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Unit 10: Gases
Unit Outline
I. Introduction
II. Gas Pressure
III. Gas Laws
IV. Gas Law Problems
V. Kinetic-Molecular Theory of Gases
VI. Real Gases
I. Opening thoughts…
Have you ever:
Seen a hot air balloon?
Had a soda bottle spray all over you?
Baked (or eaten) a nice, fluffy cake?
These are all examples of gases at work!
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Characteristics of Gases• Gases are highly compressible and occupy
the full volume of their containers.
• Gases exert pressure, P = F/A (force/area).
• Gases always form homogeneous mixtures
with other gases.
Properties of Gases
You can predict the behavior of gases
based on the following properties:
Pressure
Volume
Amount (moles)
Temperature
Lets review each of these briefly…
Pressure
Pressure is defined as the force the gas
exerts on a given area of the container in
which it is contained. The SI unit for
pressure is the Pascal, Pa.
• If you’ve ever inflated a tire,
you’ve probably made a
pressure measurement in
pounds (force) per square inch
(area).
Volume
Volume is the three-dimensional space inside
the container holding the gas. The SI unit for
volume is the cubic meter, m3. A more common
and convenient unit is the liter, L.
Think of a 2-liter bottle of soda to get
an idea of how big a liter is.
(OK, how big two of them are…)
Amount (moles)
As we’ve already learned, the SI unit for amount of
substance is the mole, mol. Since we can’t count
molecules, we can convert measured mass to the
number of moles, n, using the molecular or formula
weight of the gas.
By definition, one mole of a substance contains
approximately 6.022 x 1023 particles of the
substance.
Temperature
Temperature is the measurement of heat…or how
fast the particles are moving. Gases, at room
temperature, have a lower boiling point than things
that are liquid or solid at the same temperature.
Remember: Not all substance freeze, melt or
evaporate at the same temperature.
Water will freeze at zero degrees
Celsius. However Alcohol will not
freeze at this temperature.
II. Pressure
• Pressure is simply a force exerted over a
surface area.
760 mm
Hg
Pressure
If a tube is inserted into a container of mercury
open to the atmosphere, the mercury will rise 760
mm up the tube (at sea level).
(at sea level)
Atmospheric Pressure and the Barometer
. Standard atmospheric pressure is the pressure
required to support 760 mm of Hg in a column.
Units:
1 atm = 760 mmHg = 760 torr = 1.01325 105 Pa = 101.325 kPa.
II. Atmospheric Pressure
• Patm is simply the weight
of the earth’s
atmosphere pulled
down by gravity.
• Barometers are used to
monitor daily changes
in Patm.
• Torricelli barometer was
invented in 1643.
II. Units of Pressure
• The derived SI unit for pressure is N/m2,
known as the pascal (Pa).
• Standard conditions for gases (STP)
occurs at 1 atm and 0 °C. Under these
conditions, 1 mole of gas occupies 22.4 L.
How do they all relate?
Some relationships of gases may be easy to predict. Some are more subtle.Now that we understand the factors that affect the behavior of gases, we will study how those factors interact.
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III. Gas Laws
Robert Boyle
1627-1691.
Boyle’s Law.
Jacques Charles
1746-1823.
Charles’ Law.
J. Charles 1783.
First ascent in
hydrogen balloon.
III. Gas Laws
• A sample of gas can be physically
described by its pressure (P),
temperature (T), volume (V), and
amount of moles (n).
• If you know any 3 of these variables,
you know the 4th.
• We look at the history of how the ideal
gas law was formulated.
Boyle’s Law• This law is named for Charles Boyle, who
studied the relationship between pressure, p, and volume, V, in the mid-1600s.
• Boyle determined that for the same amountof a gas at constant temperature, results in an inverse relationship:
•when one goes up, the othercomes down.
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The Pressures-Volume Relationship: Boyle’s Law
(P vs. V at constant T)
What does Boyle’s Law
mean?
Suppose you have a cylinder with a piston in the
top so you can change the volume. The cylinder
has a gauge to measure pressure, is contained so
the amount of gas is constant, and can be
maintained at a constant temperature.
A decrease in volume will result in increased
pressure.
Boyle’s Law at Work…
Doubling the pressure reduces the volume by half. Conversely, when the volume doubles, the pressure
decreases by half.
NEXTPREVIOUSMAIN
MENU
Breathe Deeply!
It’s Boyle’s Law!
• When the diaphragm contracts,
the volume of the thoracic cavity
increases
• The lungs expand and pressure
decreases. Since Pair>Plungs, air
enters.
• When the diaphragm relaxes,
the volume of the thoracic cavity
decreases.
• The lungs contract and the
pressure in the lungs increases.
Plungs>Pair, so air is exhaled.
III. Volume and Temperature –
Charles’s Law
• The volume of a gas is directly related to its
temperature, i.e. if T is increased, V will
increase.
Charles’ Law• This law is named for Jacques Charles, who
studied the relationship volume, V, and temperature, T, around the turn of the 19th
century.
• This defines a direct relationship: With the same amount of gas he found that as the volume increases the temperature also increases. If the temperature decreases than the volume also decreases.
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tem
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Charles’s Law
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The Temperature-Volume Relationship: Charles’
Law
The Absolute Temperature
Scale• Temperature measures average
Kinetic Energy of particles:
• When motion stops, particles have
no kinetic energy.
• This means there must be an
absolute “zero” temperature!
• The Kelvin temperature scale starts
at Absolute Zero:
3 KE = RT
2
K = C + 273.15
What does Charles’ Law
mean?Suppose you have that same cylinder with a piston
in the top allowing volume to change, and a
heating/cooling element allowing for changing
temperature. The force on the piston head is
constant to maintain pressure, and the cylinder is
contained so the amount of gas is constant.
An increase in temperature results in increased
volume.
Charles’ Law at Work…
As the temperature increases, the volume increases. Conversely, when the temperature decreases, volume
decreases.
III. The Combined Gas Law
• Boyle’s and
Charles’s Laws can
be combined into a
convenient form.
III. Volume and Moles –
Avogadro’s Law
• The pressure of a
gas is directly
related to the
number of moles of
gas, i.e. if n
increases, V will
increase.
equal volumes of any gas at the same temperature and
pressure will contain the same number of molecules.
The Quantity-Volume Relationship: Avogadro’s Law
Same number
of particles
(same T and P)
V = constant n
at a constant P and T
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V = constant n
at a constant P and T
22.4 L = constant 1 mole
at a 1 atm and 273 K
Avogadro’s Law (n, V)
Gay-Lussac’s Law (T, P)• Avogadro showed that the
volume of a gas varies
directly with the amount
of gas (# of moles)
• Thus, a similar
relationship exists as in
Charles’s Law:
• Gay-Lussac studied how
temperature affects the
pressure of a gas.
• He discovered a direct
relationship!
1 2
1 2
V V V = k or =
n n n
1 2
1 2
P P P = k or = T T T
Pressure & Temperature
are held constant here!
Moles & Volume
are held constant here!
Gas Laws
Summary
BOYLE
CHARLES
AVOGADRO
GAY-LUSSAC
The Combined Gas Law • The gas laws can be combined into one equation.
• Volume and pressure vary inversely, while volume varies directly with moles and temperature:
• When variables are held constant, they can be deleted from the combined law – this produces all four gas laws we studied earlier.
1 1 2 2
1 1 2 2
PV P V = n T n T
Ideal Gas Law, cont’d
• We can rewrite the combined law in a form that is known
as the Ideal Gas Law:
PV = nRT• The value of the Ideal Gas Law over the previous laws is
that only ONE set of conditions is required – if 3 of the
variables are known, the 4th can be calculated.
• Use substitution and some algebra to derive the related
equations from the Ideal Gas Law:
mRT PM M = and d =
PV RT
Standard Molar Volume: 22.4 L @ STP
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The Ideal Gas Equation
• Summarizing the Gas Laws
P
nTV
P
nTRV
Boyle: V 1/P (constant n, T)
Charles: V T (constant n, P)
Avogadro: V n (constant P, T).
Combined:
Ideal gas equation
R = ideal gas constant
III. The Ideal Gas Law
• The ideal gas law is
a combination of the
combined gas law
and Avogadro’s Law.
R = 0.082058 L atm/K mole
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The Ideal Gas Equation
• Ideal gas equation: PV = nRT
R = gas constant = 0.08206 L•atm/mol-K.
We define STP (Standard Temperature and Pressure)
= 0C (273.15 K)
= 1 atm.
Volume of 1 mol of gas at STP is 22.4 L.
IV. Gas Law Problems
• There are many variations on gas law
problems.
• A few things to keep in mind:
1) Temperature must be in Kelvin
2) If problem involves a set of initial and final
conditions, use combined gas law.
3) If problem only gives information for one
set of conditions, use ideal gas law.
IV. Sample Problem
• What’s the final pressure of a sample of
N2 with a volume of 952 m3 at 745 torr
and 25 °C if it’s heated to 62 °C with a
final volume of 1150 m3?
IV. Sample Problem
• What volume, in mL, does a 0.245 g
sample of N2 occupy at 21 °C and 750
torr?
IV. Sample Problem
• A sample of N2 has a volume of 880 mL
and a pressure of 740 torr. What
pressure will change the volume to 870
mL at the same temperature?
IV. Other Uses of Ideal Gas Law
• The ideal gas law can be used to find
other physical values of a gas that are
not as obvious.
gas density, d = mass/volume
gas molar mass, MW = mass/mole
stoichiometry, via moles and a balanced
equation
IV. Sample Problem
• Find the density of CO2(g) at 0 °C and
380 torr.
IV. Sample Problem
• How many mL of HCl(g) forms at STP
when 0.117 kg of NaCl reacts with
excess H2SO4?
H2SO4(aq) + 2NaCl(s) Na2SO4(aq) + 2HCl(g)
Dalton’s Law• The total pressure of a mixture
of gases equals the sum of the
partial pressures of the
individual gases.
Ptotal = P1 + P2 + ...When a H2 gas is
collected by water
displacement, the gas in
the collection bottle is
actually a mixture of H2
and water vapor.
Dalton’s Law of Partial
Pressures• In a mixture of gases, the TOTAL pressure of gas is the
sum of the pressures caused by each gas (the partial
pressures):
PTotal = P1 + P2 + P3 + …
• The MOLE FRACTION ()of a gas in a mixture can be
calculated in two different ways, then:
A AA A A Total
Total Total
n P= = so we get: P = P
n P
Dalton’s Law Illustrated
GIVEN:
PH2 = ?
Ptotal = 94.4 kPa
PH2O = 2.72 kPa
WORK:
Ptotal = PH2 + PH2O
94.4 kPa = PH2 + 2.72 kPa
PH2 = 91.7 kPa
B. Dalton’s Law• Hydrogen gas is collected over water at
22.5°C. Find the pressure of the dry gas if the atmospheric pressure is 94.4 kPa.
Look up water-vapor pressure
for 22.5°C.
Sig Figs: Round to least number
of decimal places.
The total pressure in the collection bottle is equal to atmospheric
pressure and is a mixture of H2 and water vapor.
GIVEN:
Pgas = ?
Ptotal = 742.0 torr
PH2O = 42.2 torr
WORK:
Ptotal = Pgas + PH2O
742.0 torr = PH2 + 42.2 torr
Pgas = 699.8 torr
• A gas is collected over water at a temp of 35.0°C
when the barometric pressure is 742.0 torr.
What is the partial pressure of the dry gas?
Look up water-vapor pressure
for 35.0°C.
Sig Figs: Round to least number
of decimal places.
B. Dalton’s Law
The total pressure in the collection bottle is equal to barometric
pressure and is a mixture of the “gas” and water vapor.
V. Kinetic-Molecular Theory (KMT)
Kinetic Molecular Theory
• Particles in an ideal gas…
– have no volume.
– have elastic collisions.
– are in constant, random, straight-line
motion.
– don’t attract or repel each other.
– have an avg. KE directly related to Kelvin
temperature.
Imagining a Sample of Gas
• We imagine a sample of gas –
chaos, molecules bumping into
each other constantly.
• After a collision, 2 molecules
may stop completely until
another collision makes them
move again.
• Some molecules moving really
fast, others really slow.
• But, there is an average speed.
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Kinetic-Molecular Theory
0oC
100oC
N2
For gases, there is a range of velocities and energies at
each temperature.
Gas Molecular Speeds
• As temp increases,
avg. speed increases.
• i.e. avg. KE is related
to temp!!
• Any 2 gases at same
temp will have same
avg. KE!
Molecular Speeds The average kinetic energy per mole of
gas can be calculated in two different
ways:
We can rearrange and solve for “v”, the
velocity of a gas particle:
In order to get a velocity in “ms-1”, we
must use SI units for molar mass, kg
mol-1.
The gas constant (R) must be the SI
value, 8.314.
3RT v =
M
21 3 KE = Mv = RT
2 2
Don’t forget:
Molar Mass in Kg
for this equation!
Graham’s Law
• Diffusion
– Spreading of gas molecules throughout a
container until evenly distributed.
• Effusion
– Passing of gas molecules through a tiny
opening in a container
Graham’s Law
KE = ½mv2
• Speed of diffusion/effusion
– Kinetic energy is determined by the
temperature of the gas.
– At the same temp & KE, heavier molecules
move more slowly.
• Larger m smaller v
Graham’s Law• Graham’s Law
– Rate of diffusion of a gas is inversely related to the square root of its molar mass.
– The equation shows the ratio of Gas A’s speed to Gas B’s speed.
A
B
B
A
m
m
v
v
• Determine the relative rate of diffusion
for krypton and bromine.
1.381
Kr diffuses 1.381 times faster than Br2.
Kr
Br
Br
Kr
m
m
v
v2
2
A
B
B
A
m
m
v
v
g/mol83.80
g/mol159.80
Graham’s Law
The first gas is “Gas A” and the second gas is “Gas B”.
Relative rate mean find the ratio “vA/vB”.
• A molecule of oxygen gas has an average speed of 12.3 m/s at a given temp and pressure. What is the average speed of hydrogen molecules at the same conditions?
A
B
B
A
m
m
v
v
2
2
2
2
H
O
O
H
m
m
v
v
g/mol 2.02
g/mol32.00
m/s 12.3
vH2
Graham’s Law
3.980m/s 12.3
vH2
m/s49.0 vH 2
Put the gas with
the unknown
speed as
“Gas A”.
• An unknown gas diffuses 4.0 times faster than
O2. Find its molar mass.
Am
g/mol32.00 16
A
B
B
A
m
m
v
v
A
O
O
A
m
m
v
v2
2
Am
g/mol32.00 4.0
16
g/mol32.00 mA
2
Am
g/mol32.00 4.0
g/mol2.0
Graham’s Law
The first gas is “Gas A” and the second gas is “Gas B”.
The ratio “vA/vB” is 4.0.
Square both
sides to get rid
of the square
root sign.
Why is Diffusion so Slow??
• If molecular speeds are so incredibly fast,
why does a gas take so long to diffuse?
• The answer is in the completely random
path a gas particle takes as it diffuses.
• The gas particle constantly changes
direction when it collides with another
particle
• This slows down its outward diffusion
immensely!
• The MEAN FREE PATH is the distance a
particle travels before colliding with
another particle.
Effusion• Gas moving through
a pin-hole into a
vacuum
• The rate of effusion:
• Temp in Kelvin
• Molar mass in “kg
mol-1”
• Rate in “ms-1”
3RT v =
M
Graham’s Law of Effusion
• Graham compared the rates of effusion for two gases at the
same temperature.
• He derived the equation:
• Graham’s law is important because it can be used to
determine the Molar Mass of an unknown gas – if you
compare its rate of effusion with the rate of a known gas
under the same conditions!
1 2
2 1
v M =
v M
Here, Molar Mass can be
left in “grams”
Can you explain why??
Why Do Gas Laws Work So Well?
• Recall that the gas laws apply to any
gas – the chemical identity is not
important.
• Gas particles only interact when they
collide. Since this interaction is so
short, chemical properties don’t have
time to take effect!!
VI. Deviations from PV=nRT
• Under extreme
conditions (high P or
low T), gases
deviate from ideal
gas law predictions.
• Why? What’s so
different about these
conditions?
Real Gases
• Particles in a REAL gas…
– have their own volume
– attract each other
• Gas behavior is most ideal…
– at low pressures
– at high temperatures
– in nonpolar atoms/molecules
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Real Gases: Deviations from Ideal Behavior(Temperature and Pressure Effects)
• As temperature increases, the gas molecules move faster and
further apart.
• Also, higher temperatures mean more energy available to
break intermolecular forces.
• Therefore, the higher the temperature, the more ideal the gas.
• As pressure increases, gas molecules are closer together making
the gas less ideal.
• Therefore, the lower the pressure, the more ideal the gas.
Gas Particle Volume
• Gas molecules do take up space! When very close
to one another, entire volume of container is not
available for travel, so actual volume of gas is larger.
Intermolecular Forces
• Gas molecules interact if they are very close
to one another…
VI. van der Waals Equation
• Under extreme conditions, ideal gas law
cannot be used.
correction terms for P and V
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Real Gases: Deviations from Ideal
Behavior
The van der Waals Equation
• We add two terms to the ideal gas equation, one to
correct for volume of molecules, and the other to
correct for intermolecular attractions
• The correction terms generate the van der Waals
equation:
2
2
V
an
nb
V
nRTP
where a and b are empirical constants.