The kinetic theory of gases, which derives the macroscopic properties of gases (pressure, temperature, and volume) from the microscopic motion of molecules, is the microscopic foundation of thermodynamics.
Basic Knowledge
Equation of state for an ideal gas: pV = nRT (p: pressure [Pa], V: volume [m³], n: amount of substance [mol], R = 8.314 J/(mol·K), T: absolute temperature [K])
Or pV = NkT (N: number of molecules, k = 1.38×10⁻²³ J/K: Boltzmann constant)
Results of the kinetic theory of gases:
・Pressure: p = (1/3)(N/V)mv̄² (m: molecular mass, v̄²: mean-square speed)
・Mean kinetic energy of a molecule: (1/2)mv̄² = (3/2)kT
・Internal energy of a monatomic ideal gas: U = (3/2)NkT = (3/2)nRT
Boyle–Charles law: pV/T = constant (constant amount of substance)
Boyle’s law (isothermal): pV = constant; Charles’s law (constant pressure): V/T = constant
Ideal gas (a hypothetical gas with no intermolecular forces, no molecular volume, and only elastic collisions)
Equation of state pV = nRT (the fundamental equation relating pressure and volume using n, R, and T)
Gas constant R = 8.314 J/(mol·K)
Boltzmann constant k = R/NA (NA: Avogadro constant)
Root-mean-square speed v̄rms (v̄rms = √(3kT/m); proportional to the square root of temperature)
Further Exploration (Background and Significance)
The kinetic theory of gases clarifies the nature of temperature: absolute temperature is proportional to the mean kinetic energy of molecules. Classically, molecular motion stops at 0 K, which is the meaning of absolute zero. Quantum mechanically, zero-point vibration remains even at 0 K, but the classical treatment is sufficient in high-school physics.
Real gases have intermolecular forces and molecular volume, so their equations of state require correction (van der Waals equation). Deviations from ideal-gas behavior become greater at high pressure and low temperature.
Maxwell–Boltzmann speed distribution: The speeds of gas molecules are not constant but distributed, and as temperature rises, the distribution broadens toward higher speeds.
- pV = nRT. T must always be absolute temperature [K]
- Mean kinetic energy of a molecule = (3/2)kT
- U = (3/2)nRT (monatomic ideal gas) → depends only on T
- Boyle: pV = constant (isothermal); Charles: V/T = constant (constant pressure)
- Root-mean-square speed v̄rms ∝ √T
Points to Note (Common Confusions)
① T in pV = nRT must always be absolute temperature in K. ② Molecular kinetic energy is (1/2)mv² = (3/2)kT, using the mean square of the speed. ③ Boyle’s law is isothermal, while Charles’s law is at constant pressure. At constant volume, p/T = constant (Gay-Lussac’s law). ④ Do not confuse n (number of moles) with N (number of molecules). R = k×NA.
Practice
- Find the volume of 2.0 mol of an ideal gas at 27 ℃ and 1.0×10⁵ Pa (R = 8.3 J/(mol·K)).
- If the temperature becomes four times as high, how many times greater does the root-mean-square speed of the gas molecules become?
- Find the internal energy of 3.0 mol of a monatomic ideal gas at 300 K (R = 8.3 J/(mol·K)).