Recall and use the equation p V = constant for a fixed mass of gas at constant temperature, including a graphical representation of this relationship

2.1.2 Particle Model – Pressure & Volume

What You’ll Learn

Recall and use the equation \$pV = \text{constant}\$ for a fixed mass of gas at constant temperature, and draw a simple graph to show this relationship.

The Equation in Action

For a given amount of gas kept at a constant temperature, the product of its pressure (\$p\$) and volume (\$V\$) never changes:

\$pV = \text{constant}\$

Think of it like a rubber balloon: if you squeeze it (reduce \$V\$), the air inside gets more crowded, so the pressure rises. If you let it expand, the pressure drops.

Graphical Representation

Below is a simple table that you can plot on graph paper. The x‑axis is volume (\$V\$) and the y‑axis is pressure (\$p\$). The points will fall on a hyperbola.

\$V\$ (m³)\$p\$ (kPa)
0.5200
1.0100
1.566.7
2.050

When you plot these points, you’ll see a curve that goes steeply downwards – that’s the hyperbola showing how \$p\$ and \$V\$ are inversely related.

Analogy: The Balloon & the Squeeze

Imagine a water balloon (the gas). If you squeeze it, the volume decreases but the pressure inside rises – just like \$pV\$ staying the same. If you let the balloon expand, the pressure falls. This everyday experience helps you remember the inverse relationship.

Example Problem

Suppose a gas occupies \$V1 = 1.0\,\text{m}^3\$ at a pressure of \$p1 = 100\,\text{kPa}\$. If the volume is suddenly doubled to \$V2 = 2.0\,\text{m}^3\$, what is the new pressure \$p2\$?

  1. Calculate the constant: \$p1V1 = 100 \times 1.0 = 100\,\text{kPa}\cdot\text{m}^3\$

  2. Use the constant for the new state: \$p2V2 = 100\$

  3. Solve for \$p_2 = \frac{100}{2.0} = 50\,\text{kPa}\$

??

Result: The pressure halves when the volume doubles.

Exam Tips

  • 🔍 Remember the form: \$pV = \text{constant}\$ – always keep the product constant for a fixed mass and temperature.

  • 📐 Graph questions: Sketch a hyperbola. Label the axes and show that as \$V\$ increases, \$p\$ decreases.

  • 🧪 Units matter: Pressure in kPa or Pa, volume in m³. Check units when calculating constants.

  • 💡 Use analogies: The balloon example can help you explain the inverse relationship in a written answer.

  • 📝 Show work: Write the constant, set up the equation, and solve step by step. This earns full marks.

Quick Check

Fill in the blanks: If the pressure of a gas is \$p1\$ and the volume is \$V1\$, then the pressure when the volume is doubled is .

Answer: \$p_1/2\$ (because \$pV\$ stays constant).