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Analysis of matter in terms of atoms in continuous motion
is called Kinetic theory
The theory is based on ideal gas but real gases follow the
theory at low pressures & at temperature far away from
liquefaction point
The kinetic theory has postulates:
There are a large number of molecules (N), each of mass
m, moving in random directions with varying speeds
Molecules are far apart from each other i.e. average
separation is much greater than diameter of each molecule
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Molecules are assumed to observe classical laws of mechanics
& only interact with each other when they collide. Although
molecules exert weak attractive forces on each other during
collisions, PE associated with these forces are small as
compared to KE of movement i.e. it is negligible
Collisions with another molecule or the wall of the vessel are
assumed perfectly elastic. The collisions are of very short
duration compared to time between collisions. This allows us
to neglect PE during collisions
Kinetic theory can be used to explain the gas laws.
For Boyle’s law: pressure exerted on a wall of a container of a
gas is due to constant bombardment of molecules. If V is
halved, molecules are closer together & twice as many will be
striking a given area of the wall per second i.e. pressure
doubles
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Kinetic theory can also be used to calculate pressure exerted by
gas:
𝑃=
1 𝑁𝑚𝑣ത 2
3 𝑉
(see Giancoli for derivation)
where 𝑣ҧ is average speed of molecules
After further manipulation:
1
𝑚𝑣ҧ 2
2
(see Giancoli for derivation)
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𝐾𝐸 =
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Making speed subject of formula:
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2
𝑣ҧ =
3𝑘𝑇
𝑚
=
=
3
𝑘𝑇
2
3𝑅𝑇
𝑀
(after mathematical manipuilation)
Average translational KE of molecules in random motion in an
ideal gas is directly proportional to absolute T of the gas
The higher the T, the faster the molecules (next slide)
Speed distribution
Maxwell
distribution of
speeds
• Temperature is a measure of the average kinetic energy of
gas molecules
1. A beam of electrons moving in the +ve x-direction impacts a target in a
vacuum chamber.
a)
If 1.25x1014 electrons travelling at a speed of 3x107m/s strike the target
surface during each brief pulse lasting 5x10-8s, what average force is
exerted on the target during each pulse?
b)
What average pressure is exerted on the beam spot, which has a radius of
4mm?
Mass of the electron = 9.11x10-31kg
NOTES:
Assume that all the electrons impacting on the surface are absorbed by the
surface
The bean spot is the target struck by the beam.
By Newton’s 3rd Law: Average force exerted is the reaction force (its negative)
a)
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F1e = -change in momentum /time = -∆p/∆t = -mvf – mvi/t
Fav = -N ∆p/∆t = -N m(vf-vi)/∆t = -Nm (0-vi)/∆t
Fav = - (1.25x1014)x(9.11x10-31)x (-3x107)/(5x10-8) N
Therefore Fav = 0.0683N
b)
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P = F/A = 0.0683N/πr2 = 0.0683/ π x (4x10-3)2
P = 1.36x103Pa
What is the average translational KE of molecules in an
ideal gas at 37 oC?
Solution
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𝐾𝐸 =
1
𝑚𝑣ҧ 2
2
3
= 𝑘𝑇
2
−21
=
3
2
1.38𝑥10−23 37 + 273
𝐾𝐸 = 6.42𝑥10
J
NB:
A mole of molecules would have a total KE of:
𝐾𝐸 = 6.42𝑥10−21 6.02𝑥1023 = 3900 J
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Root-mean-square velocity is the measure of the velocity of
particles in a gas which is most convenient for problem solving
within the kinetic theory of gases. It is defined as the square
root of the average velocity-squared of the molecules in a gas. It
is given by the formula.
Remember from vector notation, velocity becomes speed once
you average it out, because averaging takes care of both
magnitude and direction
So vrms is also written as urms where u is the average speed of the
molecules
Velocity Distributions - Distribution of
Molecular Speeds
u rms
3RT
= u =
M
ump =
2
2RT
M
ump : uavg : urms = 1.000 : 1.128: 1.225
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Thermodynamic temperature is
the variable needed for subjects
like heat transfer, because it is
the translational kinetic energy
which leads to energy transfer
from a hot area (larger
kinetic Energy, Higher
temperature, higher molecular
speeds) to a cold area (lower
molecular speeds) in direct
collisional transfer.
At a certain speed, the root-mean-square-speed of the
molecules of H2 gas in a sample of gas is 1055 m/s. Compute
the root-mean square speed of molecules of O2 gas at the same
temperature.
Solution
1. Find T for the H2 gas with a vrms = 1055 m/s
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2.
𝑇=
𝐻2
𝑣𝑟𝑚𝑠
2
𝑀𝐻2
3𝑅
=
1055 2 2
3 8.314
= 89249.1 K
Find vrms of O2 gas at the same temperature
𝑂2
 𝑣𝑟𝑚𝑠
=
3𝑅𝑇
𝑀𝑂2
=
3 8.314 89249.1
32
= 263.75 m/s
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Kinetic theory can also be used to explain the 4 states of
matter & accompanying phase changes
Homework: Explain states of matter using Kinetic theory
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Melting – commonly used to indicate changing from solid to
liquid
Normal melting point – temperature at which vapour pressure
of solid & vapour pressure of the liquid are equal
Freezing – changing from a liquid to a solid.
Melting and freezing occur at the same temperature
Boiling/evaporation – used to indicate changing from a liquid
to a gas/vapour
Normal boiling point – temperature at which vapour pressure
of liquid is equal to standard atmospheric pressure (101.325
kPa)
Boiling point is a function of pressure, at lower pressures,
boiling point is lower
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Evaporation takes place only at surface of a liquid or solid while
boiling takes place throughout the body of a liquid
What is the difference between a Gas & a Vapour?
Gas – state of matter that exists at normal room temperature
Vapour – produced by particles escaping from a state of matter that
is normally liquid or solid at room temperature
Condensation – used to indicate changing from a vapour to a liquid
Liquefaction – turning a gas to a liquid
Only happens at low temperature & high pressure situations
Sublimation – used to indicate when a substance changes directly
from a solid to a vapour e.g. "dry ice", solid CO2
Deposition – when a substance changes directly from a vapour to a
solid (opposite of sublimation) e.g. formation of frost
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A cooling curve is a line graph that represents change of phase
of matter, typically from a gas to a solid or a liquid to a solid
The independent variable (X-axis) is time & the dependent
variable (Y-axis) is temperature
See an example of a
cooling curve of castings
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The initial point of the graph is the starting temperature of
the matter noted as pouring temperature
When the phase change occurs there is a "thermal arrest”
i.e. temperature stays constant
Amount of energy required for a phase change is known as
latent heat
“Cooling rate" is the slope of the cooling curve at any point
Next slide give a practical cooling curve for a well known
substance
Cooling curve of Naphthalene
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Watch Video: - Water the great mystery on Sakai. Answer
Questions provided on the handout
Explain the change of phase using the Kinetic Theory of matter.
Indicate two areas where water does not seem to obey the tenets
of the kinetic theory. Explain these exceptions
Does the fact that water does not obey all the tenets of the kinetic
theory invalidate the theory? Discuss