Topic 3.4 Notes – Ideal Gas Law

1. The Ideal Gas Law

The Ideal Gas Law connects the four measurable properties of a gas:

$$PV = nRT$$

Where
- P = pressure
- V = volume
- n = moles of gas
- R = universal gas constant
- T = temperature in Kelvin

This one equation contains all the simpler gas laws you’ve learned.

What the Variables Mean Physically

These aren’t just letters. They describe particle behavior.

• Pressure (P) → collisions of gas particles with container walls.
  More frequent or stronger collisions = higher pressure.
• Volume (V) → space available for particle motion.
• Temperature (T) → average kinetic energy. Higher T means faster particles.
• Moles (n) → number of particles present.

The equation works because pressure comes from collisions, and collisions depend on how many particles there are, how fast they move, and how much space they have.

Units on the AP Exam

Using $R = 0.08206 \text{ L}\\cdot\\text{atm/mol}\\cdot\\text{K}$:

• Pressure → atm
• Volume → L
• Temperature → K
• Moles → mol

Always convert °C to K using
$K = °C + 273.15$

If your units don’t match R, the math will betray you.

2. How the Variables Are Related

The ideal gas law simplifies when some variables stay constant.

Boyle’s Law (P and V)

If n and T are constant:

$$P_{1}V_{1} = P_{2}V_{2}$$

• Inverse relationship
• Decrease V → particles hit walls more often → P increases.

Charles’s Law (V and T)

If n and P are constant:

$$\frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}}$$

• Direct relationship
• Increase T → particles move faster → container expands to keep pressure constant.

Gay-Lussac’s Law (P and T)

If n and V are constant:

$$\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}$$

• Direct relationship
• Rigid container + heating → pressure increases.

Avogadro’s Law (V and n)

If P and T are constant:

$$\frac{V_{1}}{n_{1}} = \frac{V_{2}}{n_{2}}$$

• Direct relationship
• More particles → volume must increase to keep pressure constant.

All of these fall naturally out of $PV = nRT$. You don’t need to memorize them separately if you understand the full equation.

3. Using the Ideal Gas Law in Problems

Typical uses on tests:

• Solve for moles to connect to stoichiometry.
• Find molar mass using $n = \frac{m}{M}$.
• Determine gas density.

For density, rearrange:

$$PV = nRT$$

Substitute $n = \frac{m}{M}$:

$$PV = \frac{m}{M}RT$$

Rearrange to get:

$$\text{Density} = \frac{PM}{RT}$$

This shows density increases with pressure and molar mass, and decreases with temperature. That relationship shows up in conceptual questions.

A common AP move is giving you a reaction, making you find moles of gas from stoichiometry, and then plugging into PV = nRT.

4. Gas Mixtures and Partial Pressures

In a mixture of ideal gases, each gas behaves as if the others aren’t there.

Dalton’s Law

$$P_{\text{total}} = P_A + P_B + P_C + \dots$$

Each gas contributes its own partial pressure.

Mole Fraction

$$X_A = \frac{\text{moles A}}{\text{total moles}}$$

$$P_A = X_A \cdot P_{\text{total}}$$

Why does this work?
At constant T and V, pressure is proportional to moles. More moles of a gas → larger share of the total pressure.

If one gas makes up 70% of the moles, it exerts 70% of the pressure.

This is often tested when gas is collected over water or when you must subtract water vapor pressure.

5. Graphical Relationships Between Variables

Seeing the shape helps you recognize the relationship instantly. The four graphs below summarize the most common gas law relationships you are expected to recognize.

Common gas law graphs at constant variables

Key patterns:

• Inverse curve → P vs V (top left)
• Straight line through origin → direct proportionality (top right and bottom left)
• Temperature must be in Kelvin for linear graphs.

Slope meaning:

• V vs T slope = $\frac{nR}{P}$
• P vs T slope = $\frac{nR}{V}$

If you know what’s constant, you can predict the graph before seeing it.