Kinetic Molecular Theory of Gases/Ideal Gases
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CHEMISTRY 2000
Topic #2: Intermolecular Forces – What Attracts Molecules to Each Other?
Spring 2009
Prof. René Boeré
In an ‘ideal gas’:
 The distance between gas particles (atoms or molecules) is much larger than the
size of the particles. The gas particles therefore occupy a negligible fraction of
the volume.
 Gas particles are in constant motion.
motion
 Gas particles do not interact except when they collide.
 When gas particles collide with each other (or the walls of their container),
collisions are elastic (no energy is lost).
 Gas particles in a sample move at
different speeds, but the average
speed is proportional to temperature.
All gases have the same average
kinetic energy at a given temperature.
Law of Large Numbers (as applied here)
The behaviour of a system containing many molecules is
unlikely to deviate significantly from the behaviour predicted
from the statistical average of the properties of the
individual molecules.
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Kinetic Molecular Theory of Gases/Ideal Gases
Kinetic Molecular Theory of Gases/Ideal Gases
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The kinetic energy of a particle depends on its mass and speed:
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K  mv 2
2
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where
h K is
i ki
kinetic
ti energy, m iis mass andd v iis speed.
d
Speeds of molecules are distributed statistically in
a Maxwell-Boltzmann distribution:
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There are three different ways to look at the speed of gas particles in a sample:
 vmp = ‘most probable’ speed
 vav = ‘average’ speed
 vrms = ‘root mean square’
q
speed
p
((square
q
each pparticle’s speed
p
then calculate the
average then take the square root)
Because kinetic energy is proportional to v2, it makes most sense to use vrms in kinetic
energy and temperature calculations.
1 2
3
K  mvrms
 RT
2
2
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When the temperature increases, the average speed of the gas particles increases and
the distribution of speeds spreads out.
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where K is the average kinetic energy of gas particles, m is the mass of one particle, R
is the ideal gas constant and T is temperature (in K).
N t that,
Note
th t att a constant
t t ttemperature:
t
 particles with __________ mass will have __________ vrms
 particles with __________ mass will have __________ vrms
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Recall that pressure is force exerted on an area. (P=F/A)
Kinetic Molecular Theory of Gases/Ideal Gases
Kinetic Molecular Theory of Gases/Ideal Gases
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The Maxwell-Boltzmann equation can be derived from the kinetic-molecular theory of
gases:
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2
Nmvrms
Nmv rms
PV 
P
3
or
3V
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where P is pressure, V is volume, N is the number of gas particles, m is the mass of
one particle, and vrms is the root-mean-square speed.
Since the Maxwell-Boltzmann equation and the ideal gas law both describe ideal
gases, they can be related:
PV 
Therefore:
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Nmvrms
3
and
The mass of one particle (m) multiplied by the number of particles (N) gives the total
mass of gas particles. When this total mass is divided by the number of moles of gas
particles (n), we get the molar mass for the gas particles (M).
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2
Nmvrms
Nmvrms
Since
`  nRT
can be rewritten as
 3RT
n
3
3RT
2
vrms 
We get:
or
Mvrms
 3RT
M
Note that, in order for the units to cancel out, M must be converted to kg/mol. e.g.
Calculate vrms for N2 at 20 °C.
PV  nRT
2
Nmvrms
 nRT
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Kinetic Molecular Theory of Gases/Ideal Gases
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Kinetic Molecular Theory of Gases/Ideal Gases
You have a sample of helium gas at 0.00 ˚C. To what temperature should the gas be
heated in order to increase the root-mean-square speed of helium atoms by 10.0%?
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Molar heat capacity (Cp at constant pressure) refers to the amount of heat required to
raise the temperature of a substance:
H
Cp 
T
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If all of the internal energy of a monoatomic ideal gas is translational kinetic energy,
energy show
that all monoatomic ideal gases (e.g. noble gases under normal conditions) have the
same molar heat capacity.
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Nonideal Gases
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Nonideal Gases
The ideal gas law and kinetic molecular theory are based on two simplifying assumptions:
 The actual gas particles have negligible volume.
 There are no forces of attraction between gas particles.
These two assumptions are not always valid.
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Under what conditions would you expect these assumptions to no longer apply?
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Under ideal conditions, the volume of a container of gas is almost 100% empty space – so
the volume available for gas particles to move into is the entire volume of the container.
Under nonideal conditions, there is less “available volume” in the same sized container. We
can correct for this “excluded
excluded volume”
volume by adding a term to the ideal gas law:
P V  nb   nRT
where b is an experimentally determined constant specific to the gas. This constant tends to
______________ as the size of the gas particles increases.
The effect of this “excluded volume” is to _________________ the pressure of the gas.
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Nonideal Gases
Real gases: van der Waals Equation of State
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Under ideal conditions, particles in a gas sample don’t experience intermolecular forces.
Under nonideal conditions, the gas particles are close enough together that they do
experience intermolecular forces. We can correct for these attractive forces by adding a

term to the ideal gas law:
n2 
 P  a 2 V  nRT
V 

where a is an experimentally determined constant specific to the gas. This constant tends to
______________ as the polarity of the gas particles increases.
The effect of the intermolecular forces is to ________________ the pressure of the gas.
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The Kinetic Molecular theory ignores: (1) the actual volume occupied by the gas molecules
(2) intermolecular forces of attraction
The latter means that molecules can affect each other when they pass closely, not just when
they directly collide; this acts like gravitational forces between moving planets, etc.
Johannes van der Waals (1837-1923) developed an empirical correction to the ideal gas law
The van der Waals equation of state has correction factors for inherent volume (b) and
intermolecular forces (a)
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
n 
P

a

V   (V  bn )  nRT


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Nonideal Gases
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Nonideal Gases
When both corrections are applied, we get the van der Waals equation:

n2 
 P  a 2  V  nb   nRT
V 

The table below gives some sample values for a and b
a (kPa·L2·mol-2)
b (L·mol-1)
Hydrogen
H2
24.8
0.02661
Methane
CH4
228
0.04278
Ammonia
i
NH3
423
0.03707
Water
H2O
554
0.03049
Sulfur dioxide
SO2
680
0.05636
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a (kPa·L2·mol-2)
b (L·mol-1)
Hydrogen
H2
24.8
0.02661
Ammonia
NH3
423
0.03707
You have two 1.00 L containers at 298 K. One contains 1.00 mol of hydrogen gas while the
second contains 1.00 mol of ammonia gas.
(a)
Calculate the pressure in each container according to the ideal gas law.
(b)
Calculate the actual pressure in each container.
container
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Brownian Motion
Brownian Motion
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The kinetic-molecular theory doesn’t just apply to gases. Brownian motion is the
random motion of small particles (e.g. micrometer-sized) in a fluid.
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Einstein showed that the mean squared displacement is given by:
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The HIV-1 virus has a radius of about 50 nm. The viscosity coefficient of water at 20 °C
is 1.00 mPa·s. How far will a particle of HIV-1 typically travel in 1 minute?
x 2  2Dt
where D is the diffusion coefficient of the particle in that fluid. It can be found using the
equation below
D
kBT
6 r
where r is the radius of the particle and  is the viscosity coefficient of the solvent
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Properties of Liquids: Surface Tension
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Properties of Liquids: Surface Tension
Several properties of liquids can be explained by looking at intermolecular forces. For
example, we know that small samples of liquid tend to be round (e.g. water droplets).
Why?
Molecules within a liquid are attracted to all of the molecules around them via
intermolecular forces. Molecules on the surface of a liquid can only interact with the
molecules immediately below them and beside them. As such, the surface molecules
feel a net attraction toward the interior of the liquid and act as a “skin” for the liquid.
The energy required to break through this surface “skin” is referred to as surface
tension.
H
H
O
H
O
H
H
H
O
H
O
H
H
H
O
O
H
H
H
H
O
smooth surface;
lots of intermolecular attractions
H
H
O
H
O
H
H
H
O
O
H
H
O
H
H H
H
O
O
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A sphere is the shape with the smallest surface area-to-volume ratio – giving the
smallest ratio of surface molecules to bulk molecules (i.e. non-surface molecules).
This maximizes the intermolecular forces:
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This effect can also be seen when looking at the convex (“inverted”) meniscus of
mercury (Hg) in a glass test tube:

H
H
surface tension broken;
intermolecular attractions interfered with
Some insects take advantage of surface tension to literally walk on water!
Note that water behaves differently
because water molecules also
experience intermolecular forces
attracting them to the glass!
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Properties of Liquids: Surface Tension
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Properties of Liquids: Capillary Action
What happens if we add a solute to the liquid (now a solvent)?
 If the intermolecular forces between the solute and the solvent are stronger than
the intermolecular forces between solvent molecules, the solute will be pulled into
the solution away from the surface.
 If the intermolecular forces between the solute and the solvent are weaker than
the intermolecular forces between solvent molecules, the solute will tend to stay at
the surface and is referred to as a surfactant (“surface active agent”).
Surfactants reduce surface tension.
 Detergents are important surfactants (in water). They consist of a long nonpolar
“tail” and a small polar “head” (often ionic).
e.g. dodecylsulfate ion
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When a liquid is in contact with a solid surface, we must also consider intermolecular
forces between the liquid molecules and the molecules of the solid.
 If the liquid molecules are more strongly attracted to the surface than to each
other, wetting will be observed as the liquid sample spreads over the surface to
maximize those attractions.
e.g. water in a glass test tube
 If the liquid molecules are more strongly attracted to each other than to the
surface, non-wetting will be observed as the liquid sample appears to ‘bead’ on
the surface.
e.g. mercury in a glass test tube
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Properties of Liquids: Capillary Action
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The familiar concave meniscus of water on glass is an example
of wetting leading to capillary action – specifically, capillary rise.
 Glass consists of polar Si-O bonds so, if a narrow glass
tube is placed in water, the water molecules are attracted to
the glass. Other water molecules are attracted to the
initially “stuck” molecules, and are pulled up by surface
tension. This continues until the force of gravity pulling
water molecules down is equal to the force of surface
tension pulling water molecules up.
 Because the force of attraction between the water and the
glass has to support the weight of the column of water (i.e.
counteract the force of gravity), narrow capillaries allow the
liquid to rise higher (compared to wide capillaries).
A non-wetting liquid-surface combination will give capillary
depression. How do you think that works?
Properties of Liquids: Viscosity
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H
H
O
H
O
O
H O
O
H
O
O
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Si
O
Si
H
O
Viscosity is a liquid’s resistance to fluid flow. Liquids with high viscosities tend to be
thick and difficult to pour.
Viscosity increases as intermolecular forces increase. As a result, viscous liquids tend
to be either large or polar (or both).
eg
e.g.
gasoline
vs
vs.
mineral oil
(nonpolar, ~8 carbons)
(nonpolar, ~15-40 carbons)
O
H
O
Si
H
H
O O
O
H
H H
H
O
H
simplified structure for glass
(it's much more 3-dimensional!)
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