How does the Ideal Gas equation explain real gases?

1. Topic 3 - Thermodynamics
3.3 – The Ideal Gas

2. Gas Pressure
● Pressure is defined as the force per unit area.
● Pressure is measured in Pascals (Pa)
Gas pressure arises because of collisions
between particles (causing the force) and the wall
of the container (the area over which they act)
p=
F
A

3. An Ideal Gas
● Physicists made a set of observations of gases
from which 4 assumptions are made to define
the “ideal gas.”
● A pure gas contains identical molecules in
continuous random motion. (no one particle is more
special than another)
● All collisions are elastic (energy is conserved)
● The volume of the particles is negligible compared
to the volume of the container. (it is compressible)
● There are no forces on the molecules except during
collisions (the particles are very far apart)

4. An Ideal Gas
● Consider a box of
dimensions x, y & z as
shown
● A single ideal gas
particle mass m is
moving in the box with
speed u parallel to the
x direction.
x
z
y u

5. An Ideal Gas
● The molecule collides
with the blue wall as
shown.
● Its initial momentum is
mu, and its final
momentum in -mu
● Its change in
momentum is
therefore 2mu x
z
y u

6. An Ideal Gas
● The molecule travels a
distance 2x between
collisions with the blue
wall.
● The time between
collisions is therefore:
● 2x/u
x
z
y u

7. An Ideal Gas
● The force exerted on
the blue wall is the
rate of change of
momentum.
●
x
z
y u
F=
2mu
2x/u
=
mu
2
x

8. An Ideal Gas
● The pressure on the
blue wall is therefore:
● Where V is the volume
x
z
y u
P=
F
A
=
mu
2
xyz
=
mu
2
V

9. An Ideal Gas
● In a real gas there are N
molecules moving
randomly. On average
only 1/3 of these move in
the x direction.
● The molecules are not all
moving with speed u but
have an average (mean
square) speed <c2
>
x
z
y u
pV =
1
3
N m 〈c
2
〉

10. Molecular Speed
● For one mole of gas, the equation becomes:
● This could be written as:
● Where ½ m<c2
> is the average kinetic energy
pV =
1
3
N A m 〈c2
〉
pV =
2
3
N A×
1
2
m〈c
2
〉

11. Molecular Speed
● From other macroscopic experiments it can be
shown that:
● These two equations for ideal gases must
equate.
● Therefore:
pV =nRT
nRT =pV =
2
3
NA
1
2
m〈c
2
〉
1
2
m〈c
2
〉=
3
2
R
N A
T

12. Molecular Speed
● The ratio of the two constants (R over NA
) is
known as the Boltzmann constant k
● k=1.38 x 10-23
JK-1
● That is Kinetic Energy is proportional to
absolute temperature
1
2
m〈c2
〉=
3
2
k T

13. Summary of Ideal Gases
● For a real gas, the ideal gas rules can be used to give
approximate answers.
● An increase in volume will cause a longer time between
collisions, so the collisions will be less frequent, so the
pressure will decrease.
● An increase in temperature, will cause a higher KE, so the
time between collisions will increase and the force with
which they strike the container will increase. The pressure
will therefore increase.
● An increase in volume at constant pressure will cause the
particles to slow down, therefore causing a decrease in
temperature.