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6/29/2012
1
Gases
© 2009, Prentice-Hall, Inc.
Chapter 10
Gases
John Bookstaver
St. Charles Community College
Cottleville, MO
Chemistry, The Central Science, 11th edition
Theodore L. Brown; H. Eugene LeMay, Jr.;
and Bruce E. Bursten
Gases
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Chapter 10 Problems
• Problems 16, 19, 26, 33, 39,49, 57, 61
Gases
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THREE STATES
OF MATTER
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Gases
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General Properties of Gases
• There is a lot of “free” space
in a gas.
• Gases can be expanded
infinitely.
• Gases occupy containers
uniformly and completely.
• Gases diffuse and mix
rapidly.
Gases
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Gases: What Are They Like?
Flow readily and occupy
the entire volume of their
container
Vapor is the term used to denote the gaseous state
of a substance existing more commonly as a liquid
e.g., water is a vapor, oxygen is a gas
EOS
Many low molar mass molecular compounds are
gases – methane (CH4), carbon monoxide (CO)
Composed of widely separated particles in constant,
random motion
Gases
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Common Gases
EOS
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Gases
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Properties of Gases Gas properties can be
modeled using math. Model
depends on—
• V = volume of the gas (L)
• T = temperature (K)
• n = amount (moles)
• P = pressure
(atmospheres)
Gases
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• The gaseous states of three
halogens.
• Most common gases are colorless
–H2, O2, N2, CO and CO2
Properties of Gases: Gas Pressure
Gases
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The Concept of Pressure
• The pressure
exerted by a solid.
–Both cylinders have
the same mass
–They have different
areas of contact
P (Pa) = Area (m2)
Force (N)
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Gases
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Gas Pressure
SI units express pressure in Newtons (N) per square
meters (m2) -- or N m–2
a.k.a. – Pascals (Pa)
Pressure is the force per unit area – consider the
unit pounds per square inch
EOS
A barometer is an instrument
used to measure atmospheric
pressure
Gases
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Pressure
Pressure of air is
measured with a
BAROMETER
(developed by
Torricelli in
1643)
Gases
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Units of Pressure
• mm Hg or torr
–These units are literally
the difference in the
heights measured in mm
(h) of two connected
columns of mercury.
• Atmosphere
–1.00 atm = 760 torr
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Gases
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Pressure
Hg rises in tube until
force of Hg (down)
balances the force
of atmosphere
(pushing up).
P of Hg pushing
down related to
• Hg density
• column height
Gases
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Pressure
Column height
measures P of
atmosphere
• 1 standard atm
= 760 mm Hg
= 29.9 inches Hg
= about 34 feet of
water
SI unit is PASCAL,
Pa, where 1 atm =
101.325 kPa
Gases
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Examples of Pressure Units
Given these values, one can
generate conversion factors to
switch between units:
e.g., 760 mmHg = 1.01325 bar
EOS
mmHg
baror
bar
mmHg
760
01325.1
01325.1
760
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Gases
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Liquid Pressure
• The pressure exerted
by a liquid depends
on:
– The height of the
column of liquid.
– The density of the
column of liquid. P = g ·h ·d
Gases
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Barometers
Used to measure atmospheric pressure
The pressure exerted by a column
of mercury exactly 760 mm high is
defined as 1 atmosphere (atm)
EOS
Gases tend to settle under the
effects of gravity – pressure as
altitude
Gases
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Barometric Pressure Standard Atmospheric Pressure
1.00 atm, 760 mm Hg, 760 torr, 101.325 kPa, 1.01325 bar
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Gases
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Manometer
This device is used to
measure the difference
in pressure between
atmospheric pressure
and that of a gas in a
vessel.
Gases
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Open-Ended Manometers
Open-ended manometers
compare gas pressure to
barometric pressure
EOS
Column height differences
are proportional to gas
pressure
Pgas = Pbar + Dh
Gases
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Manometers
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Gases
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Standard Pressure
• Normal atmospheric pressure at sea level
is referred to as standard pressure.
• It is equal to
– 1.00 atm
– 760 torr (760 mm Hg)
– 101.325 kPa
Gases
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Simple Gas Laws
• Boyle 1662
P 1
V PV = constant
Gases
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Pressure-Volume Relationship:
Boyle’s Law
For a given amount of a gas at
constant temperature, the
volume of the gas varies
inversely with its pressure
i.e., if V , then P
EOS
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Gases
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Boyle’s Law
The volume of a fixed quantity of gas at
constant temperature is inversely proportional
to the pressure.
Gases
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As P and V are
inversely proportional
A plot of V versus P results in a curve.
Since
V = k (1/P)
This means a plot of
V versus 1/P will be
a straight line.
PV = k
Gases
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Relating Gas Volume and Pressure – Boyle’s Law. The
volume of a large irregularly shaped, closed tank can be
determined. The tank is first evacuated and then connected to a
50.0 L cylinder of compressed nitrogen gas. The gas pressure in
the cylinder, originally at 21.5 atm, falls to 1.55 atm after it is
connected to the evacuated tank. What is the volume of the
tank?
EXAMPLE
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Gases
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EXAMPLE
P1V1 = P2V2 V2 = P1V1
P2
= 694 L Vtank = 644 L
Gases
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Charles’s Law
• The volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature.
A plot of V versus T will be a straight line.
• i.e., V
T = k
Gases
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Temperature-Volume
Relationship: Charles’s Law
The volume of a fixed
amount of a gas at constant
pressure is directly
proportional to its Kelvin
temperature
i.e., if V , then T
or V/T = k
EOS
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Gases
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Temperature-Volume
Relationship: Charles’s Law
Absolute zero is the
temperature obtained by
extrapolation to zero
volume
EOS
Absolute zero on the
Kelvin scale = –273.15 °C
and ...
273.15 K = 0 °C
Gases
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Avogadro’s Law
• The volume of a gas at constant temperature
and pressure is directly proportional to the
number of moles of the gas.
• Mathematically, this means V = kn
Gases
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Avogadro’s Hypothesis Equal volumes of gases at the same T
and P have the same number of
molecules.
V = n (RT/P) = kn
V and n are directly related.
twice as many
molecules
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Gases
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Molar Volumes and Standard
Pressure and Temperature
Standard Temperature and
Pressure (STP) is defined
as
T = 0 oC and P = 1 atm
EOS
The molar volume of a gas is the
volume occupied by one mole of the
gas at STP
Gases
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Avogadro’s Law
V n or V = c n
At STP
1 mol gas = 22.4 L gas
At an a fixed temperature and pressure:
Gases
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Ideal-Gas Equation
V 1/P (Boyle’s law)
V T (Charles’s law)
V n (Avogadro’s law)
• So far we’ve seen that
• Combining these, we get
V nT
P
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Gases
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The Ideal Gas Law
The constant of proportionality (k) is given the
symbol R
RknT
PV
EOS
For 1 mol of an ideal gas at STP …
R = 0.08206 L atm mol–1 K–1
Gases
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Ideal-Gas Equation
The constant of
proportionality is
known as R, the
gas constant.
Gases
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Ideal-Gas Equation
The relationship
then becomes
nT
P V
nT
P V = R
or
PV = nRT
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Gases
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The Combined Gas Law
Given the various gas laws, all can be combined
into a single form …
V = k/P, V = kT, and V = kn
V a (nT)/P
knT
PV
For initial and final conditions:
EOS
2
22
1
11
nT
VPk
nT
VP
Gases
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The General Gas Equation
R = = P2V2
n2T2
P1V1
n1T1
= P2
T2
P1
T1
If we hold the amount and volume constant:
Gases
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Using the Gas Laws
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Gases
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6-4 Applications of the Ideal Gas
Equation
PV = nRT and n = m
M
PV = m
M
RT
M = m
PV
RT
Molar Mass Determination
Gases
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Determining a Molar Mass with the Ideal Gas Equation.
Polypropylene is an important commercial chemical. It is used
in the synthesis of other organic chemicals and in plastics
production. A glass vessel weighs 40.1305 g when clean, dry
and evacuated; it weighs 138.2410 when filled with water at
25°C (δwater = 0.9970 g cm-3) and 40.2959 g when filled with
propylene gas at 740.3 mm Hg and 24.0°C. What is the molar
mass of polypropylene?
Strategy:
Determine Vflask. Determine mgas. Use the Gas Equation.
EXAMPLE
Gases
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Determine Vflask:
Vflask = mH2O dH2O = (138.2410 g – 40.1305 g) (0.9970 g cm-3)
Determine mgas:
= 0.1654 g
mgas = mfilled - mempty = (40.2959 g – 40.1305 g)
= 98.41 cm3 = 0.09841 L
EXAMPLE
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Gases
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Use the Gas Equation:
PV = nRT PV = m
M
RT M = m
PV
RT
M = (0.9741 atm)(0.09841 L)
(0.6145 g)(0.08206 L atm mol-1 K-1)(297.2 K)
M = 42.08 g/mol
EXAMPLE
Gases
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Densities of Gases
If we divide both sides of the ideal-gas
equation by V and by RT, we get
n
V
P
RT =
Gases
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• We know that
– moles molecular mass = mass
Densities of Gases
• So multiplying both sides by the
molecular mass ( ) gives
n = m
P
RT
m
V =
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Gases
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Densities of Gases
• Mass volume = density
• So,
Note: One only needs to know the
molecular mass, the pressure, and the
temperature to calculate the density of
a gas.
P
RT
m
V = d =
Gases
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GAS DENSITY PV = nRT
n
V =
P
RT
m
M •V =
P
RT
where M = molar mass
d = m
V =
PM
RT
d and M proportional
and density (d) = m/V
Gases
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Gas Densities
Gases are much less dense than liquids and solids
Because of the magnitude of the value, densities of
gases are reported in g/L
At STP, L
Md
4.22
EOS
At other conditions, use the combined gas law …
RT
MPd
the density of a gas is directly proportional
to molar mass and pressure, and inversely
proportional to its Kelvin temperature
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Gases
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Gases in Chemical Reactions
• Stoichiometric factors relate gas
quantities to quantities of other reactants
or products.
• Ideal gas equation relates the amount of
a gas to volume, temperature and
pressure.
• Law of combining volumes can be
developed using the gas law.
Gases
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Gases and Stoichiometry 2 H2O2(liq) ---> 2 H2O(g) + O2(g)
Decompose 1.1 g of H2O2 in a flask
with a volume of 2.50 L. What is the
pressure of O2 at 25 oC? Of H2O? Bombardier beetle
uses decomposition
of hydrogen peroxide
to defend itself.
Gases
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Gases and Stoichiometry
2 H2O2(liq) ---> 2 H2O(g) + O2(g)
Decompose 1.1 g of H2O2 in a flask
with a volume of 2.50 L. What is the
pressure of O2 at 25 oC? Of H2O?
Solution Strategy:
Calculate moles of H2O2 and then
moles of O2 and H2O.
Finally, calc. P from n, R, T, and V.
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Gases
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Gases and Stoichiometry 2 H2O2(liq) ---> 2 H2O(g) + O2(g)
Decompose 1.1 g of H2O2 in a flask with a
volume of 2.50 L. What is the pressure of O2 at
25 oC? Of H2O?
Solution
1.1 g H2O2 • 1 mol
34.0 g 0.032 mol
0.032 mol H2O2 • 1 mol O2
2 mol H2O2
= 0.016 mol O2
Gases
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Gases and Stoichiometry 2 H2O2(liq) ---> 2 H2O(g) + O2(g)
Decompose 1.1 g of H2O2 in a flask with a
volume of 2.50 L. What is the pressure of O2 at
25 oC? Of H2O?
Solution
P of O2 = nRT/V
= (0.016 mol)(0.0821 L •atm/K •mol)(298 K)
2.50 LP of O2 = 0.16 atm
Gases
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Gases and Stoichiometry
What is P of H2O? Could calculate as
above. But recall Avogadro’s hypothesis.
V n at same T and P
P n at same T and V
There are 2 times as many moles of H2O
as moles of O2. P is proportional to n.
Therefore, P of H2O is twice that of O2.
P of H2O = 0.32 atm
2 H2O2(liq) ---> 2 H2O(g) + O2(g)
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Gases
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Mixtures of Gases
• Partial pressure
– Each component of a gas mixture exerts a
pressure that it would exert if it were in the
container alone.
• Gas laws apply to mixtures of gases.
• Simplest approach is to use ntotal, but....
Gases
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Dalton’s Law of
Partial Pressures
• The total pressure of a mixture of gases
equals the sum of the pressures that
each would exert if it were present
alone.
• In other words,
Ptotal = P1 + P2 + P3 + …
Gases
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Dalton’s Law
John Dalton
1766-1844
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Gases
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Partial Pressure
Ptot = Pa + Pb +…
Va = naRT/Ptot and Vtot = Va + Vb+…
Va
Vtot
naRT/Ptot
ntotRT/Ptot
= = na
ntot
Pa
Ptot
naRT/Vtot
ntotRT/Vtot
= = na
ntot
na
ntot = a Recall
Gases
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Mole Fraction
EOS
The mole fraction (x1) is the fraction of all the
molecules in a mixture that are of a given type
Consider the ratio of a component’s partial pressure
to total pressure
V, R, and T are all constant and drop out
1 1 11
1 2 ...tot tot
P n nx
P n n n
Gases
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Collection of Gases over
Water
EOS
As essentially insoluble gas is passed into a container
of water, the gas rises because its density is much less
than that of water and the water must be displaced
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Gases
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Collection of Gases over
Water
Assuming the gas is saturated with water vapor, the
partial pressure of the water vapor is the vapor
pressure of the water.
EOS
Pgas = Ptotal – PH2O(g) = Pbar – PH2O(g)
Ptotal = Pgas + PH2O(g)
Gases
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Pneumatic Trough
Ptot = Pbar = Pgas + PH2O
Gases
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Partial Pressures
• When one collects a gas over water, there is
water vapor mixed in with the gas.
• To find only the pressure of the desired gas,
one must subtract the vapor pressure of
water from the total pressure.
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Gases
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Vapor Pressure as a
Function of Temperature
The combined gas law shows the
relationship between P and T at
constant n and V:
EOS
P nR
T V
As with Charles’s law for V and T,
P and T are directly proportional
Gases
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Kinetic-Molecular Theory
This is a model that
aids in our
understanding of what
happens to gas
particles as
environmental
conditions change.
Gases
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Main Tenets of Kinetic-
Molecular Theory
Gases consist of large numbers of
molecules that are in continuous,
random motion.
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Gases
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Main Tenets of Kinetic-
Molecular Theory
The combined volume of all the
molecules of the gas is negligible
relative to the total volume in which the
gas is contained.
Gases
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Main Tenets of Kinetic-
Molecular Theory
Attractive and
repulsive forces
between gas
molecules are
negligible.
Gases
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KINETIC MOLECULAR THEORY (KMT)
Theory used to explain gas laws. KMT
assumptions are
• Gases consist of molecules in constant,
random motion.
• P arises from collisions with container
walls.
• No attractive or repulsive forces between
molecules. Collisions elastic.
• Volume of molecules is negligible.
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Gases
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Kinetic Molecular Theory
Because we assume molecules are
in motion, they have a kinetic
energy.
KE = (1/2)(mass)(speed)2 At the same T, all gases have the same average KE.
As T goes up for a gas, KE also increases — and so does speed.
Gases
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Kinetic Molecular Theory
At the same T, all gases have the same
average KE.
As T goes up, KE also increases — and
so does speed.
Gases
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Main Tenets of Kinetic-
Molecular Theory
Energy can be transferred between molecules during collisions, but the average kinetic energy of the molecules does not change with time, as long as the temperature of the gas remains constant.
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Gases
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Kinetic Molecular Theory
where u is the speed and M
is the molar mass.
• speed INCREASES with T
• speed DECREASES with M
Maxwell’s equation
root mean square speed
2uM
3RT
Gases
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Distribution of Gas
Molecule Speeds
•Boltzmann plots
•Named for Ludwig
Boltzmann doubted
the existence of atoms.
• This played a role in
his suicide in 1906.
Gases
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Velocity of Gas Molecules
Molecules of a given gas have a
range of speeds.
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Gases
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Velocity of Gas Molecules Average velocity decreases with increasing
mass.
Gases
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Main Tenets of Kinetic-
Molecular Theory
The average kinetic
energy of the
molecules is
proportional to the
absolute
temperature.
Gases
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Effusion
Effusion is the
escape of gas
molecules
through a tiny
hole into an
evacuated
space.
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Gases
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Effusion
The difference in the
rates of effusion for
helium and nitrogen,
for example,
explains a helium
balloon would
deflate faster.
Gases
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Diffusion
Diffusion is the
spread of one
substance
throughout a space
or throughout a
second substance.
Gases
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GAS DIFFUSION AND EFFUSION
Graham’s law governs
effusion and
diffusion of gas
molecules.
Thomas Graham, 1805-1869.
Professor in Glasgow and London.
Rate of effusion is
inversely proportional
to its molar mass.
M of A
M of B
Rate for B
Rate for A
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Gases
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Effusion
At a given temperature, the rates
of effusion of gas molecules are
inversely proportional to the
square roots of their molar
masses
EOS
11 2
2 1
2
3
3
RT
Mrate M
rate MRT
M
Gases
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Graham’s Law
• Only for gases at low pressure (natural escape, not a
jet).
• Tiny orifice (no collisions)
• Does not apply to diffusion.
A
BA
Brms
Arms
M
M
3RT/MB
3RT/M
)(u
)(u
Bofeffusionofrate
Aofeffusionofrate
• Ratio used can be:
– Rate of effusion (as above)
– Molecular speeds
– Effusion times
– Distances traveled by
molecules
– Amounts of gas effused.
Gases
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Real Gases Under many conditions, real gases do not follow the
ideal gas law ...
-- Intermolecular forces of attraction cause the
measured pressure of a real gas to be less than
expected
-- When molecules are close together, the volume
of the molecules themselves becomes a
significant fraction of the total volume of a gas
EOS
2
2
n aP V nb nRT
V
van der Waals equation
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Gases
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Deviations from Ideal Gas Law
• Real molecules
have volume.
• There are
intermolecular
forces.
–Otherwise a gas
could not become
a liquid.
Gases
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Deviations from Ideal Gas Law
Account for volume of
molecules and intermolecular
forces with VAN DER
WAALS’s EQUATION. Measured V = V(ideal) Measured P
intermol. forces
vol. correction
J. van der Waals,
1837-1923,
Professor of
Physics,
Amsterdam.
Nobel Prize 1910.
nRT V - nb V
2
n 2 a
P + ----- ) (
Gases
© 2009, Prentice-Hall, Inc.
Deviations from Ideal Gas Law
Cl2 gas has a = 6.49, b = 0.0562
For 8.0 mol Cl2 in a 4.0 L tank at 27 oC.
P (ideal) = nRT/V = 49.3 atm
P (van der Waals) = 29.5 atm
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Gases
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van der Waals Equation
Corrections for real gas behavior are made using the
parameters a and b
a – accounts for intermolecular attractions in real gases
b – accounts for the real volumes of gases
EOS
Gases
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Real Gases
In the real world, the
behavior of gases
only conforms to the
ideal-gas equation
at relatively high
temperature and low
pressure.
Gases
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Real Gases
Even the same gas
will show wildly
different behavior
under high pressure
at different
temperatures.
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Gases
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Deviations from Ideal Behavior
The assumptions made in the kinetic-molecular model (negligible volume of gas molecules themselves, no attractive forces between gas molecules, etc.) break down at high pressure and/or low temperature.
Gases
© 2009, Prentice-Hall, Inc.
Corrections for Nonideal
Behavior
• The ideal-gas equation can be adjusted
to take these deviations from ideal
behavior into account.
• The corrected ideal-gas equation is
known as the van der Waals equation.
Gases
© 2009, Prentice-Hall, Inc.
The van der Waals Equation
) (V − nb) = nRT n2a
V2 (P +