Gases
Chapter 5
5.1
Substances that exist as gases
5.2
Pressure of a gas
5.3
The Gas Laws
5.4
The Ideal Gas Equation
5.5
Gas Stoichiometry
5.6
Dalton’s Law of Partial Pressures
5.7
The Kinetic Molecular Theory of Gases
5.8
Deviation from Ideal Behavior
1
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5.1: Elements that exist as gases at 250C and 1 atmosphere
2
3
Physical Characteristics of Gases
•
Gases assume the volume and shape of their
containers.
•
Gases are the most compressible state of matter.
•
Gases will mix evenly and completely when confined to
the same container (infinitely miscible).
•
Gases have much lower densities than liquids and
solids.
•
Gases are thermally expandable When a gas sample is heated, its
volume increases, and when it is
cooled its volume decreases.
•
Gases have low viscosity - gases
flow much easier than liquids or
solids.
NO2 gas
4
5.2: Pressure of a gas
Gas molecules are constantly in
motion, they exerts pressure on any
surface they come in contact with.
Force
Pressure = Area
(force = mass x acceleration)
Units of Pressure
1 pascal (Pa) = 1 N/m2
1 atm = 760 mmHg = 760 torr
1 atm = 101,325 Pa
5
Pressure of the Atmosphere
• Called “atmospheric
pressure,” or the force exerted
upon us by the atmosphere
above us.
• A measure of the weight of the
atmosphere pressing down
upon us.
• Measured using a barometer a device that can weigh the
atmosphere above us.
A Mercury Barometer
The pressure exerted
by a column of air
extending from the
earth’s surface (sea
level) to the upper
atmosphere is the
atmospheric pressure
10 miles
4 miles
Sea level
0.2 atm
The actual value of
atm pressure
depends on
location,
0.5 atm temperature and
weather conditions
1 atm
Atmospheric
pressure is
measured with a
barometer
5.2
Manometers Used to Measure Gas Pressures
closed-tube
open-tube
8
A manometer – a device used to
measure the pressure of gases other than
the atmosphere.
Type I:
Closed-tube
manometer (closed
end manometer) is
used to measure
pressures below
atmospheric
pressures.
Pgas = Ph
Type II:
Open-tube manometer (open end manometer) is
used to measure pressures equal to or greater than
atmospheric pressures.
QUESTION #1:
What is the pressure (in atm) in a container of gas
connected to a mercury-filled, open-ended manometer
if the level in the arm connected to the container is
24.7 cm higher than in the arm open to the
atmosphere and atmospheric pressure is 0.975 atm?
Answer:
QUESTION #2:
What is the pressure of
the gas (in mm Hg)
inside the apparatus
shown on the right if the
external pressure is 750
mm Hg?
Answer:
5.3: The Gas laws
Boyle’s Law
P  1/V
Avogadro’s Law
Vn
Charles’s Law
Gay-Lussac’s Law
VT
Apparatus for Studying the Relationship Between
Pressure and Volume of a Gas
As P (h) increases
V decreases
14
Boyle’s Law
P a 1/V
P x V = constant
P1 x V1 = P2 x V2
Constant temperature
Constant amount of gas
15
Question #3: A sample of chlorine gas occupies a volume
of 946 mL at a pressure of 726 mmHg. What is the
pressure of the gas (in mmHg) if the volume is reduced at
constant temperature to 154 mL?
16
Charles’ & Gay-Lussac’s Law
Variation in Gas
Volume with
Temperature at
Constant
Pressure
As T increases V increases
17
Variation of gas volume with temperature
at constant pressure.
VaT
V = constant x T
V1 = V2
T1
T2
Pressure increases
from P1 to P4
Temperature must be
in Kelvin
T (K) = t (0C) + 273.15
5.3
Question #4: A sample of carbon monoxide gas occupies
3.20 L at 125 0C. At what temperature will the gas occupy
a volume of 1.54 L if the pressure remains constant?
19
Avogadro’s Law
V a number of moles (n)
Constant temperature
Constant pressure
V = constant x n
V1 / n1 = V2 / n2
20
Question #5: Ammonia burns in oxygen to form nitric
oxide (NO) and water vapor. How many volumes of NO
are obtained from one volume of ammonia at the same
temperature and pressure?
21
Summary of Gas Laws
Boyle’s Law
22
Charles Law
23
Avogadro’s Law
24
5.4: The Ideal Gas Equation
• An ideal gas is defined as one for which both
the volume of molecules and forces between
the molecules are so small that they have no
effect on the behavior of the gas.
• The ideal gas equation is:
PV = nRT
R = Ideal gas constant
R = 8.314 J / mol K = 8.314 J mol-1 K-1
R = 0.08206 l atm mol-1 K-1
Ideal Gas Equation
Boyle’s law: P a 1 (at constant n and T)
V
Charles’ law: V a T (at constant n and P)
Avogadro’s law: V a n (at constant P and T)
Va
nT
P
V = constant x
nT
P
=R
nT
P
R is the gas constant
PV = nRT
26
The conditions 0 0C and 1 atm are called standard
temperature and pressure (STP).
Experiments show that at STP, 1
mole of an ideal gas occupies
22.414 L.
PV = nRT
(1 atm)(22.414L)
PV
R=
=
nT
(1 mol)(273.15 K)
R = 0.082057 L • atm / (mol • K)
27
Question #6: What volume will be
occupied by 0.833 mole of fluorine (F2) at
645 mmHg and 15.0 oC?
Answer:
Question #7: What is the volume (in liters) occupied by
49.8 g of HCl at STP?
29
Question #8: Argon is an inert gas used in lightbulbs to
retard the vaporization of the filament.
A certain
lightbulb containing argon at 1.20 atm and 18 0C is
heated to 85 0C at constant volume. What is the final
pressure of argon in the lightbulb (in atm)?
30
Using ideal gas equation for:
Density (d) Calculations
PM
m
d=
=
V
RT
m is the mass of the gas in g
M is the molar mass of the gas
Molar Mass (M ) of a Gaseous Substance
dRT
M=
P
d is the density of the gas in g/L
31
Question #9: A 2.10-L vessel contains 4.65 g of a gas at
1.00 atm and 27.0 0C. What is the molar mass of the gas?
32
QUESTION #10: A sample of natural gas is collected at
25.0 oC in a 250.0 ml flask. If the sample had a mass of
0.118 g at a pressure of 550.0 Torr, what is the molecular
weight of the gas?
Answer:
5.5: Gas Stoichiometry
Mass
Atoms (Molecules)
Avogadro’s
Number
6.02 x 1023
Reactants
Molarity
moles / liter
Solutions
Molecules
Moles
Molecular
g/mol
Weight
Products
PV = nRT
Gases
Question #11: What is the volume of CO2 produced at 37 0C
and 1.00 atm when 5.60 g of glucose are used up in the
reaction: C6H12O6 (s) + 6O2 (g)
6CO2 (g) + 6H2O (l)
35
Question #12:
What total gas volume (in liters) at 520oC and 880
torr would result from the decomposition of 33 g of
potassium bicarbonate according to the equation:
2KHCO3(s)
Answer:
K2CO3(s) + CO2(g) + H2O(g)
QUESTION #13: A slide separating two containers is removed,
and the gases are allowed to mix and react. The first container
with a volume of 2.79 L contains ammonia gas at a pressure of
0.776 atm and a temperature of 18.7 oC. The second with a
volume of 1.16 L contains HCl gas at a pressure of 0.932 atm
and a temperature of 18.7 oC. What mass of solid ammonium
chloride will be formed, and what will be remaining in the
container, and what is the pressure?
Answer:
5.6: Dalton’s Law of Partial Pressures
Our atmosphere is a mixture of nitrogen, oxygen,
argon, carbon dioxide, water vapour and small
amounts of several other gases.
The atmospheric pressure is actually the sum of
the pressures exerted by all this individual gases.
The same condition is true for every gas mixture.
For gas mixtures:
• gas behavior depends on the number, not the identity, of
gas molecules.
• ideal gas equation applies to each gas individually and
the mixture as a whole.
• all molecules in a sample of an ideal gas behave exactly
the same way.
Definition of Dalton’s Law of
Partial Pressures:
“In a mixture of gases, each gas
contributes to the total pressure
the amount it would exert if the
gas were present in the container
by itself.”
To obtain a total pressure, add all of the partial
pressures: Ptotal = P1 + P2 + P3 +... + Pi
Dalton’s Law of Partial Pressures
V and T are constant
P1
P2
Ptotal = P1 + P2
40
Consider a case in which two gases, A and B, are in a
container of volume V.
nART
PA =
V
nA is the number of moles of A
nBRT
PB =
V
nB is the number of moles of B
PT = PA + PB
PA = XA PT
nA
XA =
nA + nB
nB
XB =
nA + nB
PB = XB PT
Pi = Xi PT
mole fraction (Xi ) =
ni
nT
41
Fig. 5.16
Question #14: A sample of natural gas contains 8.24 moles
of CH4, 0.421 moles of C2H6, and 0.116 moles of C3H8. If the
total pressure of the gases is 1.37 atm, what is the partial
pressure of propane (C3H8)?
43
QUESTION #15:
A 2.00 L flask contains 3.00 g of CO2 and 0.10 g of
helium at a temperature of 17.0 oC. What are the
partial pressures of each gas, and the total
pressure?
Answer:
Collecting a Gas over Water: Application of Dalton’s Law
2KClO3 (s)
2KCl (s) + 3O2 (g)
PT = PO2 + PH2 O
45
Fig. 5.12
Vapor of Water and Temperature
47
Question #16: Hydrogen gas generated when calcium
metal reacts with water is collected over water. The
volume of gas collected at 30 oC and pressure of 988
mmHg is 641 mL. Calculate the mass in grams of the
hydrogen gas obtained?
The pressure of water vapor at 30 oC is 31.82 mmHg.
QUESTION #17:
Calculate the mass of hydrogen gas collected over water if
156 ml of gas is collected at 20oC and 769 mm Hg. The
vapor pressure of water at 20oC is 17.54 mm Hg.
Answer:
QUESTION #18:
A sample of hydrogen gas collected by
displacement of water occupied 30.0 mL at 24oC
on a day when the barometric pressure was 736
torr. What volume would the hydrogen occupy if it
were dry and at STP? The vapor pressure of water
at 24.0oC is 22.4 torr.
Answer:
5.7: Kinetic Molecular Theory of Gases
The nanoscale behaviour of gas molecules can be used to
explain why gases behave as they do – i.e. understanding
the macroscopic properties of the gas in terms of the
behavior the microscopic molecules making up the gas.
There are 5 (five) postulates that
describe the behavior of molecules in
a gas. These postulates are based
upon some simple, basic scientific
notions, but they also involve some
assumptions
that
simplify
the
calculations.
5.7
The 5 postulates are:
1. The gas consists of tiny particles of negligible
volume.
A gas is composed of molecules whose size is much
smaller than the distances between them. The molecules
can be considered to be points; that is, they possess
mass but have negligible volume.
2. Gas molecules move randomly at various speed and
in every possible directions.
The gas molecules are in constant motion. They
frequently collide with one another or with the walls of the
container.
5.7
3. Collisions among molecules are perfectly elastic.
The collisions of the gas molecules, either with other
molecules or with the walls of the container, are elastic.
There is no loss of energy. If the collisions are inelastic,
the motion of the gas molecules would become slower
and slower, and finally the molecules would remain at the
bottom of the container – this situation has never
occurred! Therefore the molecular collision must be
elastic.
4. Gas molecules exert neither attractive nor repulsive
forces on one another.
Intermolecular forces of attraction do not exist between
gas molecules, hence, the gas molecules travel in a
straight line and are not influenced by other gas
molecules.
5.7
5. The average kinetic energy of the molecules is
proportional to the temperature of the gas in Kelvins.
Any two gases at the same temperature will have the
same average kinetic energy.
The average kinetic energy of a molecules is given by:
1
KE  mu 2
2
u12  u22  ....  u N2
u 
N
2
Since
whereby:
m = mass
u = speed
N = number of molecules
KE  T
1
mu 2  T
2
1
 mu 2  CT
2
whereby:
C = proportionality constant
T = absolute temperature
5.7
Kinetic theory of gases and …
• Compressibility of Gases
• Boyle’s Law
P a collision rate with wall
Collision rate a number density
Number density a 1/V
P a 1/V
• Charles’ Law
P a collision rate with wall
Collision rate a average kinetic energy of gas molecules
Average kinetic energy a T
PaT
55
Kinetic theory of gases and …
• Avogadro’s Law
P a collision rate with wall
Collision rate a number density
Number density a n
Pan
• Dalton’s Law of Partial Pressures
Molecules do not attract or repel one another
P exerted by one type of molecule is unaffected by the
presence of another gas
Ptotal = SPi
56
Distribution of Molecular Speed
At any given time, what fraction of the molecules in a particular
sample have a given speed?
Some of the molecules will be moving more slowly than
average, and some will be moving faster than average, but
how many?
Apparatus for studying molecular speed distribution:
Maxwell Boltzmann solved the problem mathematically.
He used statistical analysis to obtain an equation that
describes the distribution of molecular speed at a given
temperature.
The distribution of speeds
for nitrogen gas molecules
at three different
temperatures.
At the higher temperature,
more molecules are
moving at faster speed.
urms =
M
3RT
5.7
The distribution of speeds of
three different gases at the
same temperature.
At a given temperature,
the lighter molecules
are moving faster, on
the average.
5.7
Molecular Mass and Molecular Speeds
Molecule
Molecular
Mass (g/mol)
Kinetic Energy
(J/molecule)
Velocity
(m/s)
H2
CH4
2.016
16.04
6.213 x 10 - 21
1,926
6.0213 x 10 - 21
683.8
CO2
44.01
6.213 x 10 - 21
412.4
Graham’s Law: Diffusion & Effusion of Gases
Gas diffusion is the gradual mixing of molecules of one
gas with molecules of another by virtue of their kinetic
properties.
NH4Cl
A path traveled by a
single gas molecule.
Each change in direction
represents a collision
with another molecule.
r1
r2
=

M2
M1
NH3
17 g/mol
HCl
36 g/mol
NH3 is lighter & diffuses faster,
therefore NH4Cl appears nearer to HCl
bottle.
5.7
Graham's Law of Diffusion:
The rate at which gases diffuse is inversely proportional to
the square root of their densities.
If the number of moles per liter at a given T and P is
constant, then, the density of a gas is directly proportional to
its molar mass (MM).
In comparing two gases at the same T and P, the formula
becomes:
Rate1

Rate 2
MM 2
MM 1

MM 2
MM 1
Gas effusion is the is the process by which gas under
pressure escapes from one compartment of a container to
another by passing through a small opening (pinhole).
r1
r2
=
t2
t1
=

M2
M1
According to Graham’s Law, the rate of
effusion of a gas is inversely
proportionate to the square root of its
mass. The lighter the molecule, the
more rapidly it effuses.
Thomas Graham
(1805-1869)
63
QUESTION #19:
Which gas in each of the following pairs
diffuses more rapidly, and what are the relative
rates of diffusion?
(a) Kr & O2
(b) N2 & acetylene, C2H2
Question #20: Example #Nickel forms a gaseous
compound of the formula Ni(CO)x What is the value of
x given that under the same conditions methane (CH4)
effuses 3.3 times faster than the compound?
65
5.8: Deviations from Ideal Behavior
The deviation from ideal behavior is large at high pressure
and low temperature.
At lower pressures and high temperatures, the deviation from
ideal behavior is typically small, and the ideal gas law can be
used to predict behavior with little error.
Remember the conditions assumed for an Ideal Gas?:
•Molecules are perfectly elastic
(no INTERMOLECULAR FORCES).
•Molecules are point masses
(NEGLIGIBLE volume).
•Molecules move at random.
5.8
The first two of these assumptions is clearly WRONG for all
gases, because at low temperatures all gases CONDENSE, or
form a liquid phase.
This must happen because of the intermolecular forces that exist
between the molecules. Moreover, the liquid has measurable
molar volume, and this volume is simply the size of the closepacked molecules of the liquid.
At high pressures, the intermolecular distances can become quite
short, and attractive forces between molecules becomes
significant.
Neighboring molecules exert a relatively longranged attractive force on one another, which will
reduce the momentum in which they transfer to
the container walls. The observed pressure
exerted by the gas under these conditions will be
less than that for an Ideal Gas.
Deviations from Ideal Behavior
1 mole of ideal gas
PV = nRT
PV = 1.0
n=
RT
Repulsive Forces
Attractive Forces
68
Dutch physicist, Johannes van der Waals modified the ideal
gas law to describe the behaviour of real gases by including
the effects of molecular size and intermolecular forces.
Van der Waals equation
nonideal gas
2
an
( P + V2 ) (V – nb) = nRT
}
}
corrected
pressure
corrected
volume
70
QUESTION #21:
You are in charge of the manufacture of cylinders of
compressed gas at a small company. Your company
president would like to offer a 4.0-L cylinder containing
500 g of chlorine in the new catalog. The cylinders you
have on hand have a rupture pressure of 40 atm.
Use both the ideal gas law and the van der Waals
equation to calculate the pressure in a cylinder at 25oC.
Is this cylinder likely to be safe against sudden rupture
(which would be disastrous because chlorine gas is
highly toxic)?