B_-_102_F11_Notes

B_-_102_F11_Notes - B)
Gases


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Unformatted text preview: B)
Gases
 Petrucci
et
al.,
10th
Edition:
6.1‐6.9
 
 Definition
of
Gas
Pressure
 • When
talking
about
gases,
four
properties
are
often
used:
amount
(moles),
 temperature,
volume
&
pressure
 • Pressure
is
the
force
per
unit
area
=
force/area
 • Force
resulting
from
the
collision
of
the
gas
molecules
against
the
walls
of
the
 container
 • Area
of
the
walls
of
the
container
holding
the
gas
 
 
 Understanding
Pressure
 • Gas
pressure
is
usually
measured
indirectly,
by
comparing
it
with
liquid
pressure

 • Liquid
pressure
is
directly
proportional
to
the
liquid
density
and
the
height
of
the
 liquid
column
 
 















=
 
 [see
Petrucci
et
al.,
Figure
6­4]
 
 
 
 
 
 
 
 
 
 
 
 When
the
pressures
are
equal:

 • Open‐ended
Manometer
 [see
Petrucci
et
al.,
Figure
6­5]
 
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐1 
 Boyle’s
Law
(1662)
 
 [see
Petrucci
et
al.,
Figure
6­6]
 
 • “For
a
fixed
amount
of
gas
at
a
constant
temperature,
the
gas
volume
is
inversely
 proportional
to
the
gas
pressure”
 • 
 • PV
=
constant
for
a
given
temperature
and
given
amount
of
substance
 
 Question
B1
 If
gas
volume
is
doubled
but
the
temperature
remains
constant:

 A)
the
pressure
stays
the
same

 B)
the
molecules
move
faster
 C)
the
final
pressure
is
1/2
of
the
pressure
before
the
volume
change
 D)
the
final
pressure
is
twice
the
pressure
before
the
volume
change
 E)
none
of
the
above
 
 Charles’
Law
(1787)
 
 [see
Petrucci
et
al.,
Figure
6­8]
 
 • “The
volume
of
a
fixed
amount
of
gas
at
constant
pressure
is
directly
proportional
to
 the
Kelvin
(absolute)
temperature”
 • 


 • 


 
 V0
:
volume
of
the
gas
at
0ºC
 T
:



temperature
in
ºC
 c
:



a
constant
(for
all
gases
at
small
pressures,
c
=
273.15)
 
 
 Aside
–
Definition
of
Temperature
from
Charles’s
Law
 • Temperature
is
measured
using
the
ratio
of
the
volume
of
a
gas
at
the
measured
 temperature
to
the
volume
of
the
same
gas
at
0ºC
(freezing
point
of
water)
 • If
T
=
–273.15°C
(absolute
zero),
then
V
=
0
 • Lowest
temperature
possible
before
V
<
0,
which
has
no
physical
meaning
 • Kelvin
temperature
scale
is
based
on
this
concept:
 • T
(K)
=
T(°C)
+
273.15
 • Substituting
this
definition
into
Charles’
law
gives:
 
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐2 
 Avogadro’s
Law
(1808)
 • “At
a
fixed
temperature
and
pressure,
the
volume
of
a
gas
is
directly
proportional
to
 the
amount
of
gas”
 • 

 • STP:
Standard
Temperature
and
Pressure
 Temperature
=
0°C
=
273.15
K
 Pressure
=
1
atm
=
760
mm
Hg
 1
mol
=

 
 
 The
Ideal
Gas
Equation
 • Combining
all
three
gas
relationships…
 • Avogadro:

 
 • Charles:
 
 • Boyle:
 
 
 
 • Combines
to:
 
 • Or
the
Ideal
Gas
Equation:

PV
=
nRT

where
R
=
C/c
 
 
 Ideal
Gas
Constant
 • At
STP
(1
atm,
0°C)
1
mol
of
most
gases
occupies
22.4
L:
 
 
 
 
 • R
can
also
be
given
in
units
of
J/(mol·K),
since:
 
 
 
 • General
Gas
Equation
(setting
R
=
R):
 
 
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐3 
 Gas
Laws
 • Gas
laws
are
mathematical
equations
relating
the
volume
to
the
number
of
moles,
 temperature
and
pressure
of
gases:

V
=
f
(n,
T,
P)
 • This
function
is
relatively
simple
because
intermolecular
forces
in
gases
are
 generally
very
weak
 • Sometimes,
the
ideal
gas
law
is
expressed
as:
 PV
=
nRT


or


PVm
=
RT

where
Vm
is
the
molar
volume
(Vm
=
V/n).
 • Sometimes
it
is
more
convenient
to
express
the
ideal
gas
law
this
way,
since
all
 variables
are
intensive,
as
opposed
to
the
first
form
where
V
is
an
extensive
variable.
 
 
 Ideal
Gases
 • But,
the
ideal
gas
law
does
not
take
gas
type
into
account:

 PV
=
nRT
 • Idealization
assumes
that:
 • 


 • 


 
 
 Question
B2
 You
inflate
your
bicycle
tire
with
dry
air
to
a
pressure
of
35
psi
in
the
summer,
when
the
 temperature
is
35°C.

Assuming
the
volume
of
the
tire
is
constant
and
no
air
has
leaked
 out,
what
would
the
pressure
be
in
winter
when
the
temperature
drops
to
‐15°C?
 
 
 Applications
of
the
Ideal
Gas
Equation
 • Determining
the
molar
mass
(M)
of
a
gas:
 P
V
=
n
R
T





and





M
=
m
/
n
 
 
 
 
 • Determining
the
density
of
a
gas:
 P
V
=
n
R
T





and





M
=
m
/
n




and





d
=
m
/
V
 
 
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐4 
 Density
 Gas
densities
differ
from
solids
&
liquids
in
two
ways:
 1. Gas
densities
are
strongly
dependent
on
pressure
&
temperature
 • Density
increases
with
increasing
pressure
 • Density
decreases
with
increasing
temperature
 • Solid
&
liquid
density
depends
somewhat
on
temperature
but
far
less
on
 pressure
 2.


Gas
density
is
directly
proportional
to
its
molar
mass
 • No
simple
relationship
exists
between
density
and
molar
mass
for
liquids
&
 solids
 
 
 Question
B3
 A
125
mL
glass
vessel
weighs
56.10
g
when
empty
and
56.25
g
when
filled
with
a
 gaseous
hydrocarbon
at
750
mmHg
and
28°C.

What
is
the
molecular
formula
of
the
 hydrocarbon?
 
 Question
B4
 A
3.5g
sample
of
KCl‐KClO3
mixture
is
decomposed
by
heating
and
produces
250mL
O2
 (g)
measured
at
10°C
and
760
mmHg.

What
is
the
percentage
of
KClO3
in
the
mixture
by
 mass?

 
 KClO3
(s)
→
KCl
(s)
+
O2
(g)
 
 
 Gases
in
Chemical
Reactions
 Guy‐Lussac’s
Law
of
Combining
Volumes
 • Gases
react
by
volumes
in
the
ratio
of
small
whole
numbers
 For
example:
2
H2
(g)
+
O2
(g)
→
2
H2O
(g)
 
 
 Question
B5
 Three
volumes
of
oxygen
(O2)
and
one
volume
of
methane
(CH4)
are
placed
in
a
rigid,
 sealed
container.

The
temperature
and
pressure
of
this
mixture
are
120°C
and
600
kPa.

 What
is
the
pressure
in
the
same
container
after
the
complete
combustion
of
the
mixture
 and
after
it
was
cooled
to
the
initial
temperature?
 
 Mixtures
of
Gases
 • For
pure
gases
and
mixtures
of
non­reactive
gases
 
 [see
Petrucci
et
al.,
Figure
6­12]
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐5 
 Dalton’s
Law
 • Different
types
of
gasesin
a
mixture
act
independently
of
each
other
 • Partial
pressures
cannot
be
directly
measured
in
a
mixture
of
gases,
but
they
are
 the
pressures
that
each
gas
would
have
if
present
alone
in
a
container
of
the
same
 volume
and
temperature.

 
 PA
:
partial
pressure
of
gas
A

 
 PB
:
partial
pressure
of
gas
B
 
 
 
 
 
 • Notice
also
that,
 
 
 
 
 
 
 
 yA:

mol
fraction
of
gas
A
in
the
gas
mixture
 
 • Therefore,
Dalton’s
law
can
also
be
stated
as:
 
 PA
=
yAP
 
 
 Question
B6
 A
500
mL
container
holds
0.20
g
Ne
and
an
unknown
amount
of
Ar
at
35°C
with
a
total
 pressure
of
0.866
bar.

Calculate
the
mole
fraction
of
Ar
in
the
gas
mixture.
 
 
 Question
B7
 Gas
cylinder
A
has
a
volume
of
48.2
L
and
contains
N2
gas
at
8.35
atm
at
25°C.
Gas
 cylinder
B,
of
unknown
volume,
contains
He
gas
at
9.50
atm
at
25°C.

When
the
two
 cylinders
are
connected
and
the
gases
mixed,
the
pressure
in
each
cylinder
becomes
8.71
 atm
and
the
temperature
remains
unchanged.

What
is
the
volume
of
cylinder
B
in
litres?
 
 
 Question
B8
 Ethylene,
C2H4,
reacts
with
hydrogen
in
the
presence
of
a
platinum
catalyst
to
form
 ethane,
C2H6,
according
to
the
reaction
 
 C2H4
(g)
+
H2
(g)
→
C2H6
(g)
 A
mixture
of
C2H4
and
H2
known
only
to
contain
more
moles
of
H2
than
of
C2H4
has
a
 pressure
of
52
mmHg
in
an
unknown
volume.

After
the
gas
has
been
passed
over
the
 platinum
catalyst,
its
gas
pressure
is
34
mmHg
in
the
same
volume
and
at
the
same
 temperature.

What
mole
fraction
of
the
original
mixture
is
ethylene?
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐6 
 Kinetic­Molecular
Theory
of
Gases
 • The
kinetic
theory
of
gases
is
a
model
for
gas
behaviour.
 • A
model
describes
a
given
real
system
by
some
of
its
most
important
features.
 • The
ideal
gas
law
is
an
empirical
model
for
gas
behaviour.


 • The
kinetic
theory
of
gases
provides
a
theoretical
justification
(theoretical
model)
for
 the
ideal
gas
law

 
 
 Hypothesis
(Assumptions)
 • A
gas
is
composed
of
a
very
large
number
of
small
particles
in
constant,
random,
 straight‐line
motion
 • Gas
molecules
are
separated
by
a
large
distance
–
a
gas
is
mostly
empty
space
 • Molecules
collide
with
one
another
and
with
the
walls
of
the
container
 • There
are
no
intermolecular
forces,
except
during
collisions.

Molecules
act
 independently
of
one
another
 • Individual
molecules
may
gain
or
lose
energy
during
collisions
but
the
energy
of
the
 entire
gas
remains
constant.

Molecular
collisions
are
elastic
 
 
 Relationships
Between
Gas
Pressure
and
Velocity
 • 
 Translational
kinetic
energy
of
a
molecule
(ek):
 • Frequency
of
molecular
collisions
(ƒ):
 
 • Momentum
transfer
(µ)
to
the
wall
during
a
collision:
 
 • 
 
 
 
 N 2N = mu V V Gas
pressure
(P):
 P ∝ µ × f ∝ ( mu) × u This
equation
assumes,
___________________,
that
 all
molecules
of
the
gas
have
the
same
speed
 € where

 u
=
speed
of
gas
molecules
 
 m
=
mass
 
 N
=
number
of
gas
molecules
 
 V
=
volume
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐7 
 Distribution
of
Average
Velocities
 • Molecules
of
a
gas
do
not
have
the
same
translational
velocity
but
instead
a
 distribution
of
molecular
velocities
 • Therefore,
the
mean­square
speed
of
the
gas
molecules
should
be
used
to
calculate
 the
pressure
of
the
gas:
 
 u 2 + u2 2 + u3 2 + ... + uN 2 
 u2 = 1 = mean square speed N 
 
 Distribution
of
Molecular
Speeds
 um
:
most
probable
(modal)
speed
 € uav
:
average
(mean)
speed
 urms
:
root‐mean‐square
speed
 
[see
Petrucci
et
al.,
Figure
6­15]
 
 Calculation
of
Root
Mean
Square
Speed
 • Consider
1
mol
(NA
molecules)
of
gas
 
 
 
 
 • Use
the
ideal
gas
equation

 PV
=
RT

(n
=
1
mol)
 
 • Since
mNA
=
M,
then
 
 • Therefore
 
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐8 
 Distribution
of
Molecular
Speeds
 The
speeds
of
ideal
gas
molecules
obey
the
Maxwell­Boltzmann
speed
distribution:
 
 
 
 
 where
kB
is
Boltzmann’s
constant:
 
 
 Maxwell­Boltzmann
Speed
Distribution
 • A
plot
of
f
(u)
versus
u
gives
the
distribution
of
molecular
speeds
for
molecules
of
 mass
m
at
a
temperature
T

 • The
area
under
the
curve
from
speeds
u
to
u
+
∆u
corresponds
to
the
fraction
of
 molecules
that
have
speeds
varying
from
u
to
u
+
∆u

 u + Δu 
 ΔN f ( u) du = ≅ f u Δu 
 N u 
 • The
total
area
under
the
curve
equals
1.0

 
 
 € 
 
 Influence
of
Gas
Type
and
Temperature
on
Molecular
Speed
 ∫ () 
[see
Petrucci
et
al.,
Figure
6­16]
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B ‐9 
 Average
Velocities
 • Root‐mean‐square
speed:
 
 
 • Most
probable
speed:
 
 
 • Average
speed:
 
 ump
:
uav
:
urms
=
1.000
:
1.128
:
1.225
 
 Use
R
=
8.314
J/mol
K
=
8.314
kg·m2/s2·mol·K
to
obtain
velocities
in
m/s
 
 
 Question
B9
 How
many
of
the
following
statements
about
the
kinetic
theory
of
gases
are
 INCORRECT?
 i) The
translational
velocity
of
gas
particle
is
inversely
proportional
to
the
absolute
 temperature
of
the
gas
 ii) The
collision
of
two
gas
molecules
is
assumed
to
be
elastic
 iii) Boyle’s
Law
can
be
proven
using
the
kinetic
theory
of
gases
 iv) According
to
the
Maxwell‐Boltzmann
distribution
of
gas
velocities,
the
root
mean
 square
speed
is
greater
than
the
average
speed.
 v) The
absolute
temperature
of
a
gas
is
proportional
to
the
kinetic
energy
of
a
gas.
 vi) The
root‐mean‐square
speed
at
25
°C
for
gaseous
Ar
atoms
is
432
m/s.
 
 
 Gas
Properties
Relating
to
the
Kinetic­Molecular
Theory
of
Gases
 Diffusion:
migration
of
molecules
as
a
result
of
random
molecular
motion
 
 [see
Petrucci
et
al.,
Figure
6­19]
 
 Effusion:
escape
of
a
gas
molecules
through
a
tiny
orifice
or
pinhole

 
 [see
Petrucci
et
al.,
Figure
6­19]
 
 
 
 
 € rate of effusion of A ( urms ) A = = rate of effusion of B ( urms ) B ChE102
Fall
2011
Class
Notes
‐
Gases
 
 3RT 3RT MA MB = MB MA B‐10
 Question
B10
 The
hydrogen
halide
gases
all
have
the
same
general
formula,
HX,
where
X
can
be
Cl,
Br,
 I,
or
F.

If
HCl
gas
effuses
1.88
times
more
rapidly
than
one
of
the
others,
which
hydrogen
 halide
is
the
other?
 
 
 Non­Ideal
(Real)
Gases
 • Ideal
gas
law
is
valid
when
intermolecular
interactions
can
be
neglected
 • This
is
more
likely
to
be
true
for
non‐polar
gases
at
low
pressures
and
high
 temperatures

 • far
from
condensation
because
intermolecular
forces
in
liquids
are
very
 significant
 
 • For
ideal
gases
 
 
 • For
real
gases
 
 
 • Z
is
called
the
compressibility
factor

 
 Generally:
0.1
<
Z
<
1.5
 
 [see
Petrucci
et
al.,
Figure
6­20]
 
 Van
der
Waals
Equation
 • Why
isn’t
the
ideal
gas
law
always
adequate?
 
 • When
T
=
0
K
→
Vm
=
0
 • But,
we
know
that
the
volume
at
0
K
should
be
equal
to
at
least
the
volume
of
the
 molecules
since
gases
do
not
disappear
upon
cooling
but
liquefy
and
then
solidify.
 
 A
simple
correction
is:
 
 b
is
a
positive
constant
that
depends
on
gas
type
and
is
proportional
to
the
molecular
 volume
of
the
gas.
 Therefore,
 
 
 According
to
this
equation,
Z
is
always
greater
than
one.
However,
Z
can
be
less
than
one
 as
shown
in
the
plot
of
Z
versus
P.

 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B‐11
 • 
 
 • Re‐write
the
equation
so
that:
 
 If
the
gas
molecules
attract
each
other,
Preal
should
be
less
than
the
predicted
by
the
 above
equation.
The
correct
term
is
given
by:
 • a
is
a
positive
constant
that
depends
on
gas
type
and
is
 proportional
to
intermolecular
forces.
 • Vm2
is
used
in
the
denominator
of
the
new
term
since
 molecular
attractions
are
related
to
bi‐molecular
 collisions
 
 
 
 
 The
van
der
Waals
equation
is
cubic
in
volume:
 
 
 
 
 
 
 
 Question
B11
 One
mole
of
SO2
occupies
a
volume
of
1.85
L
at
0
°C
and
10
atm.

How
many
of
the
 statements
are
true?
 i) SO2
behaves
as
a
non‐ideal
gas.
 ii) The
compressibility
factor
(Z)
for
SO2
is
less
than
one.
 iii) The
density
of
SO2
is
less
than
the
density
predicted
by
the
ideal
gas
law.
 iv) SO2
molecules
are
attracting
each
other.
 v) The
van
der
Waals
equation
is
an
equation
of
state
for
real
gases
and
could
be
used
 to
calculate
the
volume
occupied
by
1
mol
of
SO2.
 ChE102
Fall
2011
Class
Notes
‐
Gases
 
 B‐12
 ...
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This note was uploaded on 12/28/2011 for the course CHE 102 taught by Professor Simon during the Winter '08 term at Waterloo.

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