13. Matter very simple
13.2 The kinetic model
What’s the relationship
Gas Law
Equation
Boyle’s Law
Charles’ Law
Pressure Law
Amount Law
Ideal Gas Law
What’s the relationship
Gas Law
Equation
Boyle’s Law
1
p
V
Charles’ Law
V T
Pressure Law
p T
Amount Law
pN
Ideal Gas Law
pV  nRT
1
p
V
V T
p T
pN
pV  nRT
Answers
0.103 m3
0.11 m3
7.14 litres
If the lab is roughly 3 m high x 10 m x 10 m,
volume = 300 m3, then about 12 kmol (with a
mass of 360 kg)
5. 2.38 kg m–3
6. 0.51 kg m–3
1.
2.
3.
4.
13. 2 The kinetic model
• Modelling of the real
world using
simplifications
• A gas is simply a
collection of fastmoving, colliding
particles, obeying the
laws of mechanics
s
v
t
s
t 
v
2x
t 
v
1
v

t 2 x
mv
F
t
2mv
F
2x / v
2
mv
F
x
V  xyz
A  yz
Calculating pressure
mv2
Fwall 
x
Awall  yz
F
p
A
2
mv
p
xyz
mv2
p
V
Improve model: add particles
Nmv
p
V
2
Improve model: random directions
1 Nmv
p
3 V
2
Improve model: random speeds
1 Nmv
p
3 V
2
1 Nmv
p
3 V
2
1 Nmv
p
3 V
2
1 Nmv
p
3 V
2
The ideal gas
• For the model to work
– Sample of gas must be large enough
– Gas molecules are moving randomly in all
directions with variable speeds
– Collisions are elastic
– Makes better predictions when:
• Density of gas is low (space molecules take up is zero)
• Energy of particles is large (no interactions)
The end of the clockwork Universe
• Maxwell and Boltzmann were using
Newtonian mechanics but in a new way
Temperature and Energy
pV  NkT
pV  Nmv
1
3
1
3
Nmv  NkT
2
mv  3kT
2
1
2
2
mv  kT
2
3
2
Temperature and Energy
mv  3kT
2
1
2
Nmv  NkT
1
2
N A mv  N A kT
2
3
2
2
R  N Ak
U  RT
3
2
3
2
Energy in a gas
U  Nmv  NkT  pV
1
2
2
3
2
3
2
We can determine the total internal
energy of a monatomic gas just by
knowing the pressure and volume,
which are easy to measure!
O2
H2
N2
Speed of a molecule
• If you know the average kinetic energy then
you can determine the average speed
Seeing primordial living motion
Let’s flip a coin
How far will a molecule go after N steps?
N
Worked examples
• If a particles follows a random walk of 10-7 m,
and made 1010 steps.
• What is the distance it has travelled?
7
d  10 10  1000m
10
• What is the displacement?
7
s  10 10  0.01m
10
The real stuff
• Treating real gases as ideal is a simplification
but a very useful one
• This is what physics is, useful simplification!
Quick Check 1
• A ball of mass 0.2 kg travels at 2 ms-1 towards
a ball and bounces back at the same speed.
Show that it gives a momentum of 0.8 kg ms-1
to the wall
Quick Check 2
• Show that 20 mol of ideal gas contains 12 x
10^24 molecules.
Quick Check 3
• A mole of hydrogen occupies a volume of
0.024 m3 at a pressure of 105 Pa and a
temperature of 300 K. Show that the mean
kinetic energy of a hydrogen molecules is
about 10-20 J.
Quick Check 4
• Show that the root mean square speed of H2
molecules (m = 3x10-27 kg) is about 2.6 kms 1. This is less than the escape velocity from
the Earth, which is about 11 ms-1. Explain why
hydrogen molecules do nethertheless escape
from the atmosphere.
Quick Check 5
• Show that a typical molecule travelling at
about 500 ms-1 will take only 20 ms to cross a
room 10 m wide. Explain why a given
molecule will actually take very much longer
than this to travel only a few cm.
Quick check 6
• Show that in a mixture of oxygen (Mr = 32)
and hydrogen (2) molecules the root mean
square speeds of the two kinds of molecules
will differ by a factor of four. Which is the
fastest?