Kinetic Theory · Lesson 6 of 6
Chapter Summary and Practice
“Connect the molecular model, gas laws, pressure, thermal energy, heat capacity and collision length scales in one complete framework.”
• Connect molecular counts and measurable gas variables through the ideal-gas and pressure equations. • Select the correct energy and heat-capacity model for a particle’s active modes. • Distinguish velocity averages and the molecular size, spacing and free-path scales. • Solve mixed gas-state and collision problems with correct units, conditions and approximations. • Check answers against dimensions, limiting trends and the assumptions behind each formula.
A chain from molecules to measurable behaviour
Begin with a sample containing N particles in volume V. Their count per volume is n. Momentum transferred during collisions gives pressure, and average translational energy sets absolute temperature. A fixed amount therefore satisfies the ideal-gas law. Active rotations and vibrations add internal energy without changing the universal translational relation, explaining why heat capacities differ between gases. Finite molecular size and relative motion determine collisions and mean free path, explaining why high thermal speed does not imply rapid directed travel across a room.
Every link has conditions. Real gases become closer to ideal when interactions and occupied-volume effects are sufficiently small. Equilibrium isotropy is needed to replace one directional mean square by a third of the total. Classical equipartition applies to thermally active quadratic terms. The simple mean-free-path formula uses a dilute gas and effective identical hard spheres. Keeping these conditions visible is as important as remembering the algebra.
Formula reference with symbols, units and conditions
| Quantity | Typical value or unit | Use |
|---|---|---|
| R | 8.314 J mol⁻¹ K⁻¹ | Molar gas equation and molar energies |
| kB | 1.38×10⁻²³ J K⁻¹ | Per-particle gas equation and energies |
| NA | 6.022×10²³ mol⁻¹ | Moles to particle count |
| 1 Å | 10⁻¹⁰ m | Molecular dimensions |
| 1 L;1 cm³ | 10⁻³ m³;10⁻⁶ m³ | Container and bubble volumes |
| Textbook STP | 273.15 K and 1 atm | Ideal molar volume≈22.4 L mol⁻¹ |
Compare the quantities that look alike
| Do not confuse | Distinction |
|---|---|
| ν and n | ν is mole number; n is molecular number density. |
| M and m₀ | M is molar mass; m₀ =M/NA is one particle’s mass. |
| Mean velocity and mean speed | Opposite velocity vectors cancel, but speeds remain nonnegative. |
| 〈v²〉 and 〈v〉² | Average the squares before taking the rms root; they generally differ. |
| Etr and total U | Rotation and vibration may add to U beyond translation. |
| One vibration and one quadratic term | A vibration has kinetic and potential terms, giving kBT when fully active. |
| Molecular diameter, centre spacing and ℓ | Size, density-based separation and collision-free path length answer different questions. |
| τ and collision duration | τ is the time between encounters; the close encounter is much briefer. |
| Gauge and absolute pressure | The gas equation uses pressure relative to vacuum, so add atmospheric pressure to a gauge reading. |
The common molar heat-capacity limits follow directly from the count: monatomic Cᵥ =3 R/2 and γ =5/3; rigid diatomic Cᵥ =5 R/2 and γ =7/5; nonlinear rigid polyatomic Cᵥ =3 R and γ =4/3. Adding active vibrational terms raises heat capacity and lowers γ. These are models with stated active modes, not permanent labels fixed solely by the number of atoms in a molecule. In a classical solid each atom has three vibrations, giving the approximate molar capacity 3 R when those modes are active.
Physical checks before accepting a numerical result
In PV =NkBT, pressure times volume has energy units, matching kBT. In P =ρvᵣₘₛ²/3, kg m⁻³ times m² s⁻² gives pascals. In vᵣₘₛ =√(3 RT/M), kg mol⁻¹ is required for M. In ℓ =1/(√2πnd²), the denominator has inverse-length units. A molecular-volume calculation cubes a length, so a radius-versus-diameter mistake produces a factor of eight, while the same mistake in a collision cross-section produces a factor of four.
At fixed temperature, heavier particles have smaller rms speed but equal average translational energy. At fixed n and molecular mass, larger mean-square speed means larger pressure. At fixed d, increasing n shortens mean free path. Equal P, V and T mean equal molecular counts for ideal gases, not equal gas masses. For a fixed sample PV/T tends toνR in the low-pressure limit; dividing additionally byν gives R. These simple limiting comparisons often reveal mistakes before a long calculation is finished.
A gas cylinder that loses molecules cannot be treated as one fixed amount across the two measurements. An expanding bubble requires the pressure at each depth, including atmospheric pressure. A nonlinear heat-capacity count cannot be assigned automatically to every linear polyatomic molecule. A water-molecule size inferred from liquid volume is only a geometric estimate, whereas a collision question may prescribe a particular effective diameter. Select the model from the stated conditions rather than making the symbols fit a memorised answer.
Worked synthesis: one gas, several measurable properties
Problem
A 0.028 kg nitrogen sample occupies 24.6 L at 300 K. Treat it as an ideal rigid diatomic gas with effective molecular diameter 2.0×10⁻¹⁰ m. Use R =8.314 J mol⁻¹ K⁻¹, NA =6.022×10²³ mol⁻¹ and kB =1.38×10⁻²³ J K⁻¹. Find amount, molecular count, pressure, number density, rms speed, translational and total internal energies, and mean free path.
- 1.The molar mass is 0.028 kg mol⁻¹, soν =0.028/0.028 =1.00 mol and N≈6.022×10²³ particles.
- 2.Convert V =24.6×10⁻³ m³. P =νRT/V = (8.314)(300)/0.0246 ≈1.014×10⁵ Pa. Number density n =N/V≈2.45×10²⁵ m⁻³.
- 3.vᵣₘₛ =√[(3)(8.314)(300)/0.028]≈517 m s⁻¹.
- 4.Total translational energy Etr =3νRT/2 =3741.3 J. The rigid diatomic model includes two rotations, giving total internal energy U =5νRT/2 =6235.5 J relative to the chosen model energy zero.
- 5.The ratio Etr/U is 3/5. Equal temperature determines the translational part, while the active rotations explain the larger U.
- 6.ℓ =1/[√2π(2.45×10²⁵)(2.0×10⁻¹⁰)²]≈2.30×10⁻⁷ m. It is much larger than the diameter but tiny compared with the vessel dimensions.
- 7.Rounded constants make R and NAkB differ slightly in their last digits; numerical agreement is expected to the accuracy of the supplied values, not arbitrary precision.
Worked synthesis: energy storage and pressure response
Problem
One mole of rigid diatomic ideal gas warms from 300 K to 330 K. Compare fixed-volume heating with fixed-pressure heating. Use R =8.314 J mol⁻¹ K⁻¹. In each case find ΔU, Q and W, and the pressure or volume ratio.
- 1.For either route, the active-mode model gives Cᵥ =5 R/2 and ΔU =νCᵥΔT = (2.5)(8.314)(30) =623.55 J.
- 2.At fixed volume W =0, so Q =623.55 J. Since P/T is constant, P₂/P₁ =330/300 =1.10.
- 3.At fixed pressure, Cₚ =7 R/2, so Q = (3.5)(8.314)(30) =872.97 J. Expansion work W =νRΔT =249.42 J.
- 4.Q−W =872.97−249.42 =623.55 J, checking the same internal-energy change. Charles’ law gives V₂/V₁ =330/300 =1.10.
- 5.The extra constant-pressure heat supplies expansion work. No change in the gas’s degree-of-freedom count is needed to explain the difference between the two heat inputs.
Choose a method for each kind of problem
Molecular-volume questions combine particle counts with geometry and density. State-equation questions require the correct sample amount, pressure reference and temperature scale. Speed questions use energy per particle and an inverse-square-root mass dependence, including isotope-bearing molecules. Heat-capacity questions count thermally active modes and identify what is held fixed. Mean-free-path questions use number density, collision diameter and relative motion. Across all these cases the common task is to connect a measurable bulk property to a specific molecular model, then check its limits.
Pressure is present inside a gas as well as at the walls. Thermal equilibrium is dynamic, and gravity produces only a small density variation across an ordinary room because m₀gh is small compared with thermal energy. Averages of squares differ from squares of averages, and a fully active vibration counts twice in the quadratic-term tally. These conceptual points prevent many apparently plausible but incorrect shortcuts.
Quick Quiz
Quiz
Two different ideal gases have equal P, V and T. What must be equal?
PV =NkBT fixes N. Mass and active internal modes can differ.
An ideal-gas sample has zero mean velocity in its bulk rest frame. Which can still be nonzero?
Random opposite directions cancel velocity vectors without removing the positive squared speeds or kinetic energies.
Which pair is correct for a rigid diatomic gas with active translation and rotation?
There are five active quadratic terms, so Cᵥ =5 R/2, Cₚ =7 R/2 and γ =7/5.
At unchanged number density, doubling molecular diameter changes corrected ℓ by a factor
ℓ is inversely proportional to d², so it becomes one quarter.
For 1 mol rigid diatomic ideal gas warmed by 30 K at fixed pressure, how much heat exceeds the fixed-volume input?
Cₚ−Cᵥ =R per mole, so the excess heat isνRΔT =30 R, which supplies expansion work.
Practice Problems
- For 0.010 kg helium in 20.0 L at 300 K, calculate mole number, particle count, pressure, number density and rms speed. Use M =0.0040 kg mol⁻¹, R =8.314 J mol⁻¹ K⁻¹ and NA =6.022×10²³ mol⁻¹.
- A mixture contains equal masses of neon and oxygen at one temperature. Find their particle-count ratio and rms-speed ratio using molar masses 20.2 and 32.0 g mol⁻¹. Compare average translational energies per particle and explain which ratios depend on mass composition.
- An oxygen sample in a rigid vessel has gauge pressure 5 atm at 300 K. After withdrawal of gas it has gauge pressure 3 atm at 290 K. For a 10.0 L vessel, find the removed mass using Patm =1 atm =1.013×10⁵ Pa, M =0.032 kg mol⁻¹ and R =8.314 J mol⁻¹ K⁻¹.
- A gas has number density 2.0×10²⁵ m⁻³ and molecular diameter 3.0 Å. Estimate mean free path. If the mean speed is 400 m s⁻¹, find mean flight time and collision frequency. Explain why the collision cross-section is not π(d/2)².
- Calculate molar Cᵥ, Cₚ and γ for(a) a monatomic gas,(b) a rigid diatomic gas and(c) a nonlinear polyatomic model with one fully active vibration. State the assumptions that make these classical counts valid.
- One mole of rigid diatomic gas warms by 50 K at fixed volume and another identical sample warms by 50 K at fixed pressure. Using R =8.314 J mol⁻¹ K⁻¹, find ΔU, Q and W for both. Relate the difference in heat to molecular energy storage and expansion work.
- A gas has number density 8.0×10²⁵ m⁻³. Estimate its representative spacing n^(−1/3). Compare that length with a molecular diameter 2.0×10⁻¹⁰ m and the mean free path predicted by the corrected hard-sphere expression.
- Two representative particles have velocity vectors of equal magnitude but opposite direction. Explain what happens to mean velocity, mean speed, mean-square speed and rms speed. Then explain why unequal speed magnitudes generally make 〈v²〉 different from 〈v〉².
- A massive piston recedes from an approaching molecule while the gas expands without heat input. Explain the energy transfer using the piston rest frame, and connect the resulting temperature trend to the first law and the kinetic temperature relation.
- Choose one application of each type: a molecular occupied-volume estimate, an ideal-gas plot, an rms-speed comparison, a heat-capacity measurement and a mean-free-path estimate. For each state the measured quantity, inferred molecular property and most important model limitation.
Key Takeaways
• The ideal-gas and pressure equations connect particle counts, molecular motion and macroscopic measurements. • Temperature sets translational energy per particle; active internal modes determine total energy storage and heat capacity. • Use the right average, pressure reference, mass unit and absolute temperature before substituting numbers. • Molecular size, spacing, mean free path, flight time and encounter duration have distinct meanings. • Physical assumptions, dimensional checks and limiting trends are essential parts of a correct solution.
Previous · Lesson 5
Mean Free Path and Molecular Transport
Next
End of chapter