(or life as a molecule)

1
Kinetic Theory

• (or life as a molecule)

2
Objective

• Be able to define temperature and pressure
• Be able to perform simple calculations using the
  Ideal Gas Law

3
Reading

• Wallace and Hobbs, pp. 64, 74
• Bohren and Albrecht
• pp. 1-30

4
Kinetic Theory

• Once Upon A Time There Was A Molecule

5
Kinetic Theory

• The molecule had no size or internal structure,
  but it was a happy molecule. Her name was Point
  Mass.

m mass
6
Kinetic Theory

• One day, Point Mass decided to move. He only
  moved in one direction. He moved a a constant
  speed.

x - direction
velocity vx
7
Kinetic Theory

• He had momentum!

momemtum mvx
x - direction
velocity vx
8
Kinetic Theory

• He had so much momentum, he could not slow down
  when he saw the wall!

9
Kinetic Theory

• He hit the wall!

OUCH!
Area A
x
x
10
Kinetic Theory

• But much to his surprise, he rebounded! His
  collision was perfectly elastic! No energy was
  lost in the collision.

11
Kinetic Theory

• He had the same momentum leaving the wall as he
  had before the collision, but in the opposite
  direction.

momentum
12
Kinetic Theory

• His change in momentum was

Change in Momentum
13
Kinetic Theory

• The force exerted on Point Mass by the wall was

or
14
Kinetic Theory

• Point Mass had other friends who are molecules
  identical to himself.

15
Kinetic Theory

• They all move at the same velocity vx

16
Kinetic Theory

• The molecules do not interact between themselves
  ...

17
Kinetic Theory

• but they all interacted with the wall

18
Kinetic Theory

• The number of molecules (N) in a given volume (V)
  is the number density (n)

V Volume
N of molecules
n number density
19
Kinetic Theory

• The flux of molecules headed toward the wall is
  ...

vx
1/2 moving towards at vx
1/2 moving away at vx
vx
20
Kinetic Theory

• The number of molecules striking the wall (A)
  during a time period (t) is ...

A
vx
vx
t
21
Kinetic Theory

• The total time integrated force on the wall (A)
  is ...

A
22
Kinetic Theory

• The time-averaged force on the wall is ...

A
23
Kinetic Theory

• The average force per unit area is ...

A
24
Kinetic Theory

• Which is pressure!

A
25
Kinetic Theory

• Lets modify one assumption. The molecules are
  moving at different speeds.

26
Kinetic Theory

• Lets replace vx2 with an average.

27
Kinetic Theory

• In reality, the molecules are moving in all
  directions (not just x).

28
Kinetic Theory

• Substitute back into the equation

29
Kinetic Theory

• This looks like Kinetic Energy!

Monatomic Gas
KE
30
Kinetic Theory

• Definition of Temperature
• Temperature is a measure of the average KE of the
  molecules!

where k Boltzmann Constant 1.38
x 10-23 J/K
31
Kinetic Theory

• Substitute temperature into pressure

32
Kinetic Theory

• Ideal Gas Law

where ... p pressure V volume N number of
molecules T temperature k Boltzman Constant
or
33
Kinetic Theory

• Monatomic Molecules
• Energy Is a Result of Atoms Motion Only

34
Kinetic Theory

• Polyatomic Molecules
• Energy Is a Result of
• Atoms Motion
• Rotation, Vibration and Oscillation of Molecule

35
Kinetic Theory

• Polyatomic Molecules
• Need to Account for Other Forms of Molecular
  Energy

36
Kinetic Theory

• Polyatomic Molecules
• More Complex Molecules Have More Rotational
  Vibrational Energy

37
Kinetic Theory

• Polyatomic Molecules
• More Complex Molecules Have More Rotational
  Vibrational Energy

38
Kinetic Theory

• Polyatomic Molecules
• Low Pressure
• Approximates Ideal Gas
• High Pressure
• Deviates More

39
Kinetic Theory

• Summary
• pressure is a measure of the total kinetic energy
  of molecules, the force per unit area of these
  molecules
• temperature is proportional to the average
  kinetic energy of molecules
• from this kinetic theory viewpoint, we can derive
  the perfect gas law

40
Kinetic Theory

• We will return to the perfect gas law from a
  macroscopic point of view and derive exactly the
  same relationship

where ... p pressure V volume, n number of
moles N number of molecules T temperature k
Boltzmann constant m mass, M molecular weight
or