KINETIC THEORY OF
GASES
&
IDEAL GAS LAW
Macroscopic definition of gas state: no
shape, no volume
Microscopic definition: molecules are separated by large distances,
fly freely & interact through elastic collisions only.
Is a mixture of gases always homogeneous, always heterogeneous
or either?
Why are gases compressible, while liquids & solids are
incompressible?
How does density of gases compare to that of liquids and solids?
Objectives:
- To introduce the basics of the kinetic theory of
gases:
- To relate pressure, as macroscopic, measurable parameter of
gas state, to molecular motion
- To relate temperature to the kinetic energy of moving gas
particles
- To establish the relationship between the three
macroscopic parameters of gas state:
temperature, pressure
&
volume
T [K]
P [atm] V [L]
through the state equation of ideal, or perfect gas, & to
introduce the definition of ideal gas.
Why important?
1. The gas state is the only state for which quantitative
description & prediction of properties is possible in simple
theoretical terms. The behavior of liquids & solids is more
complicated & hard to explain quantitatively in terms of the
properties of constituent particles (atoms, molecules or ions).
2. It is important in chemical calculations. Gas laws permit us to
perform stoichiometric calculations of reactions involving gases, at
arbitrary conditions (T & P).
KINETIC THEORY OF MATTER
IN APPLICATION TO GAS
1. Gases consist of large numbers of molecules that occupy a
volume at least 1000 times larger than they would occupy in solid
or liquid state. Molecules of gases are far apart. Most of volume
occupied by a gas is empty.
2. Molecules of a gas are in constant motion, traveling rapidly
along straight lines in random directions & with a random
distribution of their speeds (this is why the theory is called
"kinetic" - from greek "moving")
3. The only way gas particles interact with each other & with
the walls of a container is through elastic collisions. In elastic
collisions the particles exchange their kinetic energy, but the sum
of kinetic energy of two colliding particles is preserved & not
converted into other types of energy. There are no forces of
attraction or repulsion between the gas particles, except the elastic
repulsion during the collision.
4. The average kinetic energy of gas molecules is
proportional to absolute temperature.
Ek = const .T
Kinetic energy of a single molecule is
Ek= ½ mv
2
where m & v are mass & speed of the molecule.
In an individual gas (not a mixture of several gases!), the kinetic energy
depends on speed only (since all masses are the same).
Molecules in any gas travel with various, randomly
distributed speeds. This distribution is expressed by a
distribution curve, with a peak corresponding to the
most probable speed. The higher T of gas is, the higher
is the most probable speed: the distribution curve shifts
to the higher speeds.
Different gases at the same temperature have the same average
kinetic energy of their molecules. However their average speeds
will be different, since different are their masses: the higher
the mass is, the lower is the speed, since kinetic energy is
the same.
PRESSURE
Gas exerts pressure on the wall of the container.
PRESSURE (in general, not only for gas)
IS FORCE ACTING ON
THE UNIT AREA
p = F/s
With the same effort (force) we can make a hole in a
sheet of paper with a sharp needle, though not with a
thick stick. This is because the pressure is inverse
proportional to the area of contact.
For the same reason a sharp knife cuts well, though a
blunt one does not.
Units of Pressure
pascal, Pa;
millimeter of mercury, mm Hg or torr;
atmosphere, atm;
bar.
Conversion factors :
1 atm = 1.01325 bar = 101.325 kPa = 760 torr = 760 mm Hg
to be used in precise calculations.
Approx.:
1 atm

1 bar  100 kPa
 760 mm Hg
The pressure units are related
to atmospheric pressure
determined by the weight of the
air column above any place on
the Earth surface.
1 atm is the pressure of air at
the sea level. It goes down at
higher altitudes.
Barometer (particular case of a manometer) –
instrument to measure gas pressure
GAS PRESSURE is due to multiple impacts by the moving
molecules against the wall of the container.
Gas pressure increases when:
- the molecules hit the wall more frequently. This
happen when there are more of them in a given
volume. If the number of molecules is fixed (constant
mass of gas), the frequency of encounters of molecules
with the wall, & also the pressure, will increase when the
volume of gas is reduced (gas compressed).
- the energy of each impact increases. This energy
depends on temperature: the higher the temperature, the
stronger each impact, & higher the net pressure.
Hence, we expect that:
PRESSURE BY A GIVEN AMOUNT OF GAS, IN A
SEALED CONTAINER, INCREASES WHEN
TEMPERATURE INCREASES,
&
INCREASES WHEN VOLUME IS REDUCED.
These are the expectations from the kinetic theory of gas.
Experiments show this is true qualitatively for all gases, & there
are simple quantitative relationships between the three
variables:
T, V , P
for any gas at reasonable conditions.
These relationships are called GAS LAWS.
There are 4 of them:
Boyle's Law: V - P relationship
The pressure of a given mass of gas is inverse proportional to
its pressure, or
PV = const, at any given temperature.
P1V1 = P2V2
at T = const
Hence, if we know P & V at some temperature, & then compress
the gas to, say, half of its initial volume, the pressure will increase
two-fold.
This relationship is graphically presented as a
HYPERBOLA
This hyperbola is gas isotherm.
EXAMPLE:
A mass of hydrogen occupied a volume of 1L at a pressure of 4
atm. Then the gas was allowed to double its volume by effusion to
another container of the same volume. What will be the final
pressure in the system?
p1=4 atm, V1=1L, V2=2L, p2 ?
p1V1 = p2V2 or p2 = p1V1/V2
p2 = 4atm 1L / 2L = 2 atm
Charles' Law:
V - T relationship
The volume of a given mass of gas is directly proportional to
the temperature at any constant pressure.
V1/T1 = V2/T2
at p = const
or V/T = const at p = const.,
This is the equation for GAS
ISOBAR
This is how the absolute scale of temperature,T(K) has been
established, & why this scale is also called gas scale:
Charles measured temperature in °C, and found that for various
gases, or for one & the same gas but in different amounts, there is
one and the same temperature, (-273°C) at which the
extrapolation of the straight lines gives zero volume.
absolute zero:
0 K = -273°C
EXAMPLE:
0.500L of a gas was heated from 250 to 500K at constant pressure.
What volume it will ocupy at this new temperature?
V1 = 0.5 L, T1 = 250 K,T2 = 500 K, V2 -?
V1/T1 = V2/T2 or V2 = (V1/T1) T2
V2 = (0.5 L/250 K ) 500 K = 1.0 L
Gay-Lussac's Law: P - T relationship.
Pressure of a given volume of gas is directly proportional to
absolute temperature
p1/T1 = p2/T2
at V = const ISOSTER
All 3 relationships can be combined: For a given mass of gas,
pV/T = const.
or
p1V1/T1 = p2V2/T2
Combined Gas Law
AVOGADRO LAW
Equal volumes of gases, at the same to &
pressure, contain equal number of
molecules.
1 mole of any substance contains the same
number of molecules - Avogadro number
of molecules (by definition of mole).
Combining these two statements,
One mole of any gas, at the
o
same t & p, occupies the same
volume.
Experimentally found:
1 MOLE OF ANY GAS AT STANDARD
TEMPERATURE & PRESSURE
(0°C, 1 atm) OCCUPIES 22.4
L
This permits to extend stoichiometric calculations
over volumes of gases:
Example:
CaCO3(s)+2HCl(aq)  CaCl2(aq)+H2O+CO2
If 200. g CaCO3 were decomposed, what volume of
CO2, in L, was released at STP?
CO2 (L) =
200gCaCO3(1molCaCO3/100gCaCO3).(1molCO2/1molCaCO3) . 22.4 L/mol
= 44.8
L CO2
In gas reactions, volume ratios are the
same as mole ratios.
EXAMPLE: The pressure of a gas in a 2.0 L container is 3.0 atm
at 273 K. What will be the pressure of that gas if the volume
available is increased up to 6.0 L, and the temperature is raised up
to 546 K?
p1=3.0atm,V1 =2.0L, T1 =273K, V2=6.0 L, T2 =546K, p2-?
p1V1/T1 = p2V2/T2 or
p2 = (p1V1/T1) (T2/V2)
p2=(3.0atm 2.0 L/273 K)(546K/6L)= 2 atm
COMBINED GAS LAW
1 mole of any gas
at STP (T= 273 K & P = 1 atm)
occupies the volume of 22.4 L.
Hence, for one mole of gas:
pV/T=1atm x 22.4L/273K=
0.0821 L atm/K
This is true for 1 mole of any gas at any set of conditions.
UNIVERSAL GAS CONSTANT
.
.
R=0.0821L atm/K mol
or pV/T = R, or pV = RT
(for 1 mol of gas)
If we have n moles of gas,
pV = nRT
Mass of gas can be presented as molar mass M multiplied by the
number of moles, n, i.e. m = nM, or n = m/M, and
pV = (m/M)RT
This is the basis to determine MOLECULAR MASS of gases:
M = m RT/pV
Since m/V = D, M = DRT/p
M1/M2 = D1/D2