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Chapter 10
Gases
Characteristics of Gases
• Unlike liquids and solids, gases
– expand to fill their containers;
– are highly compressible;
– have extremely low densities.
Pressure
• Pressure is the
amount of force
applied to an area.
F
P=
A
• Atmospheric
pressure is the
weight of air per
unit of area.
Units of Pressure
• Pascals
– 1 Pa = 1 N/m2
• Bar
– 1 bar = 105 Pa = 100 kPa
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 = 101.3 kPa
Manometer
This device is used to
measure the difference
in pressure between
atmospheric pressure
and that of a gas in a
vessel.
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
Boyle’s Law
The volume of a fixed quantity of gas at
constant temperature is inversely proportional
to the pressure.
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.
Examples
• What would the final volume be if 247 mL of gas at
22ºC is heated to 98ºC , if the pressure is held
constant?
Charles’s Law
• 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.
Examples
• At what temperature would 40.5 L of gas at 23.4ºC
have a volume of 81.0 L at constant pressure?
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
Examples
• A deodorant can has a volume of 175 mL and a
pressure of 3.8 atm at 22ºC. What would the
pressure be if the can was heated to 100.ºC?
• What volume of gas could the can release at 22ºC
and 743 torr?
Ideal-Gas Equation
• So far we’ve seen that
V = 1/P (Boyle’s law)
V = T (Charles’s law)
V = n (Avogadro’s law)
• Combining these, we get
nT
V=
P
Ideal-Gas Equation
The constant of
proportionality is
known as R, the
gas constant.
Ideal-Gas Equation
The relationship
then becomes
nT
V=
P
nT
V=R
P
or
PV = nRT
Examples
• Kr gas in a 18.5 L cylinder exerts a pressure of 8.61 atm at
24.8ºC What is the mass of Kr?
• A sample of gas has a volume of 4.18 L at 29ºC and 732
torr. What would its volume be at 24.8ºC and 756 torr?
Examples
• Mercury can be produced by the following
decomposition:
• HgO 
• What volume of oxygen gas can be produced from
4.10 g of mercury (II) oxide at 400.ºC and 740 torr?
Densities of Gases
If we divide both sides of the ideal-gas
equation by V and by RT, we get
n
P
=
V
RT
Densities of Gases
• We know that
– moles  molecular mass = mass
n=m
• So multiplying both sides by the
molecular mass ( ) gives
m P
=
V RT
Densities of Gases
• Mass  volume = density
• So,
m P
d=
=
V RT
Note: One only needs to know the
molecular mass, the pressure, and the
temperature to calculate the density of
a gas.
Molecular Mass
We can manipulate the density equation
to enable us to find the molecular mass
of a gas:
P
d=
RT
Becomes
dRT
= P
Examples
• Complete and balance the following equation:
• NaHCO3 + HCl 
• calculate the mass of sodium hydrogen carbonate
necessary to produce 2.87-L of carbon dioxide at 25ºC
and 2.00 atm.
• If 27 L of gas are produced at 26ºC and 745 torr when
2.6-L of HCl are added what is the concentration of HCl?
Examples
• Consider the following reaction
• NH3 + O2  NO2 + H2O (balance first!)
• What volume of NO2 at 1.0 atm and 1000.ºC can be
produced from 10.0 L of NH3 and excess O2 at the
same temperature and pressure?
• What volume of O2 measured at STP will be
consumed when 10.0 kg NH3 is reacted?
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 + …
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.
Examples
• The partial pressure of nitrogen in air is 592 torr. Air
pressure is 752 torr, what is the mole fraction of
nitrogen?
• What is the partial pressure of nitrogen if the
container holding the air is compressed to 5.25 atm?
Examples
4.00 L
CH4
1.50 L
N2
3.50 L
O2
2.70 atm
4.58 atm
0.752 atm
• When these valves are opened, what is each partial
pressure and the total pressure?
Example
• N2O can be produced by the following reaction
NH4NO3(s)  N2O(g) + 2 H2O(l)
• What volume of N2O collected over water at a total
pressure of 94 kPa and 22ºC can be produced from
2.6 g of NH4NO3? (the vapor pressure of water at
22ºC is 21 torr)
Kinetic-Molecular Theory
This is a model that
aids in our
understanding of what
happens to gas
particles as
environmental
conditions change.
Main Tenets of KineticMolecular Theory
Gases consist of large numbers of
molecules that are in continuous,
random motion.
Main Tenets of KineticMolecular Theory
The combined volume of all the
molecules of the gas is negligible
relative to the total volume in which the
gas is contained.
Main Tenets of KineticMolecular Theory
Attractive and
repulsive forces
between gas
molecules are
negligible.
Main Tenets of KineticMolecular 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.
Main Tenets of KineticMolecular Theory
The average kinetic
energy of the
molecules is
proportional to the
absolute
temperature.
Effusion
Effusion is the
escape of gas
molecules
through a tiny
hole into an
evacuated
space.
Effusion
The difference in the
rates of effusion for
helium and nitrogen,
for example,
explains a helium
balloon would
deflate faster.
Diffusion
Diffusion is the
spread of one
substance
throughout a space
or throughout a
second substance.
• (KE)avg = NA(1/2 mu2)
• [m is mass of one gas particle in kg]
• (KE)avg = 3/2 RT
so…..
urms = (3RT/Mkg)1/2
• Where Mkg is the molar mass in kilograms, and R has
the units 8.314 kg-m2 / s2-mol-K or J/K-mol.
• The velocity will be in m/s
Graham's Law
KE1 = KE2
1/2 m1v12 = 1/2 m2v22
m1
m2
=
m1
=
m2

v22
v 12
v22
v12
=
v2
v1
Example
• Calculate the root mean square velocity of carbon
dioxide at 25ºC.
Examples
• A compound effuses through a porous cylinder 1.60
times faster than chlorine. What is it’s molar mass?
• If 0.00251 mol of NH3 effuses through a hole in 2.47
min, how much HCl would effuse in the same time?
• A sample of N2 effuses through a hole in 38
seconds. what must be the molecular weight of gas
that effuses in 55 seconds under identical conditions?
Real Gases
In the real world, the
behavior of gases
only conforms to the
ideal-gas equation
at relatively high
temperature and low
pressure.
Real Gases
Even the same gas
will show wildly
different behavior
under high pressure
at different
temperatures.
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.
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.
The van der Waals Equation
n2a
(P + 2 ) (V − nb) = nRT
V
Example
• Calculate the pressure exerted by 0.5000 mol Cl2 in a
1.000 L container at 25.0ºC
• Using the ideal gas law.
• Van der Waals equation given:
a = 6.49 atm L2 /mol2
b = 0.0562 L/mol