Transcript Kinetic Theory

Kinetic Theory
(or life as a molecule)
N2
O2
N2
N2
Prof. Fred Remer
University of North Dakota
N2
O2
Kinetic Theory
H2 O
N2
N2
Objective
• Be able to define temperature and
pressure
• Be able to perform simple
calculations using the Ideal Gas Law
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Reading
• Wallace and Hobbs,
pp. 64, 74
• Bohren and
Albrecht
– pp. 1-30
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Once Upon A Time There Was A
Molecule
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• The molecule had no size or internal
structure, but it was a happy
molecule. Her name was Point Mass.
m = mass
Prof. Fred Remer
University of North Dakota
Kinetic Theory
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
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• He had momentum!
momemtum
= mvx
x - direction
velocity = vx
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• He had so much momentum, he
could not slow down when he saw
the wall!
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• He hit the wall!
OUCH!
Area
A
x
Prof. Fred Remer
University of North Dakota
x
Kinetic Theory
Kinetic Theory
• But much to his surprise, he rebounded!
His collision was perfectly elastic! No
energy was lost in the collision.
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• He had the same momentum leaving
the wall as he had before the collision,
but in the opposite direction.
momentum
Prof. Fred Remer
University of North Dakota
Kinetic Theory
 mvx
Kinetic Theory
• His change in momentum was
Change in Momentum
 mvx  mvx  2mvx
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• The force exerted on Point Mass by
the wall was
Change in Momentum
Force 
Time
or
Momentum Change  2mvx   Fx dt
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Point Mass had other friends who are
molecules identical to himself.
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• They all move at the same velocity vx
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• The molecules do not interact
between themselves ...
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• …but they all interacted with the wall
x x
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• The number of molecules (N) in a
given volume (V) is the number
density (n)
N
n
V
V = Volume
N = # of
molecules
n = number density
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• The flux of molecules headed toward
the wall is ...
nv x
2
vx
vx
Prof. Fred Remer
University of North Dakota
1/2 moving towards at vx
1/2 moving away at vx
Kinetic Theory
Kinetic Theory
• The number of molecules striking the
wall (A) during a time period (t) is ...
A
nv x
At
2
vx
vx
Prof. Fred Remer
University of North Dakota
t
Kinetic Theory
Kinetic Theory
• The total time integrated force on the
wall (A) is ...
A
Prof. Fred Remer
University of North Dakota
nv x
 Fx dt  2mv x 2 At
Kinetic Theory
Kinetic Theory
• The time-averaged force on the wall
is ...
A
Prof. Fred Remer
University of North Dakota
1
2
Fx    Fx dt  mv x An
t
Kinetic Theory
Kinetic Theory
• The average force per unit area is ...
Fx 
1
mv x An

Fx dt 

A
At
A
2
A
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• …Which is pressure!
A
Prof. Fred Remer
University of North Dakota
Fx 
2
p
 nmv x
A
Kinetic Theory
Kinetic Theory
• Lets modify one assumption. The
molecules are moving at different
speeds.
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Let’s replace vx2 with an average.
p  nm  v x 
2
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• In reality, the molecules are moving
in all directions (not just x).
v x   v y   v z 
2
2
2
v x   v y   v z   v 
2
2
2
1 2
v x   v 
3
2
Prof. Fred Remer
University of North Dakota
Kinetic Theory
2
Kinetic Theory
• Substitute back into the equation
p  nm  v x 
2
1 2
v x   v 
3
2
1
p  nmv 2 
3
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• This looks like Kinetic Energy!
KE
Monatomic
Gas
1
KE  mv 2 
2
1
p  nmv 2 
3
2 1
2
p  n mv 
3 2
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Definition of Temperature
2 1
kT  n mv 2 
3 2
where k = Boltzmann Constant
= 1.38 x 10-23 J/K
– Temperature is a measure of the
average KE of the molecules!
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Substitute temperature into pressure
2 1
p  n mv 2 
3 2
2N 1
p
 mv 2 
3V 2
2 1
kT  n mv 2 
3 2
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Ideal Gas Law
N
p  kT
V
or
pV  NkT
Prof. Fred Remer
University of North Dakota
where ...
p = pressure
V = volume
N = number of molecules
T = temperature
k = Boltzman Constant
Kinetic Theory
Kinetic Theory
• Monatomic Molecules
– Energy Is a Result of Atom’s Motion
Only
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Polyatomic Molecules
– Energy Is a Result of
• Atom’s Motion
• Rotation, Vibration and Oscillation of
Molecule
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• Polyatomic Molecules
– Need to Account for
Other Forms of
Molecular Energy
Total
Molecular
Energy
Prof. Fred Remer
University of North Dakota
=
Kinetic Energy
Due to Motion
Kinetic Theory
+
Kinetic Energy
Due to Rotation &
Vibration
Kinetic Theory
• Polyatomic Molecules
– More Complex
Molecules Have More
Rotational &
Vibrational Energy
Total
Molecular
Energy
Prof. Fred Remer
University of North Dakota
=
Kinetic Energy
Due to Motion
Kinetic Theory
+
Kinetic Energy
Due to Rotation &
Vibration
Kinetic Theory
• Polyatomic Molecules
– More Complex
Molecules Have More
Rotational &
Vibrational Energy
Total
Molecular
Energy
Prof. Fred Remer
University of North Dakota
=
Kinetic Energy
Due to Motion
Kinetic Theory
+
Kinetic Energy
Due to Rotation &
Vibration
Kinetic Theory
• Polyatomic Molecules
– Low Pressure
• Approximates Ideal Gas
– High Pressure
• Deviates More
Total
Molecular
Energy
Prof. Fred Remer
University of North Dakota
=
Kinetic Energy
Due to Motion
Kinetic Theory
+
Kinetic Energy
Due to Rotation &
Vibration
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:
pV  NkT
Prof. Fred Remer
University of North Dakota
Kinetic Theory
Kinetic Theory
• We will return to the perfect gas law from a
macroscopic point of view and derive
exactly the same relationship:
pV  NkT
or
p  RT
where ...
p = pressure
V = volume, n = number of moles
N = number of molecules
T = temperature
k = Boltzmann constant
m = mass, M = molecular weight
m
Nk
 , R
, m  nM
V
m
Prof. Fred Remer
University of North Dakota
Kinetic Theory