GASEOUS STATE (Silberberg, Chapter 5)
What distinguishes a gas from a liquid or solid?
• A gas expands to fill its container
• A gas is easily compressed
• A gas fills a container instantly
Develop a kinetic model of gases to explain these features.
Gas Pressure
Pressure is the force exerted by the gas divided by the area on which the force is exerted:
Pressure
force
area
The pressure exerted by a gas results from the
collisions of gas molecules with the container
walls.
The pressure exerted by the atmosphere is measured
using a barometer. The greater the atmospheric
pressure the higher the column of mercury (at sea
level it’s about 760 mm)
To measure the pressure of a gas in a
container, a manometer is used:
Units of pressure :
Barometers/manometers measure in mm Hg
(often called Torr (named for Torricelli))
Often convenient to use atmosphere (atm) or bar.
SI unit for pressure is the pascal (Pa) where 1 Pa = 1 kg m-1 s-2.
Unit
Pascal (Pa)
Bar
Definition / relationship
kg m-1 s-2 (= 1 N m-2)
1 bar = 1 x 105 Pa (100 kPa)
Atmosphere (atm)
1* atm = 1.013 x 105 Pa (= 101.3 kPa)
Torr (or mm Hg)
760* mm Hg = 760* Torr = 1 atm
*This is an exact quantity; in calculations, we use as many significant figures as necessary.
The Empirical Gas Laws and the Ideal Gas Law
Boyle’s law relates pressure and volume: Robert Boyle
studied the compressibility of gases (ca. 1660). Adding more
mercury to a J-tube (increasing pressure on gas) causes the
volume of gas to decrease:
Boyle’s law: the volume of a sample of gas at a given temperature varies inversely with pressure.
PV = constant
Charles’ law relates volume and temperature.
Paris, 21 November 1783
Empirical Gas Laws
Charles’ law: For a fixed amount of gas under
constant pressure, the volume varies linearly
with temperature.
Volume/temperature plots for different gases at constant pressure can all be extrapolated to the same
point (-273 C). Since volume can’t be negative, this must be the lowest possible temperature: it’s
given the value 0 K and is called “absolute zero”
Therefore, if we use temperatures on the Kelvin scale we can
write Charles’ law as: V T or V = constant x T
Empirical Gas Laws
Relationship between pressure and temperature (Amonton’s law) :
For a sample of gas of fixed volume the pressure is directly
proportional to the temperature,
P T or P = constant x T (provided the Kelvin scale is used)
Avogadro’s principle relates volume and amount.
In 1811 Amadeo Avagadro recognised that under the same conditions of temperature and pressure, a given
number of molecules occupy the same volume regardless of their chemical identity.
Avogadro’s principle:
One mole of any gas contains 6.02 x 1023 molecules and occupies the same volume at a given temperature
and pressure.
By convention,
Standard temperature and pressure (STP) (0 C and 1 atm) are often used.
At STP, the molar gas volume is found to be 22.4 dm3.
The ideal gas law combines all these empirical laws.
Boyle’s law, Charles’ law and Avogadro’s principle can all be combined into one simple relationship
between pressure (P), volume (V), temperature (T) and the number of moles (n) of a gas:
PV = nRT
ideal gas law
where R is the gas constant with values as given in the table:
Value of R
0.082058 L atm K-1 mol-1
8.3145 J K-1 mol-1
8.3145 kg m2 s-2 K-1 mol-1
8.3145 kPa dm3 K-1 mol-1
Stoichiometry problems using the gas law
Problem 1: Estimate the pressure (in atmospheres) inside a television picture tube, given that its volume is
5.0 L, its temperature is 23 C and it contains 0.010 mg of nitrogen gas.
(Ans: P = 1.7 x 10-6 atm)
Problem 2: What is the density of carbon dioxide, CO2, in grams per litre at 25 C and 0.500 atm?
(Ans: = 0.901 g dm-3)
Problem 3: Carbon dioxide must be removed from the air supply on submarines (and spacecraft). One
method is to use potassium superoxide, KO2, because this compound reacts with carbon dioxide and
releases oxygen:
4 KO2 (s) + 2 CO2 (g)
2 K2CO3 (s) + 3 O2 (g)
Calculate the mass of KO2 needed to react with 50 dm3 of CO2
at 25 C and 1.0 atm.
(Ans: Mass KO2 = 290.9 g)
Gas mixtures and partial pressures
•All gases behave in the same way under the same
conditions of pressure, temperature and volume.
•A mixture of gases that don’t react with one
another behaves as a single, pure gas
• John Dalton observed (1801) : a mixture of two
gases has the same pressure as the sums of the
pressures of the individual gases.
Dalton’s law of partial pressures:
P = PA + PB + …
where PA and PB are the partial
pressures of components A and B
Each component acts as though it were the only gas present, and obeys the ideal gas law: PA V = nA R T
The composition of a gas mixture is sometimes described in terms of the mole fractions of its gases: The
mole fraction of a component is the fraction of moles of that component in the total moles of gas in the
mixture :
mole fraction of gas A
XA
nA
n total
nA
nA
nB
...
Example : Calculating partial pressures
The composition of gases in a “neon” advertising sign of volume
0.75 dm3 is 0.10 g of neon and 0.20 g of xenon. What is the mole fraction of Ne and Xe? Calculate their
partial pressures and the total pressure (in atmospheres) when the tube is operating at 40 C.
Collecting gases over water:
• Useful for gases that don’t dissolve in water
• Gas (eg. H2 produced by reaction of Zn and HCl) is
collected in a burette (the gas displaces the water in the
burette)
• As gas bubbles through water, the gas mixes with
molecules of water vapour
• The partial pressure of the water vapour (the vapour
pressure) depends only on T and can be looked up in a
table (Table 5.3, Silberberg 4th ed)
Example: If 156 cm3 of H2 gas is collected at 19 C and 769 mm Hg total pressure, what is the mass of
hydrogen collected? (The partial pressure of water vapour at 19 C is 16.5 mm Hg)
(0.013 g)
Kinetic molecular theory of an ideal gas
The kinetic molecular model of a gas helps us to explain the features of gases that we have described.
The model is based on 5 postulates (basic assumptions):
• Gases are made up of molecules whose size is negligible compared with the average distance
between them.
• Molecules move randomly in straight lines in all directions.
They only change direction after a collision.
• The forces of attraction or repulsion between two molecules in a gas are very weak
• All collisions between molecules (molecules-molecules or molecules-walls) are elastic; the total
kinetic energy is constant.
• The average kinetic energy of a molecule is proportional to the absolute temperature.
Interpretation of gas laws in terms of kinetic molecular theory
According to KMT, gas pressure results from collisions by molecules with container walls : pressure
depends on the number of molecules (concentration) and the average speed of the molecules (because
this determines the average momentum of collision)
Boyle’s law:
Charles’ law:
Ideal gas law:
P
(frequency of collision) x (average momentum)
Average momentum = m u
Frequency
PV
of collision
(u x
[m : mass of molecule; u: average speed]
1
V
x N)
[where
N
# molecules]
N m u2
Average Ek = ½ m u2
PV
average Ek
Also, average Ek absolute temperature
& number of molecules N number of moles of molecules, n
PV
nT
PV = n R T
where R is a proportionality constant
Results derived from KMT:
Molecular speeds & diffusion and effusion
A. Molecular speed
Ek = ½ m u 2
u 2 is the root-mean-square speed
KMT assumes molecules in constant random motion, and Ek
½ m u2
T.
T
From this one can derive that the average speed of molecules in a gas is related to the temperature
and the molar mass by
u
3RT
rms
M
R = 8.314 J mol-1 K-1
The distribution of molecular speeds shows the fraction of gas molecules moving at each speed at any
given instant.
(Although we can talk of average speeds of gas molecules, individual molecules may be moving at quite different
speeds: a molecule may be brought to a stop by a collision, then move off again at great speed again.)
Relationship between molecular speed and molar mass:
At a given temperature, the average speed of gas molecules is inversely proportional to molar mass
Relationship between molecular speed and temperature:
Doubling the temperature of any gas increases the average
speed of its molecules by a factor of 2 (=1.4)
B. Diffusion and Effusion
Diffusion : dispersion of one substance through another (eg. perfume through air)
Effusion : escape of one substance through a small hole into a vacuum
Graham’s law of effusion :
Rate of effusion
1
molar mass
Since rate of effusion is inversely proportional to molar mass, time taken for a given number of moles of
molecules to effuse through a barrier is directly proportional to molar mass.
time to effuse
molar mass
On average, heavy molecules escape through a hole more slowly than light molecules so the ratio of
the time taken by two gases, A and B, to effuse is:
t
( A)
molar mass A
effuse
t effuse ( B )
molar mass B
Example: If it takes a certain number of helium atoms 10 s to effuse through a porous barrier, how long
does it take the same number of methane molecules under the same conditions?
K-25 gaseous diffusion plant, Oak Ridge, Tennessee, USA.
Structure is ca. 800 m long with about 4000 effusion stages.
Diffusion of a gas particle through a
space filled with other particles:
Effusion used to separate U-235 (for nuclear
power) from U-238 (more abundant). Make
UF6 (volatile) and allow it to effuse through a
series of barriers. UF6 molecules containing
U-235 move faster, but the ratio is only 1.004
so many effusion steps are needed for
meaningful enrichment.
Diffusion follows a similar relationship to effusion, (a) with heavy
molecules diffusing more slowly than light molecules, and (b)
occurring faster at higher temperatures.
Real gases
PV=nRT explains the behaviour of gases provided the gas is ideal (obeys all the postulates of KMT)
Real gases behave like ideal gases at
• low pressures (< 2 atm) and
• moderate temperatures.
At higher pressures (or lower temperatures), real gas behaviour deviates from ideal behaviour :
measured pressure is higher or lower than predicted by the ideal gas law.
This can be understood in terms of KMT: we assumed that molecules are infinitely small and do not
interact with one another.
At low pressures assumption works because molecules are far apart (their volumes are small
compared to the total volume and they don’t interact because they’re too far away from one another).
At higher pressures (smaller volume), molecules are forced closer together. Intermolecular forces
become more important: as a gas molecule is about to collide with the container wall, neighbouring
molecules pull it away from the wall (forces of attraction) so the pressure measured is lower.
Also, in smaller volumes the molecules’ own volumes becomes significant relative to container volume.
The van der Waals equation adjusts the pressure and volume terms of the ideal gas equation to
fit a real gas:
2
(P
n a
V
2
)( V
nb )
nRT
a and b are constants, experimentally determined for a given gas.