Kinetic Theory

The One Thing

Kinetic theory explains the macroscopic behaviour of gases through the random motion, collisions and energy of their molecules. Gas pressure arises from molecular collisions with container walls, while temperature measures the average translational kinetic energy of the molecules.

Reactions, Processes and Experiments

Key Terms

• Kinetic theory: A model that explains the macroscopic properties of gases using the motion and collisions of their microscopic molecules.
• Ideal gas: A gas whose molecules are treated as point particles with negligible volume and no intermolecular forces, except during collisions.
• Random motion: Continuous molecular motion in all directions with no preferred direction.
• Elastic collision: A collision in which total kinetic energy and total momentum are conserved.
• Pressure: The force exerted per unit area by gas molecules colliding with the walls of a container.
• Root-mean-square speed: The square root of the mean of the squares of the speeds of all molecules; it represents a useful average molecular speed.
• Mean free path: The average distance travelled by a molecule between two successive collisions.
• Degrees of freedom: The independent ways in which a molecule can store energy, including translational, rotational and vibrational motion.
• Equipartition of energy: At thermal equilibrium, energy is equally distributed among all active quadratic degrees of freedom, with an average energy of one-half $k_{B}T$ per degree of freedom per molecule.
• Boltzmann constant: The constant relating temperature to molecular energy, kB = 1.38 × 10^-23 J K^-1.
• Absolute temperature: Temperature measured in kelvin, required for kinetic-theory and ideal-gas relationships.
• Internal energy: The total kinetic energy associated with the active degrees of freedom of the molecules; intermolecular potential energy is neglected for an ideal gas.
• Molar specific heat at constant volume, $C_v$: The molar heat capacity at constant volume, given by Cv = (f/2)R.
• Molar specific heat at constant pressure, $C_p$: The molar heat capacity at constant pressure, related to $C_v$ by Cp - Cv = R.
• Gas constant, $R$: The constant appearing in the molar ideal-gas equation and molecular-energy relationships.
• Avogadro constant, $N_A$: The constant linking the number of molecules $N$ to the amount of gas $n$ through N = nNA.

Easily Confused

• Temperature and molecular speed: Temperature measures average translational kinetic energy, whereas root-mean-square speed also depends on molecular mass.
• Average translational kinetic energy and total internal energy: Average translational kinetic energy concerns translational motion only; total internal energy includes all active translational, rotational and vibrational degrees of freedom.
• Monatomic and diatomic gases: A monatomic gas generally has 3 translational degrees of freedom, whereas a diatomic gas at ordinary temperatures usually has 5 active degrees of freedom.
• $C_v$ and $C_p$: $C_v$ is the molar specific heat at constant volume, while $C_p$ is the molar specific heat at constant pressure; for an ideal gas, Cp - Cv = R.
• Boyle’s law and Charles’ law: Boyle’s law applies at constant temperature, pV = constant; Charles’ law applies at constant pressure, V/T = constant.
• Celsius and kelvin temperature: Kinetic-theory equations require kelvin temperature, using T(K) = temperature in degrees Celsius + 273.15.
• Random motion and directed motion: Random molecular motion has no preferred direction, producing equal average contributions along three mutually perpendicular directions.
• Elastic collisions and energy loss: Elastic collisions conserve total kinetic energy and total momentum; the kinetic-theory model does not treat wall collisions as permanently dissipative.

What Gets Asked

• Derive or apply the kinetic-theory pressure equation: questions use pV = (1/3)Nm c_rms^2. The main mark-bearing point is the factor $1/3$, which follows from equal average contributions in three perpendicular directions.
• Calculate molecular speed: questions use c_rms = √(3kBT/m) = √(3RT/M). A common error is using Celsius rather than kelvin or confusing molecular mass $m$ with molar mass $M$.
• Relate temperature to molecular energy: questions use (3/2)kBT for one molecule or (3/2)RT for one mole. The distinction between one molecule and one mole must be retained.
• Apply the ideal-gas equation or gas laws: questions may use pV = NkBT = nRT, pV = constant, or V/T = constant. Boyle’s law requires constant temperature, while Charles’ law requires constant pressure and absolute temperature.
• Determine internal energy and specific heats: questions use U = (f/2)nRT, Cv = (f/2)R, Cp - Cv = R, and gamma = Cp/Cv = 1 + 2/f. Marks depend on identifying the correct number of active degrees of freedom.
• Compare gases using equipartition: questions distinguish a monatomic gas with 3 translational degrees of freedom from a diatomic gas at ordinary temperatures with 5 active degrees of freedom. Vibrational modes may not contribute fully at lower temperatures, so they cannot automatically be counted as active.