Download Kinetic Theory

Kinetic Theory
Microscopic Analysis

1)
2)
3)
4)
The behavior of a gas should be described by the
molecules.
The gas consists of a large number of identical
particles of mass m.
The particles have negligible size and no internal
structure.
The particles are moving in random dirctions with
speeds independent of direction
Collisions between particles and with the walls are
perfectly elastic.
Collision Force

L

Fx
Follow a collision with a wall
perpendicular to x.
Particle has mass m and a
velocity vx.
• Strike the wall: Dt = 2L/vx.
• Impulse: Dp = 2mvx.
v
m

The force from one particle:
mvx2
Dp 2mvx
F


Dt 2 L / vx
L
Pressure Force

The pressure on the wall
comes from all the particles.
2
Fx  Fxi m vx
P


A
A
AL

The volume is V = AL.
Find the value for N
particles.
L
A

P
m v x2
AL
mN

V
2
v
x
N

mN 2
vx
V
Three Dimensions

V
The pressure come from all
three dimensions, and is
equal in all three.
v 2  v x2  v y2  v z2  3v x2
P
N

Relate the pressure to the
average speed.
mN v 2 2 N 1 2
P

( 2 mv )
V 3 3V
Theory Match

V
P
The expression from a
particle level relates the
average kinetic energy.
PV  N
N

2 1 2
( 2 mv )
3
This almost matches the
ideal gas law.
PV  N (kT )
Temperature as Energy

For the particle-level theory to match the
experimental law we equate them.
2 1 2
PV  N ( 2 mv )  N (kT )
3
1
2


mv 2  32 kT
This is kinetic theory.
Temperature measures the average kinetic energy.
Average Energy


What is the average energy
of an air molecule at room
temperature (293 K)?

What is average speed for a
nitrogen molecule (28
g/mol)?

Energy directly relates to
temperature.
 (3/2)kT = 6.07 x 10-21 J.
To get the speed requires
the mass.
 m = (0.028 kg/mol) / (6.022
x 1023 /mol) = 4.65x10-26 kg
3kT
v
 511 m/s
m
Distribution of Speeds

Kinetic theory used the
average speed.

Actual atoms fall into a range
of speeds.

The Maxwell-Boltzmann
distribution describes the
probability of a molecule
having a particular speed.
Real Gases

The assumptions for kinetic theory are approximately
true.
• Non-zero molecular size
• Non-zero force between molucules

For the Van der Waals force in air, the effect is about
1% difference from an ideal gas.
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