2.2 Pressure, Temperature, and RMS Speed

Gas behavior is all about molecular motion. Pressure comes from molecules hitting container walls, while temperature relates to their average kinetic energy. The ideal gas law connects these macroscopic properties to the microscopic world of molecules.

Gases in mixtures exert partial pressures, adding up to the total pressure. Molecules constantly collide, with their mean free path and collision frequency depending on conditions. The Maxwell-Boltzmann distribution describes the range of molecular speeds in a gas.

Kinetic Theory of Gases

Microscopic vs macroscopic gas properties

• Pressure caused by gas molecules colliding with container walls
  • Directly proportional to number of collisions per unit time and average force per collision
  • Example: higher pressure in a tire due to more frequent and forceful collisions of air molecules with tire walls
• Temperature related to average kinetic energy of gas molecules
  • Higher temperature means higher average kinetic energy and faster moving molecules
  • Example: molecules in a hot air balloon move faster than those in a cold room
  • Systems in thermal equilibrium have the same temperature and average kinetic energy
• Root-mean-square (RMS) speed ($v_{rms}$) measures average speed of gas molecules
  • Calculated using $v_{rms} = \sqrt{\frac{3kT}{m}}$, where $k$ is Boltzmann constant, $T$ is absolute temperature, and $m$ is mass of a single molecule
  • Example: RMS speed of nitrogen molecules at room temperature (20℃) is about 511 m/s
• Ideal gas law $PV = nRT$ relates pressure ($P$), volume ($V$), number of moles ($n$), gas constant ($R$), and absolute temperature ($T$)
  • Describes behavior of ideal gases under various conditions
  • Example: increasing temperature of a gas in a closed container increases its pressure
  • The number of molecules in one mole of any substance is given by Avogadro's number

Partial pressures in gas mixtures

• Dalton's law states total pressure of a gas mixture is sum of partial pressures of each component gas
  • $P_{total} = P_1 + P_2 + ... + P_n$, where $P_1, P_2, ..., P_n$ are partial pressures of each gas in mixture
  • Example: in air, partial pressure of oxygen is about 21% of total atmospheric pressure
• Partial pressure of a gas in a mixture is pressure gas would exert if it occupied entire volume alone
  • Calculated using $P_i = x_i P_{total}$, where $P_i$ is partial pressure of gas $i$, $x_i$ is molar fraction of gas $i$, and $P_{total}$ is total pressure of mixture
  • Example: in a 60% nitrogen, 40% oxygen mixture at 1 atm total pressure, partial pressure of nitrogen is 0.6 atm and oxygen is 0.4 atm
• Molar fraction is ratio of number of moles of a particular gas to total number of moles in mixture
  • Calculated using $x_i = \frac{n_i}{n_{total}}$, where $n_i$ is number of moles of gas $i$ and $n_{total}$ is total number of moles in mixture
  • Example: in a mixture of 3 moles of helium and 1 mole of neon, molar fraction of helium is 0.75 and neon is 0.25

Molecular motion in gases

• Mean free path ($\lambda$) is average distance a molecule travels between collisions
  • Calculated using $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$, where $d$ is diameter of molecule and $n$ is number density (molecules per unit volume)
  • Example: mean free path of air molecules at room temperature and pressure is about 68 nm
• Collision frequency ($z$) is average number of collisions per unit time for a single molecule
  • Calculated using $z = \sqrt{2} \pi d^2 n v_{rms}$, where $d$ is diameter of molecule, $n$ is number density, and $v_{rms}$ is root-mean-square speed
  • Example: collision frequency of nitrogen molecules at room temperature and pressure is about $7 \times 10^9$ collisions per second
• Higher pressure and temperature lead to:
  1. Shorter mean free paths due to more molecules in a given volume
  2. Higher collision frequencies due to faster moving molecules
  • Example: in a high-pressure gas cylinder, molecules have shorter mean free paths and higher collision frequencies compared to the same gas at atmospheric pressure

Statistical distribution of molecular speeds

• The Maxwell-Boltzmann distribution describes the probability distribution of molecular speeds in a gas
• The equipartition theorem states that energy is equally distributed among all degrees of freedom in a system
• Diffusion is the net movement of particles from regions of high concentration to low concentration
• Effusion occurs when gas molecules escape through a small hole in a container