volume). Check that your plotted points and curve reflect the expected inverse relationship.

Key concepts

Inverse Proportionality

Inverse proportionality is a mathematical concept that describes a specific type of relationship between two variables. When two variables are inversely proportional, this means that as one variable increases, the other decreases, and vice versa. This is often represented by the equation \( y = \frac{k}{x} \), where \( y \) is inversely proportional to \( x \) and \( k \) is a constant. Such relationships create a hyperbolic curve when graphed.
In contexts such as Boyle's Law, which applies to ideal gases at a constant temperature, the pressure of the gas rises as the volume decreases. This is a real-world example of inverse proportionality. As you decrease the space available for the gas molecules (volume), they collide more frequently with the walls of their container, increasing the pressure. On a graph, if we plot pressure against inverse volume (\( 1/V \)), the curve will typically slope downward, visually representing an inverse relationship. This shows that pressure and volume move in opposite directions.

Pressure-Volume Relationship

The pressure-volume relationship is central to Boyle's Law, a fundamental principle in the study of gases. Boyle's Law states that the pressure of a gas is inversely proportional to its volume when the temperature remains constant. This means if you double the volume of the gas (while keeping the temperature constant), the pressure will halve.
Understanding this relationship requires recognizing that pressure (\( P \)) and volume (\( V \)) are tied together through a constant \( k \), as given by the equation \( PV = k \).
To conceptualize this, imagine a balloon filled with air:

• As you squeeze the balloon, reducing its volume, the pressure inside the balloon increases.
• Similarly, if you let the balloon expand (increase its volume), the pressure inside decreases.

On a graph where pressure is plotted on the y-axis and inverse volume on the x-axis, this inverse relationship becomes apparent as a smoothly descending curve.

Gas Laws

Gas laws are a set of rules defining the behavior of gases based on temperature, volume, and pressure. These include Boyle's Law, Charles's Law, and Avogadro's Law, all of which describe how gases respond to changes in conditions.
Boyle's Law is particularly important when considering circumstances where the temperature stays constant. This law underpins many real-life applications, such as the functionality of syringes or pneumatic systems, where gas compression and expansion play critical roles.
To understand and apply gas laws:

• Recognize that gases will expand to fill their containers, meaning their volume is variable.
• Pressure in a gas is caused by collisions of the gas molecules against the container walls.
• Temperature affects the energy of these molecules, causing changes in pressure and volume according to the specific gas laws.

By mastering these principles, one can predict how a gas will behave under different conditions. Graphs depicting these relationships, such as those following Boyle's Law, help visualize and internalize the changes that occur within a closed system.