The Ideal Gas Law
The ideal gas law connects pressure, volume, amount of gas and absolute temperature for a model gas with random particle motion, negligible particle volume, elastic collisions and negligible forces except during collisions.
The Ideal Gas Law: quick answer
The ideal gas law connects pressure, volume, amount of gas and absolute temperature for a model gas with random particle motion, negligible particle volume, elastic collisions and negligible forces except during collisions. The ideal gas law is an equation of state: it connects equilibrium state variables, but it does not describe every microscopic detail or every nonequilibrium process. Strong solutions first verify the ideal-gas assumptions, identify what is fixed, and use absolute temperature.
Use the ideal-gas model only when its assumptions are appropriate, and use kelvin for all gas-law temperature ratios and proportional reasoning.
What you should be able to explain and do
Use these as retrieval prompts before looking at notes.
State the assumptions of the classical ideal-gas model.
Use PV = nRT = NkBT with consistent SI quantities.
Compare two equilibrium states using proportional reasoning.
Interpret P–V, P–T and V–T graphs and connect their shape or slope to the model.
Key terms you need for this lesson
Definitions stay next to the physics they support so unfamiliar language does not break the reasoning chain.
ideal gas
ideal gas is a model gas whose particles have negligible volume, random velocities, elastic collisions and negligible interactions except during collisions.
state variable
state variable is a macroscopic property such as pressure, volume or temperature that describes an equilibrium state.
mole
mole is an amount of substance containing Avogadro’s number of particles.
absolute temperature
absolute temperature is temperature measured on the kelvin scale from absolute zero.
equilibrium state
equilibrium state is a macroscopic state whose measured properties are stable enough to be described by state variables.
isotherm
isotherm is a set of gas states at the same absolute temperature; on a P–V graph an ideal-gas isotherm curves downward.
Know what each relationship means and when it applies
The relationship, variables, condition and physical interpretation belong together.
| Relationship / model | Variables or meaning | When to use it | Physical anchor |
|---|---|---|---|
PV = nRT = Nk_B T | P pressure (Pa); V volume (m³); n amount (mol); R gas constant (J·mol⁻¹·K⁻¹); N particle count; k_B Boltzmann constant (J/K); T absolute temperature (K) | equilibrium states of an ideal gas | For fixed n and T, doubling V halves P. |
P₁V₁/T₁ = P₂V₂/T₂ | P₁,P₂ pressure (Pa); V₁,V₂ volume (m³); T₁,T₂ absolute temperature (K) | derived comparison for the same fixed ideal-gas sample in two equilibrium states | A sealed flexible gas sample can be compared before and after simultaneous compression and heating. |
At fixed amount: P ∝ T/V | P pressure (Pa); T absolute temperature (K); V volume (m³); amount of gas held fixed | proportional reasoning from the ideal gas law | At fixed V, a P-versus-T(K) graph is linear and extrapolates toward zero pressure at zero kelvin in the ideal model. |
What a strong diagram or graph must show
Translate the same physical model across words, diagrams, graphs and equations.
P–V at fixed T
inverse relationship; an isotherm curves downward.
P–T(K) at fixed V and amount
straight line through the ideal-model origin.
V–T(K) at fixed P and amount
straight line through the ideal-model origin.
Particle picture
particle number stays fixed for a sealed sample even as P, V or T changes.
Change one variable and watch the model respond
Use the tool to make a prediction first, then move the control and explain why the output changes.
Ideal-gas state calculator + isotherm
Change amount, temperature and volume. Pressure updates from PV=nRT and the graph shows the P–V isotherm for the chosen n and T.
Two examples with the model choice shown
The goal is to make the reasoning pathway visible, not just the final number.
Rigid container heating
Situation: A fixed amount of ideal gas in a rigid container is heated from 300 K to 450 K. Initial pressure is 120 kPa.
Reasoning: Rigid means V is fixed; sealed means n is fixed. From P ∝ T, P₂/P₁ = T₂/T₁.
P₂ = 120 kPa × 450/300 = 180 kPa.
Isothermal compression
Situation: A gas at constant temperature is compressed from 4.0 L to 2.5 L. Initial pressure is 90 kPa.
Reasoning: Fixed amount and constant T give P₁V₁ = P₂V₂. Volume units may remain liters because only a ratio is used.
P₂ = 90×4.0/2.5 = 144 kPa.
Recognize the physics before reaching for an equation
Use the cue to choose the model, connect it to nearby ideas, then reject any method whose assumptions do not match the situation.
What should trigger Topic 9.2?
Use the ideal-gas state model when pressure, volume, amount of gas, and absolute temperature describe equilibrium states.
How this idea fits the course
Topic 9.1 explains the particle origin of P and T; Topic 9.4 uses P–V changes to track energy transfer and work.
Scope check before solving
Use the ideal-gas model only when its assumptions are appropriate, and use kelvin for all gas-law temperature ratios and proportional reasoning.
Turn this topic into measurable evidence
A strong investigation varies the intended cause, controls competing causes, repeats measurements and uses a graph whose features have physical meaning.
Investigation target
Collect pressure and volume data for a trapped gas at nearly constant temperature, then test whether P versus 1/V is linear; separately test P versus T(K) at fixed volume.
Reasoning errors to eliminate
If one of these appears in your work, repair the model before doing more arithmetic.
Six things you should be able to do before moving on
Use these as closed-note retrieval prompts.
- I can define the core quantities and distinguish closely related terms.
- I can choose the governing model before substituting values.
- I can explain each variable, unit and sign convention.
- I can translate the situation into an appropriate graph or diagram.
- I can predict a factor/direction change before calculating.
- I can design or critique an experiment and use evidence to justify a claim.
Practice is kept on its dedicated pages
Finish the lesson first. Then open the separate topic bank when you want exam-style practice without replacing the teaching content above.
42 multiple-choice questions
Use the exact-topic MCQ bank to test model selection, representations, calculations and boundaries after learning the concept.
Open 42 Topic MCQs4 multipart free-response tasks
Use the exact-topic FRQ set for mathematical, representation, experimental and qualitative/quantitative reasoning.
Open 4 Topic FRQsTopic 9.2 FAQs
What are the ideal-gas assumptions?
Particle velocities are random, particle volume is negligible compared with container volume, collisions are elastic, and appreciable forces act only during collisions.
Why must temperature be in kelvin?
The gas law uses absolute temperature. A proportional change from 20°C to 40°C is not a doubling of absolute temperature.
When can I use P₁V₁/T₁ = P₂V₂/T₂?
For the same fixed amount of ideal gas in two equilibrium states.
What does a P–T graph show at fixed volume?
For a fixed amount of ideal gas, P is proportional to T in kelvin, so the graph is linear in the ideal model.
What is an isotherm?
A set of states at one constant temperature. On a P–V graph for a fixed amount of ideal gas, an isotherm is a downward-curving inverse relationship.
Can I use liters and kilopascals?
Ratios can cancel compatible units, but for PV=nRT calculations use a unit system consistent with the chosen value of R. SI uses Pa and m³.