Chapter 16 Kinetic Theory of Gases

Ideal gas model 2 1. Large number of molecules moving in random directions with random speeds. 2. The average separation is much greater than the diameter of each molecule. 3. Obey the laws of classical mechanics; interact only when they collide. (ignore potential energy) 4. All collisions are perfectly elastic. “Many elastic particles move randomly, no interaction”

Pressure in a gas (1) 3 like an umbrella in rain Consider a gas in cube pressure → collisions of the molecules For one collision:.. x y z l Until next collision: Average force:

Pressure in a gas (2) 4 Average force: Other kinds of collision? For all molecules in the cube: random velocities.. x y z l......

Pressure in a gas (3) 5 Net force: Pressure on the wall: Number density: Average (translational) kinetic energy: Pressure in a gas:.. x y z l S

Pressure in explosion 6 Example1: Gunpowder explodes in 10cm 3 space. The explosion produces 0.1mol gas and 2×10 5 J energy. Estimate the instantaneous pressure. Solution: Explosion → extra pressure

Molecular interpretation of temperature 7 Pressure in a gas: Compare with the ideal gas law: The average translational kinetic energy of molecules in an ideal gas is directly proportional to the absolute temperature. Molecular interpretation of temperature:

Molecule kinetic energy 8 Example2: (a) What is the average kinetic energy of molecules in an ideal gas at 37 ℃ ? (b) If H 2 and O 2 are both at 37 ℃, which kind of molecules moves faster on average? Solution: (a) Average kinetic energy: (b) Same T → same average kinetic energy

Mean speed and rms speed 9 Example3: 5 particles have the following speeds, given in m/s: 1, 2, 3, 4, 5. Calculate (a) the mean speed and (b) the root-mean-square (rms) speed. Solution: (a) Mean speed: (b) rms speed : v rms for an ideal gas: for an ideal gas?

Distribution of molecular speeds 10 How many molecules move faster than v rms ? Distribution Gauss distribution Maxwell distribution of speeds: ▲ Ideal gas in equilibrium state at temperature T f (v) v

Maxwell distribution of speeds 11 constants: N, m, k, T f (v) v f (v) dv: number of molecules with speed between v and v+dv dv f (v) v1v1 v2v2 Number of molecules with speed v 1 < v <v 2 : Sum all molecules:

Shape of the curve 12 Most probable speed v p f (v) v vpvp Distribution for different temperature: f (v) v 300K 600K Chemical reaction & temperature

Average speed 13 Solution: How to calculate average values? Example4: Determine the average speed of molecules in an ideal gas at temperature T. f (v) dv: dN with speed between v and v+dv Sum of speeds of dN molecules: Sum of speeds of all molecules: Average speed:

Three statistical speeds 14 Most probable speed v p : Average speed : Root-mean-square speed v rms : f (v) v vpvp v rms

Using f (v) 15 Solution: (a) Most probable speed: Example5: 1mol H 2 gas at 300K. (a) What is v p ? (b) How many molecules have speed v p < v < v p +20 m/s ? (b) Speed between v p and v p +dv : dN=f (v p ) dv 2v p < v < 2v p +20m/s : 3v p < v < 3v p +20m/s :

Homework 16 If the distribution of speeds in a N-particles system is (instead of Maxwell distribution) (a) C=? (b) number of particles with f(v) > C /2. v0v0 0 v f(v)f(v)

Real gases 17 High T & low P → ideal gas law PV=nRT Real gas behavior? Shown in a PV diagram: Higher pressure molecules be closer E p (attractive force) can’t be ignored molecules get even closer

Changes of phase 18 Curve D → liquefaction Curve C → critical point c Critical temperature vapor & gas PT diagram: phase diagram boiling / freezing / sublimation

Using phase diagram 19 Solution: (a) P > 1atm → liquid; P < 1atm → gas Example6: Describe the phase of water in different pressure at (a) 100 ℃ ; (b) 0 ℃. 100 ℃ P = 1atm → boiling point (b) P > 1atm → liquid; P < P 0 → gas; P 0 < P < 1atm → solid 0℃0℃ P0P0. P = 1atm → freezing point P = P 0 → sublimation point

Mean free path 20 Collisions between molecules Mean free path: average distance traveling between collisions P 396 : Molecules → hard spheres of radius r

Collision frequency 21 Solution: (a) Number density at STP: Example7: Estimate (a) the mean free path of O 2 molecules at STP and (b) the average collision frequency. (r≈1.5× m) Mean free path: (b) Average speed average collision frequency

*Diffusion 22 Diffusion: substance moves from high concentrated region to low concentrated region Random motion of molecules: more molecules moves from 1 to 2 than from 2 to 1 concentrations become equal everywhere Diffusion equation (Fick’s law): 123

*Distribution of energy 23 All molecules of atmosphere diffuse to the space? Boltzmann’s distribution of potential energy: number density at position E p =0 In an gravity field: constant T

*Height of aircraft 24 Solution: By using the equation of pressure: Example8: Air pressure gauge on an aircraft reads 0.3atm, 0 ℃ outside. What is the height?