Kinetic Theory of Gases - Summary

Kinetic Theory of Gases

Molecular Speeds and Ideal Gas Law

• Ideal gas law: $PV = nRT$
• $P$: Force exerted by molecules on the wall during collisions.
• $V$: Available volume for molecules.
• $T$: Indicates the speed of molecules.
• Gas molecules are in random and continuous motion.
• Collisions redistribute speed among molecules.

Molecular Momentum and Force

• Change in momentum for one molecule: $Δ(mv<em>x) = 2mv</em>x$
• Time between collisions: $Δt = \frac{2l}{v_x}$
• Force due to a single molecule: $F = \frac{mv_x^2}{l}$
• Average force due to all molecules in x-direction: $F<em>x = \frac{m}{l}N \overline{v</em>x^2}$
• Mean-square of $v<em>x$: $\overline{v</em>x^2} = \frac{(v<em>{x1}^2 + v</em>{x2}^2 + v<em>{x3}^2 + … + v</em>{xN}^2)}{N}$

Pressure and Mean Squared Speed

• $\overline{v^2} = \overline{v<em>x^2} + \overline{v</em>y^2} + \overline{v<em>z^2} = 3\overline{v</em>x^2}$
• Pressure: $P = \frac{F_x}{A} = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$

Mean Translation Energy

• $PV = \frac{1}{3}Nm\overline{v^2} = \frac{2}{3}N(\frac{1}{2}m\overline{v^2})$
• $Nm = nM$ (N = number of molecules, m = mass of each molecule, n = number of moles, M = molar mass)
• Mean translation energy: $E<em>{trans} = \frac{3}{2}kT$ (per molecule), $E</em>{trans} = \frac{3}{2}RT$ (per mole)

Root-Mean-Square (rms) Speed

• $v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$
• Lighter molecules move at higher rms speed.
• Different gases have different rms speeds at a given temperature.

Equipartition of Energy

• $E_{trans} = \frac{3}{2}kT = \frac{1}{2}kT + \frac{1}{2}kT + \frac{1}{2}kT$ (x, y, z directions)
• Each degree of freedom contributes $\frac{1}{2}kT$ to the energy.
• Degrees of freedom: Translation, Vibration, Rotation.

Degrees of Freedom

• Atom: 3 (translation)
• Molecule (non-linear): 3 (translation), 3 (rotation), 3N-6 (vibration)
• Molecule (linear): 3 (translation), 2 (rotation), 3N-5 (vibration)

Maxwell Speed Distribution

• Molecular collisions redistribute speeds.
• Maxwell distribution describes speeds in x, y, and z directions.

Boltzmann Distribution Law

• $P(h) = P_0 \exp(-\frac{mgh}{kT})$ relates air pressure with altitude.
• Boltzmann factor: $\exp(-\frac{E}{kT})$ relates the number of molecules in a given state with the energy of that state.
• $\frac{N<em>i}{N</em>j} = \exp(-\frac{E<em>i - E</em>j}{kT})$
• Fraction of particles in the ith level: $p(i) = \frac{N<em>i}{N} \propto \exp(-\frac{E</em>i}{kT})$

Maxwell Speed Distribution (1D and 3D)

• $p(v<em>x) = K \exp(-\frac{m v</em>x^2}{2kT})$
• $p(v<em>x, v</em>y, v<em>z) = (\frac{m}{2\pi kT})^{3/2} \exp(-\frac{m(v</em>x^2 + v<em>y^2 + v</em>z^2)}{2kT})$
• $p(v)dv = 4\pi (\frac{m}{2\pi kT})^{3/2} \exp(-\frac{m v^2}{2kT}) v^2 dv$

Average Values

• $g(x) = \sum p(x<em>j)g(x</em>j)$
• Most probable speed: $v_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2kT}{m}}$
• Average speed: $\overline{v} = \sqrt{\frac{8kT}{m\pi}} = \sqrt{\frac{8RT}{\pi M}}$
• Root-mean-square speed: $v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$
• v{rms} > v{ave} > v_{mp}

Molecular Collisions

• Collision cross-section area: $\sigma<em>{AB} = \pi d</em>{AB}^2$
• Single-particle collision frequency: $z<em>{AB} = \eta</em>B \sigma<em>{AB} v</em>{rel}$
• Total collision frequency: $Z<em>{AB} = \eta</em>A z<em>{AB} = \eta</em>A \eta<em>B \sigma</em>{AB} v_{rel}$

Collision Frequency

• $v<em>{rel} = \sqrt{(v</em>{A,ave})^2 + (v_{B,ave})^2}$
• If A = B: $z<em>{AA} = \sqrt{2} \eta</em>A \sigma<em>{AA} v</em>{A,ave}$; $Z<em>{AA} = \frac{1}{2} \sqrt{2} \eta</em>A^2 \sigma<em>{AA} v</em>{A,ave}$

Mean Free Path

• $\lambda = \frac{v_{ave}}{z} = \frac{1}{\sqrt{2} \eta \sigma}$

Molecular Diffusion/Effusion

• Diffusion: Molecules move from high to low concentration.
• Effusion: Escape of molecules through a tiny hole.

Graham's Law

• \frac{r1}{r2} = \sqrt{\frac{M2}{M1}}