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Kinetic Theory of Gases - Summary
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
x) = 2mv
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
x = \frac{m}{l}N \overline{v
x^2}
Mean-square of v
x: \overline{v
x^2} = \frac{(v
{x1}^2 + v
{x2}^2 + v
{x3}^2 + … + v
{xN}^2)}{N}
Pressure and Mean Squared Speed
\overline{v^2} = \overline{v
x^2} + \overline{v
y^2} + \overline{v
z^2} = 3\overline{v
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
{trans} = \frac{3}{2}kT (per molecule), E
{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
i}{N
j} = \exp(-\frac{E
i - E
j}{kT})
Fraction of particles in the ith level: p(i) = \frac{N
i}{N} \propto \exp(-\frac{E
i}{kT})
Maxwell Speed Distribution (1D and 3D)
p(v
x) = K \exp(-\frac{m v
x^2}{2kT})
p(v
x, v
y, v
z) = (\frac{m}{2\pi kT})^{3/2} \exp(-\frac{m(v
x^2 + v
y^2 + v
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
j)g(x
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
{AB} = \pi d
{AB}^2
Single-particle collision frequency: z
{AB} = \eta
B \sigma
{AB} v
{rel}
Total collision frequency: Z
{AB} = \eta
A z
{AB} = \eta
A \eta
B \sigma
{AB} v_{rel}
Collision Frequency
v
{rel} = \sqrt{(v
{A,ave})^2 + (v_{B,ave})^2}
If A = B: z
{AA} = \sqrt{2} \eta
A \sigma
{AA} v
{A,ave}; Z
{AA} = \frac{1}{2} \sqrt{2} \eta
A^2 \sigma
{AA} v
{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{r
1}{r
2} = \sqrt{\frac{M
2}{M
1}}
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