chapter 9 - gases...the kinetic-molecular theory of gases • a gas consists of a large collection...
TRANSCRIPT
CHANGING CONDITIONS The Ideal Gas Law can also explain the
manner in which a gas changes when
conditions change (e.g. increase P,
decrease T...):
= R P1V1
n1T1
P2V2
n2T2
=
Subscript 1: Initial conditions
Subscript 2: Final conditions
QUESTION What is the effect of increasing the
pressure by a factor of four on an ideal
gas, if the number of moles and
temperature are held constant?
The volume increases by a factor of four
The volume decreases by a factor of four
The volume remains the same
The volume doubles
Answer
A -
B -
C -
D -
• Initially discovered through observations of
gas behavior - the changing of gas volumes.
• Macroscopic behavior of gases explained by
Kinetic-Molecular Theory.
UNDERSTANDING THE
IDEAL GAS LAW
BOYLE’S LAW The volume of an ideal gas is inversely
proportional to the external pressure - at a fixed
T and n:
P1V1 P2V2
n1T1 n2T2
= P1V1 = P2V2
V2 = (P1/P2)V1
MOLECULAR VIEW OF
BOYLE’S LAW
Pext increased
T, n constant
More frequent collisions (force) over a smaller
surface area.
CHARLES’ LAW The volume of an ideal gas is directly
proportional to the temperature of the gas - at a
fixed P and n.
P1V1 P2V2
n1T1 n2T2
=
V2 = (T2/T1)V1
V1 = V2
T1 T2
MOLECULAR VIEW OF
CHARLES’ LAW
CHARLES’ LAW V2 = (T2/T1)V1
CHARLES’ LAW AND
ABSOLUTE ZERO
• Celsius scale - arbitrary zero point
• Kelvin scale (K) - absolute
temperature scale; 0 K is the lowest
possible temperature
• -273.15°C = 0 K
• At 0 K (for ideal gases) - molecular
motion stops; zero volume for gases
QUESTION What is the effect of increasing the
temperature by a factor of two and
increasing the pressure by a factor of two on
an ideal gas, if the number of moles of the
gas is held constant?
The volume increases by a factor of four
The volume increases by a factor of two
The volume remains the same
The volume decreases by a factor of two
The volume decreases by a factor of four
Answer
A -
B -
C -
D -
E -
AVOGADRO’S LAW
The volume of an ideal gas is directly proportional to
the amount of gas - at fixed P and T.
P1V1 P2V2
n1T1 n2T2
=
V2 = (n2/n1)V1
V1 = V2
n1 n2
At a fixed P and T, equal volumes of an ideal
gas contain an equal number of particles.
MOLECULAR VIEW OF
AVOGADRO’S LAW
STOICHIOMETRIC
RELATIONSHIPS BETWEEN
GASEOUS REAGENTS
P, V, T
of gas A
moles of
gas A
moles of
gas B
P, V, T
of gas B
Ideal gas
law
Ideal gas
law
Molar ratio
from
balanced
equation
QUESTION Magnesium metal (0.100 mol) and 1.00 L of 0.500 M
hydrochloric acid are combined and react to completion.
How many liters of hydrogen gas, measured at 273.15 K
and 1.00 atm are produced?
Mg(s) + 2HCl(aq) → MgCl2(aq) + H2(g)
2.24 L of H2
4.48 L of H2
5.60 L of H2
11.2 L of H2
22.4 L of H2
Answer:
A
B
C
D
E
GAS DENSITY AND MOLAR MASS
n = mass
MW
n = PV
RT
MW = (mass)RT
PV
MW = dRT
P
d = mass
V
• The ideal gas law (PV=nRT)
can be used to determine the
density (d) and/or molecular
weight (MW) of the gas.
• The MW of a gas will always
remain the same, as it is based
on the mass of the atoms of
the gas.
• The density of a gas will
change with pressure, and
temperature.
PROBLEM
A 0.50 L bottle contains an unknown gas under a pressure of
3.0 atm at T = 300.K. The mass of the container is 267.37 g
when evacuated and 269.08 g when filled with the gas.
Which of the following could be the identity of the gas?
nitrogen
oxygen
fluorine
argon
carbon dioxide
DENSITY OF AN IDEAL GAS The density of an ideal gas depends on its
chemical identity (molar mass).
X moles of ideal gas H2(g) will occupy the
same volume as X moles of CH4(g).
Yet X moles of H2(g) has less mass than
CH4(g).
density = (molar mass)P
RT
Gases are
compressible: density
will vary greatly with
changes in T and P
CALCULATION QUESTION A 30.0 L balloon is filled with gas at 292.5 K
and has a pressure of 1.2 atm. How many
moles of gas are in the balloon?
Does it matter if the gas in the balloon is air,
helium, nitrogen or methane...?
STANDARD MOLAR VOLUME
One mole of any
idea gas will
occupy 22.4 L at
standard
temperature and
pressure.
The difference
will be in the
mass of the gas
(and the density
of the gas).
Standard Temperature and Pressure
(STP) = 0°C (273.15 K) and 1.00 atm
QUESTION List the following gases in order of increasing
density. Assume temperature and pressure are
constant.
Cl2<Kr<SO2
Kr<SO2<Cl2
SO2<Cl2<Kr
SO2<Kr<Cl2
Cl2<SO2<Kr
Answer:
A
B
C
D
E
PROBLEM What will be the density of CO2 at 4.00 atm
pressure and T = 300. K?
DALTON’S LAW OF
PARTIAL PRESSURES • When there is a mixture of gases, the total pressure of the
mixture is due to the sum of the individual gas pressures.
• These are termed partial pressures - the amount of
pressure produced by each individual type of gas.
Pair = Pnitrogen + Poxygen + Pcarbon dioxide + Pargon + ...
Ptotal = P1 + P2 + P3 + P4 + ...
Py = Xy × Ptotal
Xy = mole fraction of gas y = (moles of y)÷(total moles)
DALTON’S LAW PROBLEM What is the total pressure (in atm) when
1.00 mol of Ar, 0.400 mol of He, and 1.60
mol of N2 gases are injected into a 9.12 L
flask at 0.00°C?
What is the pressure of each gas in the mixture?
PROBLEM A 5.63 g sample of methane (CH4) is combusted with a
stoichiometric amount of oxygen. The gas produced from
the combustion is captured in a 6.48 L canister and is at a
temperature of 126°C. What is the pressure inside the
cylinder?
CH4(g) + 2O2(g) → CO2(g) + 2H2O(g)
PROBLEM Small metal cylinders, filled with 16.0 g of compressed
CO2(g) are often used by bicyclists to inflate flat tires. If
such a cylinder has a volume of 25.0 mL, and can fill a
1.25 L bicycle inner tube to a pressure of 7.05 atm at
22.4°C, what is the pressure of CO2(g) inside a sealed
cylinder at the same temperature?
DIFFUSION
The random thermal motion
of gases causes gas
particles to spread out.
Gas will diffuse from areas
of high concentration to
areas of low concentration.
Given enough time the gas
particles will be distributed
evenly (homogeneous
mixture).
DIFFUSION
• If gas molecules move so fast (~500 m/s) why is diffusion so
slow?
• Each molecule collides once every ~1 ns.
• The mean free path of a gas molecule (the average distance it
moves before it hits something) is ~70 nm (or 103 molecular
diameters).
EFFUSION • Effusion - The process by
which a gas molecule escapes
from a container through a tiny
hole(s).
• Graham’s Law of Effusion -
The rate of effusion of a gas is
inversely proportional to the
square root of its molar mass.
• Root-mean-squared speed
(urms)
rateA
rateB MA
MB
=
Rate: volume or number
of moles of gas per unit
time.
EFFUSION The escape of
molecules through a
tiny hole into a
vacuum is fastest for
smaller mass gases.
rateA
rateB MA
MB
=
QUESTION If it takes 20.0 minute for 0.350 moles of H2S(g) to diffuse
through a porous wall, how long would it take for 0.175
moles of krypton gas (Kr(g)) to diffuse through the same
barrier?
rateA
rateB MA
MB
=
Answer:
A - 10.0 min
B - 15.7 min
C - 20.0 min
D - 31.4 min
E - 42.1 min
QUESTION The rate of effusion for nitrogen gas has been
measured in an apparatus, and found to be 79 mL/s.
If measurement is repeated with sulfur dioxide, at the
same temperature and pressure, what will be the
effusion rate for sulfur dioxide?
THE KINETIC-MOLECULAR
THEORY OF GASES
• A gas consists of a large collection of individual particles that are
very small (no volume).
• Gas particles are in constant, random, straight-line, motion
(except for collisions)
• Collisions between particles are elastic - their total kinetic
energy (Ek) is constant.
• Between collisions, the gas particles do not influence each other
in any way (act independently).
HOW FAST DO THE
PARTICLES OF A GAS MOVE? Very fast!
500 m/s = 1800
km/h Speed increases with
temperature.
= (3/2)(R/NA)T Ek
Temperature is a
measure of molecular
motion.
NA=Avogadro's number
AVOGADRO’S LAW REVISITED
Why do equal numbers of particles of
an ideal gas occupy equal volumes at
constant temperature and pressure?
He (4 g/mol) Ar (40 g/mol)
= (3/2)(R/NA)T = ½mu2 Ek m = mass, u =
speed
MASS VERSUS SPEED
Gas particles with lower mass have higher speeds
Root-mean-squared speed (urms)
QUESTION Pressure is a measure of force per unit area.
What does this imply regarding the velocity of
gas particles?
•The velocity of gas particles is independent of
the mass of the particle.
•The velocity of gas particles is directly
proportional to the mass of the particles.
•The velocity of gas particles is inversely
proportional to the mass of the particles.
•All of the above statements are true.
•None of the above statements are true.
Answer
A
B
C
D
E
CALCULATION QUESTION What is the root-mean-square velocity for a
molecule of nitrogen gas at 30°C?
How fast does a molecule of SF6 travel at
the same temperature?
QUESTION A sample of an ideal gas is heated in a steel container
from 25°C to 100°C. Which quantity will remain
unchanged?
The average kinetic energy of the gas
particles.
The collision frequency.
The density.
The pressure of the gas.
Answer:
A
B
C
D
POSTULATES OF KINETIC-
MOLECULAR THEORY Postulate 1: Particle volume
Because the volume of an ideal gas particle is so small
compared to the volume of its container, the gas particles
are considered to have mass, but no volume.
Postulate 2: Particle motion
Gas particles are in constant, random, straight-line motion
except when they collide with each other or with the
container walls.
Postulate 3: Particle collisions Collisions are elastic, therefore the total kinetic energy of
the particles is constant.
REAL GASES
REAL VS IDEAL GASES
• Real gases do not act exactly as we predict ideal gases would
behave.
• Intermolecular Attractions - are much weaker than bonding, so
only seen under extreme conditions. Intermolecular attractions
reduce the force of the impact with the walls of the container.
• Molecular Volume - as the free volume (empty space) decreases,
the volume of gas molecules becomes significant.
QUESTION At very high pressures (~1000 atm) the
measured pressure exerted by a real gas is
greater than that predicted by an ideal gas.
Why is that?
•Because it is difficult to measure high pressures accurately.
•Because real gases will condense to form liquids at that pressure.
•Because gas phase collisions prevent the molecules from colliding with
the container walls.
•Because of the attractive intermolecular forces between the gas
molecules.
•Because the volume occupied by the gas molecules becomes
significant.
Answer:
A
B
C
D
E
INTERPARTICLE
ATTRACTIONS
• Interparticle attractions are very weak forces - much
weaker than the bonds in a molecule.
• At low pressures gas particles are far from each other so
the Interparticle attractions have little influence.
• At high external pressures the particles are closer
together and the Interparticle attractions becomes
significant. This also happens at very low temperatures.
INTERPARTICLE
ATTRACTION • Impact of interparticle attractions is a reduction in the velocity of the
particles.
• A lower particle velocity reduces the force of the collision with the walls.
• Overall this means the gas exerts less pressure on the container walls.
MOLECULAR VOLUME
• In the ideal model of a gas we presume that the volume of
the gas molecule is negligible in comparison to the free
space around each particle.
• As the pressure increases, the amount of free space for
particles to move is reduced, to the point where the gas
particles become a meaningful amount of that “free space”.
REAL VS IDEAL GASES
• Real gases do not act exactly as we predict ideal gases would
behave.
• Collisions are not elastic - Intermolecular attraction, though
very weak, and seen at low pressures, becomes a factor.
Intermolecular attractions reduce the force of the impact with
the walls of the container; slow down the gas particles.
• Molecular Volume - As the free volume (free space)
decreases, the volume of gas molecules becomes significant.
The gas molecules have less space to move without having a
collision, so the pressure is higher than predicted.
PVDW=45.9 atm
ADJUSTING THE IDEAL GAS LAW • To better describe real gasses we need to:
• Adjust P up, to account for inter particle
attractions.
• Adjust V down, to account to particle
volume.
• The values a and b were determined
experimentally by Johannes van der
Waals.
4.89 mol CO2 in 1.98 L at 299K:
Preal=44.8 atm
PIGL=60.6 atm