Short on time? This page is the recap. For explanations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- The ideal gas model assumes: atomic velocities are random; the atoms’ own volume is negligible compared with the gas volume; collisions are elastic; there are no forces on the atoms except during collisions.
- Real gases fit the model best at low pressure and well above their condensation temperature.
- Graphs of P, V and T test the model and give properties of the gas from gradients and intercepts.
- Absolute zero is the temperature at which an ideal gas would have zero pressure. It is found by extrapolating a P–T line, not by measurement.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| PV = nRT | P in Pa, V in m³, n in mol, R = 8.31 J/(mol·K), T in K | find any one variable from the other three |
| PV = Nk_BT | N = number of atoms, k_B = 1.38 × 10⁻²³ J/K | work with numbers of atoms |
| N = nN_A, R = N_A k_B | N_A = 6.02 × 10²³ mol⁻¹ | convert between moles and atoms |
| P₁V₁/T₁ = P₂V₂/T₂ | fixed n; any consistent P and V units; T in K | predict a change without finding n |
| Graph (fixed n) | Shape | Gradient or intercept tells you |
|---|---|---|
| P against V, constant T | curve (isotherm), P ∝ 1/V | further from origin = larger nRT |
| P against 1/V, constant T | straight line through origin | gradient = nRT |
| V against T (K), constant P | straight line through origin | gradient = nR/P |
| P against T (K), constant V | straight line through origin | gradient = nR/V |
| P or V against T (°C) | straight line, not through origin | T-intercept ≈ −273 °C |
Assumptions behind the numbers
- The gas is ideal and the amount of gas is fixed unless the question says otherwise.
- “Rigid” or “sealed rigid” means constant V. “Free piston” usually means constant P.
- Convert °C to K by adding 273; 1 L = 10⁻³ m³; 1 atm ≈ 1.0 × 10⁵ Pa.
Mistakes to avoid
- Celsius ratios. Heating from 20 °C to 40 °C does not double P; the factor is 313/293.
- Wrong units in PV = nRT. Use Pa and m³ there; L and kPa only work in the ratio form.
- n with k_B, or N with R. Pair n with R and N with k_B.
- Forgetting a second change. If T and V both change, apply both factors.
- Drawing a line through the first and last points only. Use a best-fit line before extrapolating.
- Saying ideal gas atoms never collide. They collide elastically; there are no forces between collisions.
Quick self-check
- How many moles of ideal gas fill 2.0 L at 1.0 × 10⁵ Pa and 300 K? (0.080 mol)
- A fixed amount of gas has its volume halved and its absolute temperature doubled. By what factor does its pressure change? (× 4)
- A P–T(°C) line for a gas meets P = 0 at −275 °C. What is this temperature called, and what is the accepted value? (Absolute zero; −273 °C, or 0 K)
Next: practice questions.