Ideal Gas Equation
Putting the preceding empirical laws together we
obtain:
PV
= n × constant
T
The constant in the above equation is known as the
gas constant, R = 8.314 J K mol-1.
If, instead of using number of moles (n), we use the
total number of molecules, N, the constant is
different and is called the Boltzmann constant, kB
kB = R / NA = 8.314 J K-1 mol-1 / 6.022×1023 mol-1
= 1.381×10-23 J K-1
B
Thus we obtain...
The Ideal Gas Equation:
PV = nRT
or
PV = NkBT
9. What is the molar volume of a gas at 298K
and atmospheric pressure (say 1 bar)?
Uses of the Ideal Gas Equation
Calculating the density of a gas:
Density (d) =
Mass (m)
Volume (V)
Number of moles (n) =
mass (m)
Molar Mass (M)
PV = nRT
From
→ PV = (m / M ) RT
Therefore
PM
d = m /V =
RT
Determining Molecular Mass:
If we know the mass and volume of a gas at known
temperature and pressure, we can calculate the
molar mass
We can also determine the amount of gas produced
in a chemical reaction
Mixtures of Gases
Dalton (1801) showed that in a mixture of
unreactive gases, each gas obeys the ideal gas law.
There is a pressure called the partial pressure
associated with each component gas of a gas
mixture.
Dalton’s Law:
The total pressure of a mixture of gases is the
sum of their individual partial pressures.
P = ∑ Pi
i
So, for a gas mixture made up of gases A and B:
RT
PB = nB
and
V
RT
P = PA + PB = (nA + nB )
V
RT
PA = nA
V
We can define the mole fraction, x, of a gas:
PA
nA
=
= xA
P n A + nB
Kinetic Molecular Theory
The preceding observations on the physical
properties of ideal gases can be described at a
molecular level using kinetic molecular theory:
1. Gases consist of molecules in continuous,
random motion
2. The volume of the gas molecules themselves is
negligible compared to the volume of the
container that the gas occupies
3. Collisions between gas molecules are elastic –
the energies of the colliding molecules may
change, but the total energy of both molecules
remains the same
4. Molecules in a gas do not attract or repel each
other
5. The average kinetic energy of gas molecules is
proportional to the absolute temperature
Applying the kinetic molecular theory to the
empirical gas laws:
V ∝ 1/ P
Boyle’s Law:
As the volume increases, the larger separation
between molecules means fewer collisions and
lower pressure
V ∝T
Charles’ Law:
To keep the pressure constant as temperature is
increased, the gas must be allowed to expand
P ∝T
Gay-Lussac’s Law:
An increase in temperature implies that molecules
have higher kinetic energy, thus a higher rate of
collisions and hence a higher pressure
Dalton’s Law:
Owing to negligible intermolecular forces between
gas molecules, molecules in a mixture do not
interact
Other physical properties accounted for by the
kinetic molecular theory:
Compressibility:
The large intermolecular separation allows gases to
be compressed easily
Diffusion:
The continuous, random motion of gases means that
they will rapidly occupy any enclosed volume
Real Gases
Up to now, we have assumed that gas molecules do
not attract or repel each other and their molecular
volume is infinitesimal. Gas properties have been
described by the ideal gas equation:
PV = nRT
which is only reasonable at moderate pressures.
At high pressures, the volume of the molecules
themselves becomes more important, as do the
attractive forces between gas molecules.
We can see this non-ideal behaviour most easily if
we look at the density of an ideal gas and compare
it to a real gas.
The gas density is
PM
d=
RT
Consider: How does gas density vary with pressure?
Change in density with pressure for real gases:
Effect of temperature on gas density:
We can take the non-ideal behaviour of real gases
into account using correction factors for the
pressure and volume of the gas.