Kinetic Theory Gases Physics Lesson 19 by Owen Borville 12.15.2025
The ideal gas law relates pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
A mole of any substance has a number of molecules equal to the number of atoms in a 12 gram sample of carbon 12. The number of molecules in a mole is called Avogadro's number, NA, NA = 6.02 * 10^23 mol-1. A mole of any substance has a mass in grams numerically equal to its molecular mass in unified mass units, which can be determined from the periodic table of elements.
The ideal gas law can also be written and solved in terms of the number or moles of gas: PV = nRT, where n is the number of moles and R is the universal gas constant: R = 8.31 J/mol*K
The ideal gas law in terms of molecules is PV = NkBT where P is pressure, V is volume, T is temperature, N is the number of molecules. The ideal gas law ratio if the amount of gas is constant is P1V1/T1 = P2V2/T2 The ideal gas law is generally valid at temperatures well above the boiling temperature. (kB is the Boltzmann constant, 1.38 x 10^-23 J/K)
The van der Waals equation of state for gases is valid closer to the boiling point than the ideal gas law = [P + a(n/V)^2] (V-nb) = nRT
Above the critical temperature and pressure for a given substance, the liquid phase does not exist, and the sample is supercritical.
Kinetic theory is the atomic description of gases as well as liquids and solids. Kinetic theory models the properties of matter in terms of continuous random motion of molecules.
The ideal gas law can be expressed in terms of the mass of the gas's molecules and v^2, the average of the molecular speed squared, instead of the temperature: PV = 1/3Nmv^2 where P is the pressure (average force per unit area), V is the volume of gas in a container, N is the number of molecules in the container, m is the mass of a molecule, and v^2 is the the average of the molecular speed squared.
Thermal energy is defined to be the average translational kinetic energy KE of an atom or molecule. The temperature of gases is proportional to the average translational kinetic energy of molecules. KE = 1/2mv^2 = 3/2kT or √v^2 = vrms = √3kT/m
Therefore, the speed of gas molecules vrms is proportional to the square root of the temperature and inversely proportional to the square root of the molecular mass. In a mixture of gases, each gas exerts a pressure equal to the total pressure times the fraction of the mixture that the gas makes up. The root-mean-square speed is vrms = √3RT/M = √3kBT/m
The mean free path (the average distance between collisions) and the mean free time of gas molecules between collisions are proportional to the temperature and inversely proportional to the molar density and the molecules' cross-sectional area.
The mean free path of a molecule is = λ = V/4√2πr^2N = kBT/4√2πr^2p
The mean free time of a molecule is = τ = kBT/4√2πr^2pvrms
The average kinetic energy of a molecule of monatomic ideal gas: K = 3/2kBT. The internal energy is Eint = 3/2NkBT
Every degree of freedom of an ideal gas contributes 1/2kBT per atom or molecule to its changes in internal energy. Every degree of freedom contributes 1/2R to its molar heat capacity at constant volume CV.
Degrees of freedom do not contribute if the temperature is too low to excite the minimum energy of the degree of freedom as given by quantum mechanics. Therefore, at ordinary temperatures, d = 3 for monatomic gases, d = 5 for diatomic gases, and d is approximately 6 for polyatomic gases.
The heat in terms of molar heat capacity at constant volume is Q = nCV ΔT
Molar heat capacity at constant volume for an ideal gas with d degrees of freedom is CV = d/2R
The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution. f(v) = 4/√π(m/2kBT)^3/2 * v^2 * e^-mv^2/2kBT
The average and most probable velocities of molecules having the Maxwell-Boltzmann speed distribution, as well as the rms velocity, can be calculated from the temperature and molecular mass.
Average velocity of a molecule = v = √8/π*kBT/m = √8/π*RT/M
Peak velocity of a molecule = vp = √2kBT/m = √2RT/M
The ideal gas law relates pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
A mole of any substance has a number of molecules equal to the number of atoms in a 12 gram sample of carbon 12. The number of molecules in a mole is called Avogadro's number, NA, NA = 6.02 * 10^23 mol-1. A mole of any substance has a mass in grams numerically equal to its molecular mass in unified mass units, which can be determined from the periodic table of elements.
The ideal gas law can also be written and solved in terms of the number or moles of gas: PV = nRT, where n is the number of moles and R is the universal gas constant: R = 8.31 J/mol*K
The ideal gas law in terms of molecules is PV = NkBT where P is pressure, V is volume, T is temperature, N is the number of molecules. The ideal gas law ratio if the amount of gas is constant is P1V1/T1 = P2V2/T2 The ideal gas law is generally valid at temperatures well above the boiling temperature. (kB is the Boltzmann constant, 1.38 x 10^-23 J/K)
The van der Waals equation of state for gases is valid closer to the boiling point than the ideal gas law = [P + a(n/V)^2] (V-nb) = nRT
Above the critical temperature and pressure for a given substance, the liquid phase does not exist, and the sample is supercritical.
Kinetic theory is the atomic description of gases as well as liquids and solids. Kinetic theory models the properties of matter in terms of continuous random motion of molecules.
The ideal gas law can be expressed in terms of the mass of the gas's molecules and v^2, the average of the molecular speed squared, instead of the temperature: PV = 1/3Nmv^2 where P is the pressure (average force per unit area), V is the volume of gas in a container, N is the number of molecules in the container, m is the mass of a molecule, and v^2 is the the average of the molecular speed squared.
Thermal energy is defined to be the average translational kinetic energy KE of an atom or molecule. The temperature of gases is proportional to the average translational kinetic energy of molecules. KE = 1/2mv^2 = 3/2kT or √v^2 = vrms = √3kT/m
Therefore, the speed of gas molecules vrms is proportional to the square root of the temperature and inversely proportional to the square root of the molecular mass. In a mixture of gases, each gas exerts a pressure equal to the total pressure times the fraction of the mixture that the gas makes up. The root-mean-square speed is vrms = √3RT/M = √3kBT/m
The mean free path (the average distance between collisions) and the mean free time of gas molecules between collisions are proportional to the temperature and inversely proportional to the molar density and the molecules' cross-sectional area.
The mean free path of a molecule is = λ = V/4√2πr^2N = kBT/4√2πr^2p
The mean free time of a molecule is = τ = kBT/4√2πr^2pvrms
The average kinetic energy of a molecule of monatomic ideal gas: K = 3/2kBT. The internal energy is Eint = 3/2NkBT
Every degree of freedom of an ideal gas contributes 1/2kBT per atom or molecule to its changes in internal energy. Every degree of freedom contributes 1/2R to its molar heat capacity at constant volume CV.
Degrees of freedom do not contribute if the temperature is too low to excite the minimum energy of the degree of freedom as given by quantum mechanics. Therefore, at ordinary temperatures, d = 3 for monatomic gases, d = 5 for diatomic gases, and d is approximately 6 for polyatomic gases.
The heat in terms of molar heat capacity at constant volume is Q = nCV ΔT
Molar heat capacity at constant volume for an ideal gas with d degrees of freedom is CV = d/2R
The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution. f(v) = 4/√π(m/2kBT)^3/2 * v^2 * e^-mv^2/2kBT
The average and most probable velocities of molecules having the Maxwell-Boltzmann speed distribution, as well as the rms velocity, can be calculated from the temperature and molecular mass.
Average velocity of a molecule = v = √8/π*kBT/m = √8/π*RT/M
Peak velocity of a molecule = vp = √2kBT/m = √2RT/M