Kinetic Theory and Gas
Pressure
Objectives
(d) define the term pressure and
use the kinetic model to explain
the pressure exerted by gases;
(i) state the basic assumptions of
the kinetic theory of gases;
Outcomes
ALL MUST
Be able to define pressure.
Be able to use the kinetic model to describe
pressure.
MOST SHOULD
Be able to state the basic assumptions of the
kinetic theory of gases.
Be able to use the kinetic model to explain the
pressure exerted by gases.
Be able to select and use the equation p=1/3ρv2
SOME COULD
Be able to derive the equation p=1/3ρv2
Pressure
P (Pa) = F (N) / A (m2)
Number of
molecules
Speed (c)
cp the most
probable speed
c the mean
speed (the
average speed
of all of the
molecules)
crms the rms speed
ie the root mean
square speed. A
useful concept.
Listen carefully
and watch how it
is calculated.
The Kinetic Theory of Gases
Question (We do not have enough particles here
to be realistic but it will illustrate the point!)
6 particles have the following speeds: 600, 650,
650, 700, 725, 750ms-1. Determine the most
probable speed cp, the mean speed and the root
mean square speed crms.
Thermodynamics
The Kinetic Theory of Gases
6 particles have the following
speeds: 600, 650, 650, 700,
725, 750ms-1.
The most probable speed cp = 650ms-1 as there are more
particles going at that speed than any other.
Thermodynamics
The Kinetic Theory of Gases
6 particles have the following
speeds: 600, 650, 650, 700,
725, 750ms-1.
The mean speedc
= (600 + 650 + 650 + 700 + 725 + 750) / 6 = 679.16ms-1.
Thermodynamics
The Kinetic Theory of Gases
6 particles have the following
speeds: 600, 650, 650, 700,
725, 750ms-1.
The mean square speed
2
2
2
2
2
2
= (600 + 650 + 650 + 700 + 725 + 750 ) / 6
= 2,783,125 / 6
= 463854.17
The root mean square speed crms
= 463,854.17
= 681.07ms-1
Thermodynamics
Assumptions of the kinetic theory of an IDEAL GAS.
1
1
A Gas consists of particles called molecules.
2
2
The molecules are in constant random motion. As
many travelling in one direction as any other. The centre of mass
of the gas is at rest.
3
Intermolecular forces are negligible.
4
The duration of collisions between molecules is negligible.
5
Molecules move with constant velocity in between
collisions.
6
6
The volume of gas molecules is negligible
compared with
the volume of the gas.
7
All collisions are totally elastic.
8
Newtonian mechanics can be applied to the collisions.
The Kinetic Theory of Gases
y
l
l
m,cx
If these assumptions
are correct, we
should be able to
prove the equation
of state for an ideal
gas from these
assumptions!
WOW!
l
x
z
Thermodynamics
y
The Kinetic Theory of Gases
l
l
Consider the change in momentum as the particle
hits the wall
m,cx
l
p = mcx - -mcx = 2mcx
Time interval between collisions t = 2l/cx
x
z
Now F=dp/dt from Newton’s second law so the force F
of one molecule hitting the wall is given by:
F=p/t = 2mcx / 2l/cx = mcx2 / l
2
(mcx
2
2
mcx
3
But p = F/A so p =
/ l) / l =
/l
If there are N of them then
3
2
2
2
2
p = (m / l ) (cx1 + cx2 + cx3 +...........+ cxN )
Note that
(cx12 +
3
cx22
2
+
cx32
+...........+
so p = (m / l ) Ncx = N(m/V)cx
2
cxN2)
/ N =cx
2
EQ(1)
Thermodynamics
y
The Kinetic Theory of Gases
3
2
so p = (m / l ) Ncx = N(m/V)cx
l
l
2
EQ(1)
m,cx
l
Pythagoras’ Theorem will show that
c2 = cx2 + cy2 + cz2 - considering a general direction
2
2
2
2
and so c = cx + c y + cz
2
2
2
Due to the large no. of particles,cx =c y = cz
x
z
2
so c = 3 cx
2
from EQ(1) we get p
So

1 Nm 2
c
3 V
pV

1
2
Nmc
3
Thermodynamics
Outcomes
ALL MUST
Be able to define pressure.
Be able to use the kinetic model to describe
pressure.
MOST SHOULD
Be able to state the basic assumptions of the
kinetic theory of gases.
Be able to use the kinetic model to explain the
pressure exerted by gases.
Be able to select and use the equation p=1/3ρv2
SOME COULD
Be able to derive the equation p=1/3ρv2