Connect pressure, volume, amount and temperature
You will be able to: Use PV=nRT with consistent units and stated constraints.
What changes when you squeeze a sealed gas?
A slowly squeezed syringe has less space for the same amount of gas. If you allow its temperature to stay constant, its pressure increases. The ideal gas law connects these changes instead of treating pressure, volume and temperature as unrelated.
A useful starting point: Read a spread of particle speeds →
Words and symbols before equations
- Absolute pressure P
- Pressure measured relative to vacuum, in Pa; not gauge pressure.
- Volume V
- Gas volume in m³; 1 L=0.001 m³.
- Amount n
- Number of moles; one mole contains N_A particles.
- R and k_B
- R≈8.31 J/(mol·K); k_B≈1.38×10⁻²³ J/K. Use R with moles and k_B with particle count N.
What this picture assumes
One mole of ideal gas, R=8.31 J/(mol·K). All pressures are absolute. The curve varies V from 10 to 50 L with the selected T fixed. The point shows the selected V; these are equilibrium states, not a time evolution.
Connect the picture to the physics
The ideal gas model treats particles as occupying negligible volume, with random velocities, elastic collisions and no appreciable interactions except during collisions. It is an approximation, especially poor near condensation or at very high density.
PV=nRT=Nk_B T. The two forms express the same relationship because N=nN_A and R=N_A k_B. Choose one description of amount and keep the units consistent. Pressure times volume has units Pa·m³=J.
For a sealed fixed amount, P₁V₁/T₁=P₂V₂/T₂. Before using a ratio, say what is fixed. Halving volume doubles pressure only if temperature stays fixed; rapid compression can also raise temperature.
A worked example, step by step
An ideal gas has n=1.0 mol, T=300 K and V=0.02493 m³. Find its pressure, then halve V at the same T.
- Use absolute pressure and kelvin: P=nRT/V.
- P=(1)(8.31)(300)/0.02493=100000 Pa=100 kPa.
- With n and T unchanged, P₂/P₁=V₁/V₂=2.
- P₂=200 kPa. The isothermal condition matters.
Use absolute pressure, kelvin, and m³ with the SI value of R. Gauge pressure and Celsius cannot be inserted directly.
Can pressure rise while temperature stays constant?
Compare with an explanation
Yes. Compressing a fixed amount isothermally raises particle number density and pressure.
Predict. Change one thing. Explain.
Change volume while holding T and n fixed. Then change T at fixed volume. Explain each pressure change in both particle language and algebra.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
P=99.72 kPa absolute at V=0.025 m³ and T=300 K for 1 mol. The marked equilibrium state satisfies PV=nRT. Curve comparisons use the same n and selected T.
One mole of ideal gas, R=8.31 J/(mol·K). All pressures are absolute. The curve varies V from 10 to 50 L with the selected T fixed. The point shows the selected V; these are equilibrium states, not a time evolution.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant particle, temperature or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA sealed ideal gas initially has P=100 kPa, V=2 L and T=300 K. It ends at V=1 L and T=450 K. (a) Identify the fixed quantity. (b) Write a two-state relation. (c) Calculate final pressure. (d) Explain why doubling pressure would be insufficient.
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Compare with the answer and four-point rubric
- 1 point: Amount n is fixed.
- 1 point: P₂=P₁(V₁/V₂)(T₂/T₁).
- 1 point: P₂=100(2)(450/300)=300 kPa absolute.
- 1 point: Volume reduction doubles P, and the kelvin temperature rise adds a factor 1.5.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Ideal gas assumptions?
Negligible particle volume, random motion, elastic collisions and negligible interactions except during collisions.
RECALL 2When is PV constant?
For a fixed amount at constant temperature.
RECALL 3Moles versus particle count?
Use nR or Nk_B; do not combine N with R.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Connect pressure, volume, amount and temperature
- PV=nRT=Nk_B T.
- For fixed amount: P₁V₁/T₁=P₂V₂/T₂.
- 1 L=10⁻³ m³; 1 kPa=10³ Pa.
Remember: Use absolute pressure, kelvin, and m³ with the SI value of R. Gauge pressure and Celsius cannot be inserted directly.
Conditions: One mole of ideal gas, R=8.31 J/(mol·K). All pressures are absolute. The curve varies V from 10 to 50 L with the selected T fixed. The point shows the selected V; these are equilibrium states, not a time evolution.
Refresh Kid · AP Physics 2 Unit 1 (official Unit 9) · Objectives 9.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.2, objectives 9.2.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Refresh Kid calls this the first AP Physics 2 unit; College Board numbers it Unit 9, continuing after AP Physics 1. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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