Kinetic Theory

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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 F x m v

  Follow a collision with a wall perpendicular to

x

.

Particle has mass m and a velocity

v x

.

• Strike the wall: D

t

• Impulse: D

p

= 2

L

/

v x

.

= 2

mv x

.

 The force from one particle:

F

 D

p

D

t

 2 2

L mv

/

v x x

mv L

2

x

Pressure Force

L A

 The pressure on the wall comes from all the particles.

P

F x A

 

A F xi

m

AL v x

2  The volume is

V

=

AL

.

P

 Find the value for

N

particles.

m

v x

2 

mN

v x

2

AL V N

mN v x

2

V

Three Dimensions

V P N

 The pressure come from all three dimensions, and is equal in all three.

v

2 

v x

2 

v

2

y

v z

2  3

v x

2  Relate the pressure to the average speed.

P

mN v

2

V

3  2

N

3

V

( 1 2

m v

2 )

Theory Match

V P N

 The expression from a particle level relates the average kinetic energy.

PV

N

2 3 ( 1 2

m v

2 )  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.

PV

N

2 3 ( 1 2

m v

2 ) 

N

(

kT

) 1 2

m v

2  3 2

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)?

 Energy directly relates to temperature.

 (3/2)

kT

= 6.07 x 10 -21 J

.

 What is average speed for a nitrogen molecule (28 g/mol)?

 To get the speed requires the mass.

m

= (0.028 kg/mol) / (6.022 x 10 23 /mol) = 4.65x10

-26 kg

v

 3

kT

 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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