Gases
Content
Physical Chemistry 2
1
Characteristics of Gases
• Unlike liquids and solids, gases
–
–
–
–
expand to fill their containers;
are highly compressible;
have extremely low densities
Gases form homogenous mixtures
Common properties of gases that are
measured are: Pressure, Temperature and
volume
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Characteristics of Gases
1. Gases can be compressed into smaller volumes; that
is, their densities can be increased by applying
increased pressure.
2. Gases exert pressure on their surroundings; in turn,
pressure must be exerted to confine gases.
3. Gases expand without limits, and so gas samples
completely and uniformly occupy the volume of any
container.
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Characteristics of Gases
1. Gases diffuse into one another, and so samples of gas placed in
the same container mix completely. Conversely, different gases
in a mixture do not separate on standing.
2. The amounts and properties of gases are described in terms of
temperature, pressure, the volume occupied, and the number of
molecules present. For example, a sample of gas occupies a
greater volume when hot than it does when cold at the same
pressure, but the number of molecules does not change.
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Pressure
• Pressure is defined as
the amount of force
per unit area.
F
P=
A
• Atmospheric
pressure is the
weight of air per
unit of area.
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Units of Pressure
• Pascals
– 1 Pa = 1 N/m2
• Bar
– 1 bar = 105 Pa = 100 kPa
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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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Manometer
This device is used to
measure the difference in
pressure between
atmospheric pressure and
that of a gas in a vessel.
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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
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The four variables needed to define the state of a gas are P, V, T, & n
BOYLE’S LAW: THE VOLUME–PRESSURE RELATIONSHIP
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The four variables needed to define the state of a gas are P, V, T, & n
Boyle’s Law : THE VOLUME–PRESSURE RELATIONSHIP
The volume of a fixed quantity of gas at constant
temperature is inversely proportional to the
pressure.
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As P and V are
inversely proportional
A plot of V versus P
results in a curve.
Since PV = k
V = k (1/P)
This means a plot of
V versus 1/P will be
a straight line. ©
2009, Prentice-Hall, Inc.
Let us think about a fixed mass of gas at constant temperature, but at
two different conditions of pressure and volume. For the first condition
we can write
π·π π½π = π (ππππππππ π, π»)
For the second condition, we can write:
π·π π½π = π (constant n,T)
Because the right-hand sides of these two equations are the same, the
left-hand sides must be equal, or
π·π π½π = π·π π½π
Q1. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased to 2.4 atm?
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Q1. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased to 2.4 atm?
Q1a. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased by 2.4 atm?
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Charles’s Law : THE TEMPERATURE-VOLUME RELATIONSHIP
• The volume of a fixed
amount of gas at constant
pressure is directly
proportional to its absolute
temperature.
• i.e.,
V =k
T
A plot of V versus T will be a straight line.
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Absolute Temperature is measured in kelvin. The relationship
between the Celsius and Kelvin temperature scales
is K = °C + 273.15°.
Rearranging the expression gives V/T = k, a concise
statement of Charles’s Law. As the temperature increases,
the volume must increase proportionally. If we let subscripts
1 and 2 represent values for the same sample of gas at two
different temperatures, we obtain
π½π π½π
=
π»π π»π
which is the more useful form of Charles’s Law. This relationship
is valid only when temperature,T, is expressed on an absolute
(usually the Kelvin) scale.
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Q2. A sample of nitrogen occupies 117 mL at 100.°C. At
what temperature in °C would it occupy 234 mL if the
pressure did not change?
Combined Gas Equation is the combination of Boyle’s
and Charles law. It is given by:
π·π π½π π·π π½π
=
π»π
π»π
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Q3. A sample of neon occupies 105 liters at 27°C
under a pressure of 985 torr. What volume would it
occupy at standard temperature and pressure
(STP)?
Standard temperature and pressure (STP) are, by international
agreement, exactly 0°C (273.15 K) and one atmosphere of
pressure (760. torr).
Q4. A sample of gas occupies 12.0 liters at 240.°C
under a pressure of 80.0 kPa. At what temperature
would the gas occupy 15.0 liters if the pressure were
increased to 107 kPa?
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Avogadro’s Law : THE QUANTITY-VOLUME RELATIONSHIP
Avogadro’s hypothesis: Equal volumes of gases at the same
temperature and pressure contains equal number of molecules
• The volume of a gas at constant temperature and
pressure is directly proportional to the number of
moles of the gas.
• Mathematically, this means
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V = kn
Avogadro’s law for two conditions can be expressed as
π1 π2
=
(ππππ π‘πππ‘ π, π)
π1 π2
The standard molar volume of an ideal gas is taken to be 22.414
liters per mole at STP.
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Ideal-Gas Equation
• So far we’ve seen that
V ο΅ 1/P (Boyle’s law)
V ο΅ T (Charles’s law)
V ο΅ n (Avogadro’s law)
• Combining these, we get
nT
Vο΅
P
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Ideal-Gas Equation
The relationship
then becomes
nT
Vο΅
P
nT
V=R
P
or
PV = nRT
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Ideal-Gas Equation
The constant of
proportionality is
known as R, the gas
constant.
The numerical value of R, the universal gas constant, depends
on the choices of the units for P, V, and T. One mole of an ideal
gas occupies 22.414 liters at 1.0000 atmosphere and 273.15 K
(STP).
Solving the ideal gas law for R gives R = 0.082057
In working problems, we often round R to 0.0821 Latm/molK.
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Q5. A sample of neon occupies 105 liters at 27°C under a
pressure of 985 torr. What volume would
it occupy at standard temperature and pressure (STP)?
Q6. A sample of gas occupies 12.0 liters at 240.°C
under a pressure of 80.0 kPa. At what temperature
would the gas occupy 15.0 liters if the pressure were
increased to 107 kPa?
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Q7. What is the volume of a gas balloon filled with
4.00 moles of He when the atmospheric pressure
is 748 torr and the temperature is 30.°C?
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Q8. A 0.109-gram sample of a pure gaseous compound
occupies 112 mL at 100.°C and 750. torr. What is the
molecular weight of the compound?
Q9. A 120.-mL flask contained 0.345 gram of a gaseous
compound at 100.°C and 1.00 atm pressure. What is
the molecular weight of the compound?
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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
P
=
V
RT
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Densities of Gases
• We know that
– moles ο΄ molecular mass = mass
nο΄ο=m
• So multiplying both sides by the
molecular mass (ο ) gives
m Pο
=
V RT
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Densities of Gases
• Mass οΈ volume = density
• So,
m Pο
d=
=
V RT
Note: One only needs to know the
molecular mass, the pressure, and the
temperature to calculate the density of
a gas.
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ππ = ππ π
ππ. 1
π
π
=
ππ. 2
π π π
πππ π , π
ππππ, π =
ππ. 3
πππππ πππ π , π
π π’ππ π‘ ππ. 3 intπ 2
π
π
=
ππ π π
π ππ
=
π
π π
(π₯ πππ‘β π πππ ππ¦ π
(π = π π
ππ
π=
π π
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Molecular Mass
We can manipulate the density equation to
enable us to find the molecular mass of a
gas:
Pο
d=
RT
Becomes
dRT
ο= P
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Q10. Nitric acid, a very important industrial chemical, is made
by dissolving the gas nitrogen dioxide, NO2, in water.
Calculate the density of NO2 gas, in g/L, at 1.24 atm and 50.°C.
Q11. A 0.109-gram sample of a pure gaseous compound
occupies 112 mL at 100.°C and 750. torr. What is the
molecular weight of the compound?
Q12. A 120.-mL flask contained 0.345 gram of a gaseous
compound at 100.°C and 1.00 atm pressure. What is the
molecular weight of the compound?
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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 + …
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Partial Pressure and mole fractionA
The mole fraction, XA, of component A in a
mixture is defined as
Like any other fraction, mole fraction is a dimensionless quantity. For
each component in a mixture, the mole fraction is
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Partial Pressure and mole fraction
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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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Kinetic-Molecular Theory
This is a model that aids
in our understanding of
what happens to gas
particles as
environmental
conditions change.
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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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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.
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Main Tenets of Kinetic-Molecular
Theory
Attractive and
repulsive forces
between gas
molecules are
negligible.
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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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Main Tenets of Kinetic-Molecular
Theory
The average kinetic
energy of the
molecules is
proportional to the
absolute temperature.
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Kinetic Theory in summary
1.
Gases consist of large number of molecules that are in continuous,
random, straight-line motion with varying velocities
2.
The volume of all the molecules of the gas is negligible compared
to the total volume of the container
3.
Attractive and repulsive forces between gas molecules are
negligible
4.
The collisions between gas molecules and with the walls of the
container are perfectly elastic: the average kinetic energy of the
molecule does change with time or the the total energy is
conserved during a collision; that is, there is no net energy gain or
loss
5.
The average kinetic energy of the molecule is proportional to the
absolute temperature. At any given temperature, the molecules of
all gases have the same average kinetic energy.
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The KMT hangs on these three major assumptions
1.Gas molecules are have negligible volume/size compared
to the container – point mass concept
2.There is no intermolecular interaction - attraction and
repulsion forces are negligible
3.Collision is elastic
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The KMT
So how does KMT explains the observable gas properties?
Pressure: This results from the collision of the molecules with the wall of the
container. The magnitude of pressure depends on how often (frequency) or how
forceful (speed) the molecules collide with the container.
The absolute temperature is a measure of the average kinetic energy of the
molecule. That is molecular motion increases with increasing temp
The number of collision is influenced by the concentration of the gas molecule
while the speed depends on the temperature.
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The KMT
So how does KMT explains the observable gas properties?
We can relate pressure to speed by the equation:
1 πππ΄
π=
ππ’2
3 π
π = πππ π ππ ππππππ’ππ; ππ΄ = π΄π£ππππππ ππππ tanππ; π = π£πππ’ππ;
u = ππππ‘ ππππ π ππ’πππ (πππ ) π ππππ
Rearranging,
pV ο½
1
nNA mc 2
3
Introducing kinetic energy which relates to temperature gives an equation similar
to the ideal gas equation.
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Please note that although the molecules in a sample of gas have average kinetic
energy and hence average speed, the individual molecules have varying speeds
Root mean square velocity is the speed of a molecule possessing average
kinetic energy
It is the measure of the speed of particles in a gas that is most convenient for
problem solving within the kinetic theory of gases because it is related directly to
the kinetic energy of the gas molecules
If the average kinetic energies of molecules of different gases are equal at a
given temperature. It means that smaller masses must have a higher speed to
maintain a constant value of energy.
Use KMT to explain Boyles law (constant Temp)
and effect of a temp increase at constant volume.
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From kinetic molecular theory,
π’=
3π π
π
The equation above relates the speed of a gas to its molar mass
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Because gas molecules are in constant, rapid, random motion, they
diffuse quickly throughout any container. For example, if hydrogen sulfide
(the smell of rotten eggs) is released in a large room, the odor can
eventually be detected throughout the room. If a mixture of gases is
placed in a container with thin porous walls, the molecules effuse through
the walls. Because they move faster, lighter gas molecules effuse through
the tiny openings of porous materials faster than heavier molecules.
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Effusion
Effusion is the
escape of gas
molecules
through a tiny
hole into an
evacuated
space.
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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.
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Diffusion
Diffusion is the spread
of one substance
throughout a space or
throughout a second
substance.
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Graham’s law defines the relationship of the speed of gas diffusion (mixing of
gases due to their kinetic energy) or effusion (movement of a gas through a tiny
opening) and the molecular mass (or density). In general, the lighter the gas, the
faster is its rate of effusion. Normally we use a comparison of the effusion rates of
two gases with the specific relationship being:
ππ
=
ππ
ππ
=
ππ
π΄π
π΄π
ππ
ππ
Where r1 and r2 are the rates of effusion/diffusion of gases 1 and 2,
respectively. M2 and M1 are the molecular (molar) masses of gases 2 and 1,
respectively
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ππ
=
ππ
π΄π
π΄π
In the above formula, the rate, r, can also be defined as volume/time.
If it takes 2 minutes 25 seconds for a 500 mL of methane (CH4)
gas to pass through a hole, how long will it take the same volume
of sulphur dioxide (SO2) gas to pass through the same hole?
If V cm3 of NH3 effuse at a rate of 2.25 cm3 s-1, and V cm3 of an unknown gas
X effuse at a rate of 1.40 cm3s-1 under the same experimental conditions, what
is Mr for X? Suggest a possible identity for X.
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REAL VS IDEAL GASES
What does the term ideal gas mean? An ideal gas is one whose
particles take up no space and have no intermolecular attractive
forces. An ideal gas follows the gas laws under all conditions of
temperature and pressure.
In the real world, no gas is truly ideal. All gas particles have some
volume, however small it may be, because of the sizes of their atoms
and the lengths of their bonds. All gas particles also are subject to
intermolecular interactions.
Despite that, most gases will behave like ideal gases at many
temperature and pressure levels.
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Real Gases
In the real world, the
behavior of gases only
conforms to the idealgas equation at
relatively high
temperature and low
pressure.
Nonideal gas behaviour (deviation from the predictions of the ideal gas
laws) is most significant at high pressures and/or low temperatures,
that is, near the conditions under which the gas liquefies.
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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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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. ©
2009, Prentice-Hall, Inc.
interpretation of deviations from ideal behaviour. (a) A sample of gas at a low
temperature. Each sphere represents a molecule. Because of their low kinetic
energies, attractive forces between molecules can now cause a few molecules
to “stick together.” (b) A sample of gas under high pressure. The molecules are
quite close together. The free volume is now a much smaller fraction of the
total volume.
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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.
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According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the molecules themselves becomes a significant
fraction of the total volume occupied by the gas. As a result,
the available volume, Vavailable, for any molecule to move in is
less than the measured volume by an amount that depends
on the volume excluded by the presence of the other
molecules. To account for this, we subtract a correction
factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
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According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the
molecules themselves becomes a
significant fraction of the total volume occupied by the gas.
As a result, the available volume, Vavailable, for any molecule
to move in is less than the measured volume by an amount
that depends on the volume excluded by the presence of
the other molecules. To account for this, we subtract a
correction factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
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According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the
molecules themselves becomes a
significant fraction of the total volume occupied by the gas.
As a result, the available volume, Vavailable, for any molecule
to move in is less than the measured volume by an amount
that depends on the volume excluded by the presence of
the other molecules. To account for this, we subtract a
correction factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
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The factor nb corrects for the volume occupied by the molecules
themselves. Larger molecules have greater values of b, and the
greater the number of molecules in a sample (higher n), the larger is
the volume correction. The correction term becomes negligibly small,
however, when the volume is large.
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The kinetic–molecular theory assumes that attractive forces between molecules
are insignificant (at higher temp and low pressure).
When the temperature is quite low (low kinetic energy), and at low pressure, the
molecules move so slowly and they are very close together.
The above results in (1) fewer collisions and (2) less energetic collision owing
to attraction.
As a consequence, the pressure that the gas exerts, Pmeasured, is less than the
pressure it would exert if attractions were truly negligible, Pideally exerted.
To correct for this, we subtract a correction factor, n2a/V2, from the ideal
pressure
π·ππππππππ = π· ππ πππππ πππππππ −
Physical Chemistry 2
ππ π
π½π ππππππππ
73
π·ππ πππππ πππππππ = π· ππππππππ +
ππ π
π½π ππππππππ
We can then substitute both corrections into ideal gas law, PV =nRT
π·ππππππππ +
πππ
π½π ππππππππ
πππ
π·+ π
π½
π½ππππππππ − ππ = ππΉπ»
π − ππ = ππΉπ»
This is the van der Waals equation. In this equation, P, V, T, and n represent
the measured values of pressure, volume, temperature.
When a and b are both zero, the van der Waals equation reduces to the ideal
gas equation.
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The van der Waals Equation
n2 a
(P + 2 ) (V − nb) = nRT
V
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Calculate the pressure exerted by 1.00 mole of methane, CH4, in a
500.-mL vessel at 25.0°C assuming (a) ideal behaviour and (b)
nonideal behaviour.
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Good morning, May I welcome you from the holiday with this:
30) A sample of gas (1.9 mol) is in a flask at 21 °C and
697 mm Hg. The flask is opened and more gas is added
to the flask. The new pressure is 795 mm Hg and the
temperature is now 26 °C. There are now __________
mol of gas in the flask.
34) The mass of nitrogen dioxide contained in a 4.32 L
vessel at 48 °C and 141600 Pa is __________ g.
36) The density of N2O at 1.53 atm and 45.2 °C is
__________ g/L.
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Two tanks are connected by a closed valve. Each tank is filled with gas as
shown, and both tanks are held at the same temperature. We open the
valve and allow the gases to mix.
(a) After the gases mix, what is the partial pressure of each gas, and what
is the total pressure?
(b) What is the mole fraction of each gas in the mixture?
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Hydrogen was collected over water at 21°C on
a day when the atmospheric pressure was 748
torr. The volume of the gas sample collected
was 300. mL. (a) How many moles of H2 were
present? (b) How many moles of water vapor
were present in the moist gas mixture? (c)
What is the mole fraction of hydrogen in the
moist gas mixture? (d) What would be the
mass of the gas sample if it were dry?
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A quantity of Neon originally held at 30 kPa in a 150 cm3
container at 30 °C is transferred to a 500 cm3 container at 20 °C.
A quantitiy of oxygen gas originally held at 45 kPa in a 100 cm3
container at 25 °C is transferred to this same container (i.e. 500
cm3 at 20 °C). Calculate the total pressure of the gas mixture in
the 500 cm3 container.
A gas
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1
Characteristics of Gases
• Unlike liquids and solids, gases
–
–
–
–
expand to fill their containers;
are highly compressible;
have extremely low densities
Gases form homogenous mixtures
Common properties of gases that are
measured are: Pressure, Temperature and
volume
© 2009, Prentice-Hall, Inc.
Characteristics of Gases
1. Gases can be compressed into smaller volumes; that
is, their densities can be increased by applying
increased pressure.
2. Gases exert pressure on their surroundings; in turn,
pressure must be exerted to confine gases.
3. Gases expand without limits, and so gas samples
completely and uniformly occupy the volume of any
container.
Physical Chemistry 2
3
Characteristics of Gases
1. Gases diffuse into one another, and so samples of gas placed in
the same container mix completely. Conversely, different gases
in a mixture do not separate on standing.
2. The amounts and properties of gases are described in terms of
temperature, pressure, the volume occupied, and the number of
molecules present. For example, a sample of gas occupies a
greater volume when hot than it does when cold at the same
pressure, but the number of molecules does not change.
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4
Pressure
• Pressure is defined as
the amount of force
per unit area.
F
P=
A
• Atmospheric
pressure is the
weight of air per
unit of area.
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Units of Pressure
• Pascals
– 1 Pa = 1 N/m2
• Bar
– 1 bar = 105 Pa = 100 kPa
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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
© 2009, Prentice-Hall, Inc.
Manometer
This device is used to
measure the difference in
pressure between
atmospheric pressure and
that of a gas in a vessel.
© 2009, Prentice-Hall, Inc.
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
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The four variables needed to define the state of a gas are P, V, T, & n
BOYLE’S LAW: THE VOLUME–PRESSURE RELATIONSHIP
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The four variables needed to define the state of a gas are P, V, T, & n
Boyle’s Law : THE VOLUME–PRESSURE RELATIONSHIP
The volume of a fixed quantity of gas at constant
temperature is inversely proportional to the
pressure.
© 2009, Prentice-Hall, Inc.
As P and V are
inversely proportional
A plot of V versus P
results in a curve.
Since PV = k
V = k (1/P)
This means a plot of
V versus 1/P will be
a straight line. ©
2009, Prentice-Hall, Inc.
Let us think about a fixed mass of gas at constant temperature, but at
two different conditions of pressure and volume. For the first condition
we can write
π·π π½π = π (ππππππππ π, π»)
For the second condition, we can write:
π·π π½π = π (constant n,T)
Because the right-hand sides of these two equations are the same, the
left-hand sides must be equal, or
π·π π½π = π·π π½π
Q1. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased to 2.4 atm?
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13
Q1. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased to 2.4 atm?
Q1a. A sample of gas occupies 12 L under a pressure of 1.2
atm. What would its volume be if the pressure were
increased by 2.4 atm?
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14
Charles’s Law : THE TEMPERATURE-VOLUME RELATIONSHIP
• The volume of a fixed
amount of gas at constant
pressure is directly
proportional to its absolute
temperature.
• i.e.,
V =k
T
A plot of V versus T will be a straight line.
© 2009, Prentice-Hall, Inc.
Absolute Temperature is measured in kelvin. The relationship
between the Celsius and Kelvin temperature scales
is K = °C + 273.15°.
Rearranging the expression gives V/T = k, a concise
statement of Charles’s Law. As the temperature increases,
the volume must increase proportionally. If we let subscripts
1 and 2 represent values for the same sample of gas at two
different temperatures, we obtain
π½π π½π
=
π»π π»π
which is the more useful form of Charles’s Law. This relationship
is valid only when temperature,T, is expressed on an absolute
(usually the Kelvin) scale.
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Q2. A sample of nitrogen occupies 117 mL at 100.°C. At
what temperature in °C would it occupy 234 mL if the
pressure did not change?
Combined Gas Equation is the combination of Boyle’s
and Charles law. It is given by:
π·π π½π π·π π½π
=
π»π
π»π
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17
Q3. A sample of neon occupies 105 liters at 27°C
under a pressure of 985 torr. What volume would it
occupy at standard temperature and pressure
(STP)?
Standard temperature and pressure (STP) are, by international
agreement, exactly 0°C (273.15 K) and one atmosphere of
pressure (760. torr).
Q4. A sample of gas occupies 12.0 liters at 240.°C
under a pressure of 80.0 kPa. At what temperature
would the gas occupy 15.0 liters if the pressure were
increased to 107 kPa?
Physical Chem 2
18
Avogadro’s Law : THE QUANTITY-VOLUME RELATIONSHIP
Avogadro’s hypothesis: Equal volumes of gases at the same
temperature and pressure contains equal number of molecules
• The volume of a gas at constant temperature and
pressure is directly proportional to the number of
moles of the gas.
• Mathematically, this means
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V = kn
Avogadro’s law for two conditions can be expressed as
π1 π2
=
(ππππ π‘πππ‘ π, π)
π1 π2
The standard molar volume of an ideal gas is taken to be 22.414
liters per mole at STP.
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20
Ideal-Gas Equation
• So far we’ve seen that
V ο΅ 1/P (Boyle’s law)
V ο΅ T (Charles’s law)
V ο΅ n (Avogadro’s law)
• Combining these, we get
nT
Vο΅
P
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Ideal-Gas Equation
The relationship
then becomes
nT
Vο΅
P
nT
V=R
P
or
PV = nRT
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Ideal-Gas Equation
The constant of
proportionality is
known as R, the gas
constant.
The numerical value of R, the universal gas constant, depends
on the choices of the units for P, V, and T. One mole of an ideal
gas occupies 22.414 liters at 1.0000 atmosphere and 273.15 K
(STP).
Solving the ideal gas law for R gives R = 0.082057
In working problems, we often round R to 0.0821 Latm/molK.
© 2009, Prentice-Hall, Inc.
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Q5. A sample of neon occupies 105 liters at 27°C under a
pressure of 985 torr. What volume would
it occupy at standard temperature and pressure (STP)?
Q6. A sample of gas occupies 12.0 liters at 240.°C
under a pressure of 80.0 kPa. At what temperature
would the gas occupy 15.0 liters if the pressure were
increased to 107 kPa?
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26
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27
Q7. What is the volume of a gas balloon filled with
4.00 moles of He when the atmospheric pressure
is 748 torr and the temperature is 30.°C?
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28
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29
Q8. A 0.109-gram sample of a pure gaseous compound
occupies 112 mL at 100.°C and 750. torr. What is the
molecular weight of the compound?
Q9. A 120.-mL flask contained 0.345 gram of a gaseous
compound at 100.°C and 1.00 atm pressure. What is
the molecular weight of the compound?
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30
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31
Densities of Gases
If we divide both sides of the ideal-gas
equation by V and by RT, we get
n
P
=
V
RT
© 2009, Prentice-Hall, Inc.
Densities of Gases
• We know that
– moles ο΄ molecular mass = mass
nο΄ο=m
• So multiplying both sides by the
molecular mass (ο ) gives
m Pο
=
V RT
© 2009, Prentice-Hall, Inc.
Densities of Gases
• Mass οΈ volume = density
• So,
m Pο
d=
=
V RT
Note: One only needs to know the
molecular mass, the pressure, and the
temperature to calculate the density of
a gas.
© 2009, Prentice-Hall, Inc.
ππ = ππ π
ππ. 1
π
π
=
ππ. 2
π π π
πππ π , π
ππππ, π =
ππ. 3
πππππ πππ π , π
π π’ππ π‘ ππ. 3 intπ 2
π
π
=
ππ π π
π ππ
=
π
π π
(π₯ πππ‘β π πππ ππ¦ π
(π = π π
ππ
π=
π π
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35
Molecular Mass
We can manipulate the density equation to
enable us to find the molecular mass of a
gas:
Pο
d=
RT
Becomes
dRT
ο= P
© 2009, Prentice-Hall, Inc.
Q10. Nitric acid, a very important industrial chemical, is made
by dissolving the gas nitrogen dioxide, NO2, in water.
Calculate the density of NO2 gas, in g/L, at 1.24 atm and 50.°C.
Q11. A 0.109-gram sample of a pure gaseous compound
occupies 112 mL at 100.°C and 750. torr. What is the
molecular weight of the compound?
Q12. A 120.-mL flask contained 0.345 gram of a gaseous
compound at 100.°C and 1.00 atm pressure. What is the
molecular weight of the compound?
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37
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 + …
© 2009, Prentice-Hall, Inc.
Physical Chem 2
39
Partial Pressure and mole fractionA
The mole fraction, XA, of component A in a
mixture is defined as
Like any other fraction, mole fraction is a dimensionless quantity. For
each component in a mixture, the mole fraction is
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40
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41
Partial Pressure and mole fraction
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43
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.
© 2009, Prentice-Hall, Inc.
Kinetic-Molecular Theory
This is a model that aids
in our understanding of
what happens to gas
particles as
environmental
conditions change.
© 2009, Prentice-Hall, Inc.
Main Tenets of Kinetic-Molecular
Theory
Gases consist of large numbers of molecules
that are in continuous, random motion.
© 2009, Prentice-Hall, Inc.
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.
© 2009, Prentice-Hall, Inc.
Main Tenets of Kinetic-Molecular
Theory
Attractive and
repulsive forces
between gas
molecules are
negligible.
© 2009, Prentice-Hall, Inc.
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.
© 2009, Prentice-Hall, Inc.
Main Tenets of Kinetic-Molecular
Theory
The average kinetic
energy of the
molecules is
proportional to the
absolute temperature.
© 2009, Prentice-Hall, Inc.
Kinetic Theory in summary
1.
Gases consist of large number of molecules that are in continuous,
random, straight-line motion with varying velocities
2.
The volume of all the molecules of the gas is negligible compared
to the total volume of the container
3.
Attractive and repulsive forces between gas molecules are
negligible
4.
The collisions between gas molecules and with the walls of the
container are perfectly elastic: the average kinetic energy of the
molecule does change with time or the the total energy is
conserved during a collision; that is, there is no net energy gain or
loss
5.
The average kinetic energy of the molecule is proportional to the
absolute temperature. At any given temperature, the molecules of
all gases have the same average kinetic energy.
© 2009, Prentice-Hall, Inc.
51
The KMT hangs on these three major assumptions
1.Gas molecules are have negligible volume/size compared
to the container – point mass concept
2.There is no intermolecular interaction - attraction and
repulsion forces are negligible
3.Collision is elastic
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52
The KMT
So how does KMT explains the observable gas properties?
Pressure: This results from the collision of the molecules with the wall of the
container. The magnitude of pressure depends on how often (frequency) or how
forceful (speed) the molecules collide with the container.
The absolute temperature is a measure of the average kinetic energy of the
molecule. That is molecular motion increases with increasing temp
The number of collision is influenced by the concentration of the gas molecule
while the speed depends on the temperature.
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53
The KMT
So how does KMT explains the observable gas properties?
We can relate pressure to speed by the equation:
1 πππ΄
π=
ππ’2
3 π
π = πππ π ππ ππππππ’ππ; ππ΄ = π΄π£ππππππ ππππ tanππ; π = π£πππ’ππ;
u = ππππ‘ ππππ π ππ’πππ (πππ ) π ππππ
Rearranging,
pV ο½
1
nNA mc 2
3
Introducing kinetic energy which relates to temperature gives an equation similar
to the ideal gas equation.
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54
Please note that although the molecules in a sample of gas have average kinetic
energy and hence average speed, the individual molecules have varying speeds
Root mean square velocity is the speed of a molecule possessing average
kinetic energy
It is the measure of the speed of particles in a gas that is most convenient for
problem solving within the kinetic theory of gases because it is related directly to
the kinetic energy of the gas molecules
If the average kinetic energies of molecules of different gases are equal at a
given temperature. It means that smaller masses must have a higher speed to
maintain a constant value of energy.
Use KMT to explain Boyles law (constant Temp)
and effect of a temp increase at constant volume.
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55
From kinetic molecular theory,
π’=
3π π
π
The equation above relates the speed of a gas to its molar mass
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56
Because gas molecules are in constant, rapid, random motion, they
diffuse quickly throughout any container. For example, if hydrogen sulfide
(the smell of rotten eggs) is released in a large room, the odor can
eventually be detected throughout the room. If a mixture of gases is
placed in a container with thin porous walls, the molecules effuse through
the walls. Because they move faster, lighter gas molecules effuse through
the tiny openings of porous materials faster than heavier molecules.
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Effusion
Effusion is the
escape of gas
molecules
through a tiny
hole into an
evacuated
space.
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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.
© 2009, Prentice-Hall, Inc.
Diffusion
Diffusion is the spread
of one substance
throughout a space or
throughout a second
substance.
© 2009, Prentice-Hall, Inc.
Graham’s law defines the relationship of the speed of gas diffusion (mixing of
gases due to their kinetic energy) or effusion (movement of a gas through a tiny
opening) and the molecular mass (or density). In general, the lighter the gas, the
faster is its rate of effusion. Normally we use a comparison of the effusion rates of
two gases with the specific relationship being:
ππ
=
ππ
ππ
=
ππ
π΄π
π΄π
ππ
ππ
Where r1 and r2 are the rates of effusion/diffusion of gases 1 and 2,
respectively. M2 and M1 are the molecular (molar) masses of gases 2 and 1,
respectively
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ππ
=
ππ
π΄π
π΄π
In the above formula, the rate, r, can also be defined as volume/time.
If it takes 2 minutes 25 seconds for a 500 mL of methane (CH4)
gas to pass through a hole, how long will it take the same volume
of sulphur dioxide (SO2) gas to pass through the same hole?
If V cm3 of NH3 effuse at a rate of 2.25 cm3 s-1, and V cm3 of an unknown gas
X effuse at a rate of 1.40 cm3s-1 under the same experimental conditions, what
is Mr for X? Suggest a possible identity for X.
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62
REAL VS IDEAL GASES
What does the term ideal gas mean? An ideal gas is one whose
particles take up no space and have no intermolecular attractive
forces. An ideal gas follows the gas laws under all conditions of
temperature and pressure.
In the real world, no gas is truly ideal. All gas particles have some
volume, however small it may be, because of the sizes of their atoms
and the lengths of their bonds. All gas particles also are subject to
intermolecular interactions.
Despite that, most gases will behave like ideal gases at many
temperature and pressure levels.
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Real Gases
In the real world, the
behavior of gases only
conforms to the idealgas equation at
relatively high
temperature and low
pressure.
Nonideal gas behaviour (deviation from the predictions of the ideal gas
laws) is most significant at high pressures and/or low temperatures,
that is, near the conditions under which the gas liquefies.
© 2009, Prentice-Hall, Inc.
Real Gases
Even the same gas will
show wildly different
behavior under high
pressure at different
temperatures.
© 2009, Prentice-Hall, Inc.
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. ©
2009, Prentice-Hall, Inc.
interpretation of deviations from ideal behaviour. (a) A sample of gas at a low
temperature. Each sphere represents a molecule. Because of their low kinetic
energies, attractive forces between molecules can now cause a few molecules
to “stick together.” (b) A sample of gas under high pressure. The molecules are
quite close together. The free volume is now a much smaller fraction of the
total volume.
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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.
© 2009, Prentice-Hall, Inc.
According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the molecules themselves becomes a significant
fraction of the total volume occupied by the gas. As a result,
the available volume, Vavailable, for any molecule to move in is
less than the measured volume by an amount that depends
on the volume excluded by the presence of the other
molecules. To account for this, we subtract a correction
factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
Physical Chemistry 2
69
According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the
molecules themselves becomes a
significant fraction of the total volume occupied by the gas.
As a result, the available volume, Vavailable, for any molecule
to move in is less than the measured volume by an amount
that depends on the volume excluded by the presence of
the other molecules. To account for this, we subtract a
correction factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
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According to the kinetic–molecular theory, the molecules
are so small, relative to the total volume of the gas, that
each molecule can move through virtually the entire
measured volume of the container, Vmeasured .
But under high pressures, a gas is compressed so that the
volume of the
molecules themselves becomes a
significant fraction of the total volume occupied by the gas.
As a result, the available volume, Vavailable, for any molecule
to move in is less than the measured volume by an amount
that depends on the volume excluded by the presence of
the other molecules. To account for this, we subtract a
correction factor,
π½ ππ πππππ πππππππππ = π½ ππππππππ − ππ
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The factor nb corrects for the volume occupied by the molecules
themselves. Larger molecules have greater values of b, and the
greater the number of molecules in a sample (higher n), the larger is
the volume correction. The correction term becomes negligibly small,
however, when the volume is large.
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72
The kinetic–molecular theory assumes that attractive forces between molecules
are insignificant (at higher temp and low pressure).
When the temperature is quite low (low kinetic energy), and at low pressure, the
molecules move so slowly and they are very close together.
The above results in (1) fewer collisions and (2) less energetic collision owing
to attraction.
As a consequence, the pressure that the gas exerts, Pmeasured, is less than the
pressure it would exert if attractions were truly negligible, Pideally exerted.
To correct for this, we subtract a correction factor, n2a/V2, from the ideal
pressure
π·ππππππππ = π· ππ πππππ πππππππ −
Physical Chemistry 2
ππ π
π½π ππππππππ
73
π·ππ πππππ πππππππ = π· ππππππππ +
ππ π
π½π ππππππππ
We can then substitute both corrections into ideal gas law, PV =nRT
π·ππππππππ +
πππ
π½π ππππππππ
πππ
π·+ π
π½
π½ππππππππ − ππ = ππΉπ»
π − ππ = ππΉπ»
This is the van der Waals equation. In this equation, P, V, T, and n represent
the measured values of pressure, volume, temperature.
When a and b are both zero, the van der Waals equation reduces to the ideal
gas equation.
Physical Chemistry 2
74
The van der Waals Equation
n2 a
(P + 2 ) (V − nb) = nRT
V
© 2009, Prentice-Hall, Inc.
Calculate the pressure exerted by 1.00 mole of methane, CH4, in a
500.-mL vessel at 25.0°C assuming (a) ideal behaviour and (b)
nonideal behaviour.
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Good morning, May I welcome you from the holiday with this:
30) A sample of gas (1.9 mol) is in a flask at 21 °C and
697 mm Hg. The flask is opened and more gas is added
to the flask. The new pressure is 795 mm Hg and the
temperature is now 26 °C. There are now __________
mol of gas in the flask.
34) The mass of nitrogen dioxide contained in a 4.32 L
vessel at 48 °C and 141600 Pa is __________ g.
36) The density of N2O at 1.53 atm and 45.2 °C is
__________ g/L.
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79
Two tanks are connected by a closed valve. Each tank is filled with gas as
shown, and both tanks are held at the same temperature. We open the
valve and allow the gases to mix.
(a) After the gases mix, what is the partial pressure of each gas, and what
is the total pressure?
(b) What is the mole fraction of each gas in the mixture?
Physical Chemistry 2
80
Hydrogen was collected over water at 21°C on
a day when the atmospheric pressure was 748
torr. The volume of the gas sample collected
was 300. mL. (a) How many moles of H2 were
present? (b) How many moles of water vapor
were present in the moist gas mixture? (c)
What is the mole fraction of hydrogen in the
moist gas mixture? (d) What would be the
mass of the gas sample if it were dry?
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81
A quantity of Neon originally held at 30 kPa in a 150 cm3
container at 30 °C is transferred to a 500 cm3 container at 20 °C.
A quantitiy of oxygen gas originally held at 45 kPa in a 100 cm3
container at 25 °C is transferred to this same container (i.e. 500
cm3 at 20 °C). Calculate the total pressure of the gas mixture in
the 500 cm3 container.
A gas
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