Kinetic Theory: common doubts, answered
The questions students ask most often about Kinetic Theory, each with a short answer. For the full chapter, read the Kinetic Theory notes.
Why a gas can be explained by moving molecules
Read this section in the notes →Why can gases be explained with a simple molecular model when solids cannot?
Because in a gas the molecules are far apart compared with their size and interact only during brief collisions. Between collisions each moves freely in a straight line, so the motion is easy to describe. In solids and liquids the particles are packed close and interact strongly all the time, which makes the analysis much harder.
The molecular nature of matter
Read this section in the notes →How far apart are gas molecules compared with their size?
At ordinary temperature and pressure, gas molecules are roughly ten times their own diameter apart on average, not thousands of times. This spacing is still large enough that the molecules occupy only a small fraction of the gas's volume. That is why a gas can be compressed so easily compared with a liquid or solid.
The ideal-gas equation
Read this section in the notes →What is the difference between R and k_B in the ideal gas equation?
R is the gas constant per mole, about 8.314 J mol⁻¹ K⁻¹, while k_B, Boltzmann's constant, is the same constant per molecule: k_B = R/N_A. So PV = μRT counts the gas in moles, and PV = k_B N T counts it in molecules. Both describe the same physics.
Why must temperature be in kelvin in PV = μRT?
Because the ideal gas equation makes pressure and volume proportional to absolute temperature. At 0 °C a gas still has pressure, since it is at 273 K, so putting temperature in Celsius would give zero pressure or a negative value, which is nonsense. Always convert with T = t_C + 273.15 before substituting.
Real gases, Boyle's law and Charles' law
Read this section in the notes →When does a real gas behave like an ideal gas?
At low pressure and high temperature. Then the molecules are far apart, so their own volume and the forces between them become negligible, which are exactly the assumptions of the ideal-gas model. At high pressure or near liquefaction, a real gas departs from PV = μRT noticeably.
Partial pressures and the size of a molecule
Read this section in the notes →What is Dalton's law of partial pressures?
It states that the total pressure of a mixture of ideal gases equals the sum of the pressures each gas would exert alone in the same volume at the same temperature: P = P₁ + P₂ + …. Each gas behaves as if the others were absent, because ideal gas molecules do not interact apart from brief collisions.
Pressure of an ideal gas from molecular impacts
Read this section in the notes →Why does a gas exert pressure on the walls of its container?
Because gas molecules keep hitting the walls and bouncing off. Each collision reverses part of a molecule's momentum, and by Newton's laws the wall must push on the molecule, so the molecule pushes on the wall. The combined effect of enormous numbers of collisions each second is a steady force per unit area, P = ⅓ n m ⟨v²⟩.
Is mean square speed the same as the square of the mean speed?
No. The mean of the squared speeds, ⟨v²⟩, is always at least as large as the square of the average speed, ⟨v⟩², and greater whenever the molecules have a spread of speeds. Squaring before averaging gives extra weight to fast molecules. The rms speed is the square root of ⟨v²⟩.
Kinetic interpretation of temperature
Read this section in the notes →What does temperature mean in terms of molecules?
Absolute temperature measures the mean translational kinetic energy per molecule: ½ m⟨v²⟩ = (3/2) k_B T. Double the kelvin temperature and that average energy doubles. It depends only on temperature, not on which gas it is, its pressure or its volume, so a hotter gas simply has molecules jostling about faster on average.
Do heavier molecules have more kinetic energy at the same temperature?
No. At the same temperature, all gas molecules have the same average translational kinetic energy, (3/2) k_B T, whatever their mass. Heavier molecules simply move more slowly. That is why, in a mixture, lighter molecules have a higher rms speed than heavier ones at the same temperature.
How does rms speed depend on molar mass?
The rms speed is v_rms = √(3RT/M₀), so at a fixed temperature it varies as one over the square root of the molar mass. Hydrogen molecules move four times faster than oxygen molecules at the same temperature, since oxygen is sixteen times heavier. The molar mass must be in kg mol⁻¹, not g mol⁻¹.
Worked cases: speeds, isotopes and a moving bat
Read this section in the notes →Does mixing two gases change the rms speed of each?
No. In an equilibrium mixture, each gas has the rms speed set by the common temperature and its own molecular mass. The proportions of the gases in the mix make no difference to each gas's rms speed. Only the temperature and the mass of each molecule matter.
Why can uranium isotopes be separated using diffusion?
Because at the same temperature the lighter isotope's molecules move slightly faster, since rms speed varies as one over the square root of molecular mass. In a gaseous compound passed repeatedly through porous barriers, the lighter form diffuses a little more quickly. The difference per stage is tiny, so the process needs many stages.
Degrees of freedom and equipartition of energy
Read this section in the notes →What are degrees of freedom of a gas molecule?
They are the independent ways a molecule can move and store energy. A monatomic molecule has three translational degrees of freedom. A rigid diatomic molecule adds two rotational ones, about the two axes perpendicular to its bond, giving five. Rotation about the bond axis does not count, as its moment of inertia is negligible.
What does the law of equipartition of energy say?
In thermal equilibrium, each quadratic term in a molecule's energy, translational or rotational, has an average energy of ½ k_B T. A vibrational mode gets k_B T, because it contains both kinetic and potential energy terms. This lets you find the internal energy and specific heats of gases by counting energy terms.
Why does a vibrational mode get k_B T and not ½ k_B T?
Because vibration stores energy in two squared terms: the kinetic energy of the moving atoms and the potential energy of the stretched bond. Each squared term gets ½ k_B T under equipartition, so one vibrational mode gets k_B T in total. Translation and rotation have only kinetic terms, giving ½ k_B T each.
Specific heats of gases
Read this section in the notes →What is the value of γ for monatomic and diatomic gases?
For a monatomic gas, C_v = (3/2)R and C_p = (5/2)R, so γ = 5/3, about 1.67. For a rigid diatomic gas, C_v = (5/2)R and C_p = (7/2)R, so γ = 7/5 = 1.4. If a diatomic molecule also vibrates, C_v rises and γ falls further.
Which specific heat applies to a gas heated in a sealed rigid cylinder?
Use C_v, the molar specific heat at constant volume. A rigid sealed cylinder cannot change its volume, so the gas does no work and all the heat goes into raising its internal energy. C_p applies only when the gas is free to expand at constant pressure, as under a freely moving piston.
Specific heat of solids
Read this section in the notes →Why is the molar specific heat of solids about 3R?
In a solid each atom vibrates about its fixed position in three directions, and each vibrational mode carries an average energy k_B T. That gives 3k_B T per atom and 3RT per mole, so C = 3R, about 25 J mol⁻¹ K⁻¹. Because solids expand very little, the difference between C_p and C_v is negligible.
Mean free path
Read this section in the notes →What is mean free path and what does it depend on?
Mean free path is the average distance a molecule travels between successive collisions: l = 1/(√2 nπd²). It is longer when the gas is less dense, since there are fewer molecules to hit, and shorter for larger molecules, as they present a bigger target. At normal pressure it is far smaller than the size of a room.
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