Kinetic Theory of Gases in Ideal Gas Model

Chapter 16 Kinetic Theory of Gases

Ideal gas model 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”

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

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

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

Pressure in explosion Example1: Gunpowder explodes in 10cm3 space. The explosion produces 0.1mol gas and 2×105J energy. Estimate the instantaneous pressure. Solution: Explosion → extra pressure 6

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

Molecule kinetic energy Example2: (a) What is the average kinetic energy of molecules in an ideal gas at 37℃? (b) If H2 and O2 are both at 37℃, which kind of molecules moves faster on average? Solution: (a)Average kinetic energy: (b)Same T→ same average kinetic energy 8

for an ideal gas? Mean speed and rms speed 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 : vrms for an ideal gas: 9

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

f (v) v1 v2 dv f (v) v Maxwell distribution of speeds constants: N, m, k, T f(v)dv:number of molecules with speed between v and v+dv Number of molecules with speed v1< v <v2 : Sum all molecules: 11

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

Example4: Determine the average speed of molecules in an ideal gas at temperature T. Average speed Solution: How to calculate average values? 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: 13

vrms vp Average speed : f (v) v Three statistical speeds Most probable speed vp : Root-mean-squarespeed vrms : 14

Using f(v) Example5: 1mol H2 gas at 300K. (a) What is vp? (b) How many molecules have speed vp<v<vp+20m/s? Solution: (a)Most probable speed: (b) Speed between vp and vp+dv: dN=f(vp)dv 2vp<v<2vp+20m/s: 3vp<v<3vp+20m/s: 15

f(v) v 0 v0 Homework 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. 16

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

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

0℃ 100℃ . P0 Using phase diagram Example6: Describe the phase of water in different pressure at (a) 100℃; (b) 0℃. Solution: (a)P > 1atm → liquid; P < 1atm → gas P = 1atm →boiling point (b)P > 1atm → liquid; P < P0→ gas; P0 <P< 1atm → solid P = 1atm →freezing point P = P0→sublimation point 19

Mean free path Collisions between molecules Mean free path: average distance traveling between collisions P396: Molecules → hard spheres of radius r 20

Collision frequency Example7: Estimate (a) the mean free path of O2 molecules at STP and (b) the average collision frequency. (r≈1.5×10-10m) Solution: (a)Number density at STP: Mean free path: (b)Average speed average collision frequency 21

1 2 3 *Diffusion 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): 22

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

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

Kinetic Theory of Gases in Ideal Gas Model

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