Kinetic Theory
States of Matter
(i) Solid
 Molecules in solid are closely packed in a regular
structure by strong intermolecular attraction forces.
 Molecules can only vibrate about fixed positions, they
cannot move freely from one place to another.
∴ Solid has a fixed shape.
(ii) Liquid
 Molecules in liquid are loosely packed, without

regular structure.
Molecules can move slowly from one place to
another.
∴ Liquid has no fixed shape.
(iii) Gas
 Molecules in a gas are very far apart.
 Gas molecules move randomly at a high speed.
∴ Gas occupies all the space of the container.
Kinetic theory model of an ideal gas
(i) Assumptions of the kinetic model of an ideal gas


Gas consists of molecules and they are in random motion.
The intermolecular attraction forces between molecules are negligible.


The size of a gas molecule is very small and negligible.
The collisions between the gas molecules and the walls of the container are
perfectly elastic.
The duration time of collision between gas molecules and the wall is
negligible compared with the time spent between collisions.

(ii) Qualitative explanation of pressure exerted by gas molecules
 Gas exerts a pressure when the gas molecules bombard and rebound from
the walls of the container.
 The pressure is measured by the average rate of change of momentum of the
gas molecules per unit surface area.
(iii) Derivation of the Kinetic Theory of Equation


A molecule with velocity c is resolved into 3 perpendicular components u, v
and w.
Consider the x-direction: the molecule with speed u hits the wall A and
rebounds with the same speed u but in opposite direction.
Change of momentum of the molecule during impact with wall A =
−(𝑚𝑢) − (𝑚𝑢) = −2𝑚𝑢
Time of flight of the molecule between 2 successive impacts with wall A =
2ℓ
𝑢
Rate of change of momentum of the molecule = (−2𝑚𝑢) ×
1
2ℓ
𝑢
=−
By Newton’s 2nd Law, average force on the molecule by wall A = −
By Newton’s 3rd Law, average force on wall A by the molecule = +

Average force on wall A due to the impact of N molecules:
̅̅̅2
𝑁𝑚𝑢
𝐹=
ℓ

Average pressure on wall A: 𝑝 = 𝐴 =
𝐹
̅̅̅̅
2
𝑁𝑚𝑢
𝑉
𝑚𝑢2
ℓ
𝑚𝑢2
ℓ
𝑚𝑢2
ℓ


For any molecule, 𝑐 2 = 𝑢2 + 𝑣 2 + 𝑤 2
By symmetry, mean-square speeds in 3 directions are equal as molecules
move randomly.
Pressure P =
1 2
𝑁𝑚( ̅̅̅
𝑐 )
3
𝑉
1
=3×
̅̅̅
2
𝑁𝑚𝑐
𝑉
1
, ∴ 𝑝𝑉 = 3 𝑁𝑚𝑐̅̅̅2
(iv) The Kinetic theory equation of an ideal gas



Kinetic theory equation:

1
𝑝𝑉 = 3 𝑁𝑚𝑐̅̅̅2

𝑝𝑉 = 3 𝑁𝑚𝑐 2
1
∴ ̅̅̅
𝑐2 = 𝑐2
Note that:
Root-mean-square speed of gas molecules = the average speed of gas
molecules
However, both of them follow similar changes with temperature.
Gas molecules at a certain temperature moves with different speeds. The
speed follow a distribution curve as shown below.
(v) Relation of root-mean-square speed with density
 In the kinetic theory equation, 𝑁𝑚 = 𝑀 which is the total mass of the gas.

If ρ is the density of the gas where 𝜌 =
𝑀
𝑉
Interpret temperature of an ideal gas
(i)
Average kinetic energy of gas molecules
 Average KE of gas molecules:

1 ̅̅̅2 1
̅̅̅̅
𝐾𝐸 = 𝑚𝑣
= 𝑚𝑐 2
2
2
Combine the two equations for an ideal gas:
1

𝑝𝑉 = 𝑛𝑅𝑇, 𝑝𝑉 = 3 𝑁𝑚𝑐 2

1
3𝑅𝑇
̅̅̅̅
𝐾𝐸 = 2 𝑚𝑐 2 = 2𝑁
𝐴

The average KE of gas molecules:
3𝑅
3
𝐾𝐸𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑇, ̅̅̅̅
𝐾𝐸 ∝ 𝑇, 𝐾𝐸 = 𝑘𝑇
2 𝑁𝐴
2

For an ideal gas:


Average kinetic energy of gas molecules is independent of the type of
gas
 Average kinetic energy of gas molecules depends on the absolute
temperature only
 All ideal gases have the same average kinetic energy at the same
temperature
The absolute temperature can be interpreted as the average KE of ideal gas.
̅̅̅̅ ∝ 𝑇
𝐾𝐸
(ii) Root-mean-square speed of gas molecules
 Since different gases have different molecular mass, therefore different
gases have different velocities at the same temperature.
 Combine the two equations for an ideal gas:
1

𝑝𝑉 = 𝑛𝑅𝑇, 𝑝𝑉 = 3 𝑁𝑚𝑐 2

𝑐=√ℳ

𝑐 ∝ √𝑇 for constant ℳ (same type of gas)

𝑐 ∝ √ℳ for constant T
3𝑅𝑇
1
(iii) Internal energy of an ideal gas
 Internal energy of a gas is equal to total molecular PE and total KE of the
gas molecules.
 Molecular PE arises from the intermolecular attraction forces.
 KE arises from the random motion of the gas molecules.
 Since the inter-molecular forces in an ideal gas are negligible

∴ ideal gas has no molecular PE.
Internal energy of an ideal gas = total KE of the gas molecules
3 𝑅
3
∴ 𝑈 = 𝑁 × 𝐾𝐸 = 𝑁 × 2 𝑁 𝑇 = 2 𝑛𝑅𝑇
𝐴
Relation between different gas properties
(i) Chart for analyzing gas properties
(ii) Compressing a gas at constant temperature
 Qualitative explanation of the Boyle’s Law:
 When the gas is compressed, volume decreases. Distance between two
walls then decreases.



Gas molecules bombard the walls more frequently, thus pressure
increases.
As the temperature is constant, the KE and the speed of molecules remain
unchanged.
As the volume is decreased, the gas separation is decreased and the density
is increased.
(iii) Heating a gas at constant volume

Qualitative explanation of the Pressure Law:


As temperature increases, KE of molecules increases and the
molecules move faster.
 Gas molecules then bombard the wall of container more violently and
more frequently. Thus pressure increases.
As the volume is constant, the gas separation and the gas density remain
unchanged.
(iv) Heating a gas at constant temperature

Qualitative explanation of Charles’ Law:
 When temperature increases, the KE of gas molecules increases and
thus the molecules move faster.


Since the molecules bombard the wall more violently, the volume must
increase so that the molecules then bombard the wall less frequently to
give the same pressure.
As the volume is increased, the gas separation increases and the gas density
decreases.
Real gas
(i) Departure of real gas from ideal gas behaviour
 In the assumption for ideal gas, there are 2 assumptions that real gas cannot
be obeyed:
 The intermolecular attraction forces between molecules are negligible
for an ideal gas.

 The size of a gas molecule is very small and negligible for an ideal gas.
Actually for a real gas:
 The intermolecular attraction forces exist between molecules.

The gas molecules have sizes which is not negligible.
(ii) High pressure




At high pressure (high density), the gas molecules become very close.
The total size of the molecules becomes more significant compared with the
volume of the container.
The intermolecular attractive forces would also become significant at high
pressure.
The ideal gas behaviour can no longer be held by gases at high pressure.
(iii) Low temperature

At low temperature, the molecules move slower than they do at higher

temperature. Thus, the weak attractive forces between the molecules
become more significant.
The weak attractive forces would then tend to combine the gas molecules
together and condense to liquid.
The ideal gas behaviour can no longer be held by gases at low temperature.

(iv) Conditions to approach ideal gas behaviour

A real gas approaches ideal gas behaviour only at
 Low pressure
 High temperature
Brownian Motion
(i) Observation of Brownian Motion




Brownian motion is observed when a pollen grain is placed on water surface
and viewed under microscope.
The pollen grain is found to move in a zig-zag path.
The pollen grain must be bombarded randomly by millions of water
molecules which have much smaller sizes than that of the pollen grain.
Since the bombardments are random, it gives a random resultant force. Thus
the pollen grain ,moves in the zig-zag path.
(ii) Smoke cell
 Brownian motion is observed when a smoke cell
containing smoke particles is viewed under
microscope.
 The smoke particles are found to move in a zigzag path.

By kinetic theory, the large smoke particles must be bombarded randomly

by millions of air molecules which have much smaller sizes than that of the
smoke particles.
Since the bombardments are random, it gives a random resultant force. Thus
the pollen grain ,moves in the zig-zag path.
Note that the motion of molecules is not Brownian motion.

(iii) Importance of Brownian Motion
 Brownian Motion provides an evidence for the Kinetic Theory.
 Note that Brownian Motion is a macroscopic phenomenon.
Mechanical model of the Kinetic Theory
(i) Simulation of the kinetic theory of gas





Simulation of gas molecules: ball bearing
Simulation of the volume of the container: height of column
Simulation of the temperature: motor and platform
Simulation of the pressure: load and piston
Simulation of the Brownian motion: large polystyrene ball
(ii) Explanation of the rise of gas pressure


The rise of pressure is due to the bombardment of the ball bearings onto the
piston. Thus the ball bearings experience a change of momentum which
then produces a force. The force per unit area gives the pressure.
The height of the column would not be affected by the change of
atmospheric pressure.
(iii) Simulation of the Boyle’s Law


Without changing the motor speed, add some weight to the piston, the
volume of column decreases.
When volume decreases, distance between two walls decreases, thus the
ball bearings bombard the wall more frequently (but not more violently) to
produce a greater pressure.
(iv) Simulation of the Pressure Law


The motor speed is increased to simulate the increase of temperature.
When the motor speed is increased, the ball bearings gain more KE and
move with a greater speed. Thus they bombard the wall more frequently and
more vigorously to give a greater force. Therefore, greater load should be
added to maintain the same volume. Thus pressure increases.
(v) Simulation of the Charles’ Law



Under the same load the motor speed is increased to simulate the increase of
temperature. The height of the column is found to increase, representing, the
increase of volume.
When the motor speed is increased, the ball bearings gain more KE and
move with a greater speed. Thus they bombard the wall more frequently and
more vigorously to give a greater force.
To keep pressure constant, the volume has to increase so that the ball
bearings bombard the wall of the piston less frequently to give the same
force, and thus the same pressure.
(vi) Simulation of Brownian Motion

To simulate Brownian Motion, a large polystyrene ball should be put into
the column.

The large polystyrene ball then performs Brownian motion due to the
random collisions of the ball bearings to produce random resultant force.
(vii) Derivation of the mechanical model from ideal gas model
 The collisions between the ball bearings and the wall are not perfectly
elastic.
 The ball bearings themselves actually occupy some volume.