10.2 Kinetic theory of gases

Gases are made up of tiny particles zipping around randomly. Kinetic theory explains how these particles behave, creating pressure and temperature. It's like a microscopic game of bumper cars, where molecules bounce off each other and container walls.

The theory links pressure, volume, and temperature through the ideal gas law. It also shows how molecular speed relates to temperature. Understanding these connections helps us predict gas behavior in various situations, from weather balloons to car tires.

Kinetic Theory of Gases

Postulates of kinetic theory

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• Gas consists of a large number of molecules in constant random motion
  • Molecules move rapidly in straight lines until they collide with other molecules or container walls (Brownian motion)
  • Collisions between molecules are perfectly elastic, kinetic energy is conserved during collisions (billiard balls)
• Molecules are treated as point masses with negligible volume compared to the container
  • The volume occupied by the molecules themselves is much smaller than the volume of the container (ping pong balls in a room)
• No attractive or repulsive forces exist between molecules, except during collisions
  • Molecules do not exert any long-range forces on each other, they only interact during brief collisions (people in a large crowd)
• The average kinetic energy of molecules is proportional to the absolute temperature
  • As temperature increases, molecules move faster and have higher average kinetic energy (hot air balloon rising)

Relationships in kinetic theory

• Pressure is caused by molecular collisions with the container walls
  • $P = \frac{1}{3}nm\overline{v^2}$, where $n$ is the number density, $m$ is the molecular mass, and $\overline{v^2}$ is the average of the square of the molecular velocity
  • More collisions or higher velocity collisions result in higher pressure (tire pressure increasing with temperature)
• Ideal gas law: [PV = nRT](https://www.fiveableKeyTerm:pv_=_nRT), where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature
  • The ideal gas law relates pressure, volume, temperature, and amount of gas (weather balloon expanding as it rises)
• Kinetic energy is directly proportional to absolute temperature
  • $\frac{1}{2}m\overline{v^2} = \frac{3}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature
  • Higher temperature means higher average molecular kinetic energy (faster molecule movement in boiling water vs room temperature water)

Velocity and energy of gas molecules

• Root mean square (rms) velocity: $v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{M}}$, where $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass
  • RMS velocity is a measure of the average speed of molecules in a gas (speedometer for gas molecules)
• Average kinetic energy: $\overline{KE} = \frac{1}{2}m\overline{v^2} = \frac{3}{2}kT$, where $m$ is the molecular mass, $k$ is the Boltzmann constant, and $T$ is the absolute temperature
  • The average kinetic energy of gas molecules depends on temperature and mass (heavier molecules move slower at the same temperature)

Maxwell-Boltzmann velocity distribution

• The Maxwell-Boltzmann distribution describes the probability distribution of molecular velocities in an ideal gas at thermal equilibrium
  • Gives the fraction of molecules with a specific velocity at a given temperature (bell curve of molecular speeds)
• The distribution depends on temperature and molecular mass
  • Higher temperatures result in a broader distribution and higher average velocities (faster molecules in hot gas)
  • Heavier molecules have a narrower distribution and lower average velocities compared to lighter molecules at the same temperature (oxygen vs hydrogen gas at room temperature)
• The most probable velocity ($v_p$), average velocity ($\overline{v}$), and root mean square velocity ($v_{rms}$) can be calculated from the distribution
  1. Most probable velocity: $v_p = \sqrt{\frac{2RT}{M}}$ (peak of the bell curve)
  2. Average velocity: $\overline{v} = \sqrt{\frac{8RT}{\pi M}}$ (average of all molecular speeds)
  3. Root mean square velocity: $v_{rms} = \sqrt{\frac{3RT}{M}}$ (square root of the average of the squared velocities)