Chapter 12 – Gases and Their Properties
In solids and liquids, molecules are tightly packed such that
every molecule is always interacting with other molecules. In
gases, on the other hand, the spaces between molecules are
generally much bigger than the molecules themselves:
The most common gas is air. Dry air consists of:
• nitrogen (N2, 78% by volume),
• oxygen (O2, 21% by volume),
• argon (Ar, 1% by volume),
• trace amounts of the other noble gases (He, Ne, Kr, Xe),
as well as H2, CH4 and N2O
Air can also contain water vapour (H2O, 0-7% by volume).
When describing the properties of a gas, we are interested in:
• number of moles
• volume (measured in L or m3)
• temperature (measured in K)
• pressure (measured in mm Hg, Torr, atm, Pa or kPa)
The SI unit for pressure is the pascal (Pa) but, because this is
such a small unit, other units are often used for convenience.
To convert between the different units for pressure, remember
that:
1 mm Hg = 1 Torr
1 kPa = 1000 Pa
1 Pa = 1 N/m2 = 1 J/m3
1 atm = 760 mm Hg = 101.3 kPa
Gas Laws
Centuries of studying gases have led chemists to develop a
variety of laws describing their behaviour:
Boyle’s Law: volume is inversely related to pressure
Charles’ Law: volume is directly related to temperature (in K)
Avogadro’s Law: volume is directly related to # moles
For any of these laws to apply, it is assumed that the properties
not mentioned remain constant.
e.g. If the pressure on a syringe of air is increased, the volume
of the air decreases (Boyle’s law). We assume that
temperature and the number of moles of gas do not change.
Ideal Gas Law
Often, more than one property of a gas is changed in an
experiment. (e.g. Heating the air in a balloon will cause the
volume and pressure to increase.)
By combining Boyle’s, Charles’ and Avogadro’s laws, we arrive
at the ideal gas law:
Boyle:
Charles:
combined:
Avogadro:
.
PV = nRT
therefore:
where R is the ideal gas constant whose value depends on the
units chosen for pressure and volume:
J
m3 Pa
R = 8.3145
= 8.3145
mol K
mol K
R = 0.0820574
L atm
mol K
***Note that the ideal gas law can be used to generate
any of the other gas laws.***
For any ideal gas:
Since R is a universal constant:
PV = R
nT
P1V1 = P2V2
n1T1
n2T2
Recall that, to use Boyle’s law, n and T must be constant.
Therefore, n1 = n2 and T1 = T2:
Which gives Boyle’s law:
Similarly, to use Charles’ law, n and P must be constant.
Therefore, n1 = n2 and P1 = P2:
Which gives Charles’ law:
.
Finally, to use Avogadro’s law, T and P must be constant.
Therefore, T1 = T2 and P1 = P2:
Which gives Avogadro’s law:.
Many experiments involving gases are performed at room
pressure. This has led to the term STP (Standard Temperature
and Pressure) referring to T = 0 ˚C = 273.15 K and P = 1 atm.
e.g. An empty freezer is rectangular with the inside dimensions
1.0 m × 1.0 m × 2.0 m. If it is filled with nitrogen gas at
STP, how many moles of nitrogen does it contain?
e.g. The nitrogen-filled freezer described above is cooled to
-40 ˚C. Calculate the pressure in the cooled freezer.
e.g. Calculate the density of nitrogen gas in the freezer in g/L.
Typical densities for solids/liquids are between 0.5 and 15 g/mL.
The molecules/atoms are tightly packed so they are difficult to
compress.
Gases are much less dense, so their densities are generally
reported in g/L. The molecules/atoms are disperse so they are
relatively easy to compress. (With enough pressure, a gas can
even be compressed to a solid or liquid.)
e.g. Ether is a widely used organic solvent that evaporates
readily (and used to be used as an anaesthetic). If ether
vapour has a density of 2.71 g/L at 25 ˚C and 680 mmHg,
calculate the molar mass of ether.
Gas Mixtures and Dalton’s Law of Partial Pressures
The behavior of a gas is based primarily on the number of moles
(or number of particles) rather than on the mass. This is true for
a mixture of gases as well as for pure gases.
When working with gas mixtures, we use the mole fraction, X,
as a unit of concentration:
ni
moles of i
Xi = total moles of gas =
ntotal
Each gas in a mixture contributes to the total pressure of the
system:
Ptotal = Σ Pi
where Pi is the partial pressure from gas i. This is known as
Dalton’s law of partial pressures.
The ideal gas law can be applied to each gas within the mixture
as well as to the mixture as a whole. The temperature and
volume are the same for each gas since they are all in the same
container. R is a universal constant so it is also the same.
For a mixture of two gases, A and B:
We can also relate the partial pressure of one gas in the mixture
to the total pressure of the mixture:
e.g. Flask He contains 3.0 L of helium gas at 145 mm Hg.
Flask Ar contains 2.0 L of argon gas at 355 mm Hg. The
two flasks are connected and the two gases allowed to mix.
(a) Assuming no change in temperature, calculate the partial
pressure of each gas in the mixture and the total pressure.
(b) Calculate the mole fraction of helium in the helium-argon
mixture.
e.g. Silane (SiH4) reacts with oxygen to give silicon dioxide and
water:
SiH4(g) + 2 O2(g)
→
SiO2(s) + 2 H2O(g)
Suppose that SiH4 and O2 are mixed in the correct
stoichiometric ratio, and that mixture has a total pressure of
120 mmHg.
(a) Calculate the partial pressures of SiH4 and O2.
(b) A spark is applied to initiate the reaction. Calculate the
total pressure in the flask after the reactants have been
completely consumed.
The Kinetic-Molecular Theory of Gases
Dalton’s law and the ideal gas law deal with macroscopic
properties of gases. Chemists are also interested in the
behaviour of gases at a molecular level.
The following assumptions describe the behaviour of an ‘ideal
gas’:
1. The distance between gas particles (atoms or molecules) is
much larger than the size of the particles.
2. Gas particles are in constant motion in random directions.
3. Gas particles do not interact except when they collide.
4. When gas particles collide with each other (or the walls of
their container), collisions are elastic (no energy is lost).
5. Gas particles move at different speeds, but the average
speed is proportional to temperature. All gases have the
same average kinetic energy at a given temperature.
The kinetic energy of a gas particle depends on mass and speed:
EK = ½ mv2
where EK is kinetic energy, m is mass and v is speed.
Because motion of gas
particles is random, not all
particles in a sample have the
same speed.
Instead, the
speeds
are
distributed
statistically in a MaxwellBoltzmann distribution:
When the temperature increases, the speed of the gas particles
increases and the distribution of speeds spreads out.
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’ speed (generated by squaring
each particle’s speed, calculating the average, then
taking the square root)
Because kinetic energy is proportional to v2, we use vrms in
kinetic energy and temperature calculations.
EK = ½ mvrms2 = 1.5 RT
where EK 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).
Note that, at a constant temperature:
• particles with __________ mass will have __________ vrms
• particles with __________ mass will have __________ vrms
The Maxwell-Boltzmann equation can be derived from the
kinetic-molecular theory of gases:
P = N m vrms2
or
PV = ⅓ N m vrms2.
3V
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 rootmean-square speed.
Since the Maxwell-Boltzmann equation and the ideal gas law
both describe ideal gases, they can be related:
PV = ⅓ N m vrms2.
and
PV = nRT
Therefore:
⅓ N m vrms2. = nRT
Therefore:
Nmvrms2 = 3 RT
n
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 (U). Therefore:
U vrms2 = 3 RT
To solve for root-mean-square speed, rearrange this equation:
vrms =
3 RT
U
Note that, in order for the units to cancel out, U must be
converted to kg/mol:
e.g. 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%?
Diffusion and Effusion
Diffusion and effusion are both terms used to describe the
movement of gas particles from an area of high concentration to
one of lower concentration. If left long enough, both diffusion
and effusion result in evenly distributed gas particles; however,
both processes are slower than might be anticipated because gas
particles move in random directions (not automatically toward
an area of low concentration).
Diffusion is the ‘spreading out’ of gas particles due to their
random motion.
Effusion is the ‘escape’ of gas particles through a container with
a pinhole in it.
The rate of effusion is directly proportional to the root-meansquare speed of the gas particles. Recall that:
vrms =
where U is the molar mass.
3 RT
U
We can compare the rates of effusion of two different gases at
the same temperature:
3 RT
Effusion Rate of A
vrms(A)
UA
=
=
Effusion Rate of B
vrms(B)
3 RT
UB
=
Effusion Rate of A
vrms(A)
=
Effusion Rate of B
vrms(B)
=
1
UA
1
UB
UB
UA
This is Graham’s law of effusion (“the rate of effusion is
inversely related to the square root of the molar mass of a gas”).
e.g. Argon gas effuses through a pinhole 83.74% as fast as an
unknown diatomic gas (known to be a pure element).
Calculate the molar mass of this unknown gas and name it.
Real Gases
The ideal gas law and kinetic molecular theory are based on two
simplifying assumptions:
1. The actual gas particles have no volume.
2. There are no forces of attraction between gas particles.
Neither of these assumptions is true and there is therefore no
such thing as an ideal gas.
At atmospheric pressure (and lower) and room temperature (and
higher), however, the volume of gas particles is negligible
compared to the volume of the container (assumption #1).
Similarly, the forces of attraction between particles are
negligible (assumption #2). This allows us to use the ideal gas
law and kinetic molecular theory under those conditions.
When dealing with gases at high pressure and/or low
temperature, we must use a modified version of the gas law, the
van der Waals equation of state:
P + a
n
V
2
(V - b n) = nRT
where a and b are correction factors specific to the gas in
question. Correction factor a has units of atm·L2·mol-2 and
corrects for the ‘stickiness’ of the gas molecules (how strongly
they are attracted to each other). Correction factor b has units of
L·mol-1 and corrects for the inherent volume of the gas particles.
Note that a is higher for polar gas molecules than nonpolar ones
while b is higher for large gas molecules than small ones:
b (L·mol-1)
a (atm·L2·mol-2)
Hydrogen
H2
0.2444
0.02661
Methane
CH4
2.253
0.04278
Ammonia
NH3
4.170
0.03707
Water
H2O
5.464
0.03049
Sulfur dioxide SO2
6.714
0.05636
e.g. 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 real pressure in each container.
Important Concepts from Chapter 12
• gas laws
o Avogadro’s law (V ∝ n)
o Boyle’s law (P1V1 = P2V2)
o Charles’ law (V1 / T1 = V2 / T2)
o Dalton’s law of partial pressures
o ideal gas law (PV = nRT)
• ideal gas constant
• STP (standard temperature and pressure = 0 ˚C and 1 atm)
• mole fraction as a unit of concentration
• kinetic-molecular theory and relationship between kinetic
energy, temperature and root-mean-square speed
• Maxwell-Boltzmann equation
• diffusion vs. effusion
• Graham’s law of effusion
• ideal gases vs. real gases
• van der Waals equation of state