The Gaseous State
Chapter 10
1
Objectives
1.
2.
3.
4.
5.
Understand the definition of pressure. Use the
definition to predict and measure pressure
experimentally
Describe experiments that show relationships between
pressure, temperature, volume, and moles of a gas
sample
Use empirical gas laws to predict how change in one
of the properties of a gas will affect the remaining
properties.
Use empirical gas laws to estimate gas densities and
molecular masses.
Use volume-to-mole relationships obtained using the
empirical gas laws to solve stoichiometry problems
involving gases.
2
Objectives
6. Understand the concept of partial pressure in
mixtures of gases.
7. Use the ideal kinetic-molecular model to
explain the empirical gas laws.
8. List deficiencies in the ideal gas mode3el that
will cause real gases to deviate from behaviors
predicted by the empirical gas laws. Explain
how the model can be modified to account for
these deficiencies.
3
Definition of Gas
Gas: large collection of particles moving at random
throughout a volume that is primarily empty space. Have
relatively large amount of energy.
Gas pressure: due to collisions of randomly moving
particles with the walls of the container.
Force/unit area
5
Definition of Gases
• STP: 0°C, and 1 atmosphere pressure
• Elements that exist as gases at STP: hydrogen, nitrogen,
oxygen, fluorine, chlorine and Noble Gases
• Ionic compounds are all solids
• Molecular compounds - depends on the intermolecular
forces. Most are liquids and solids. Some are gaseous
• http://www.chemistry.ohiostate.edu/betha/nealGasLaw/fr1.1.html
6
Properties of Gases
• Assume the volume and shape of their
container
• Compressible
• Mix evenly and completely when confined
to the same container
• Lower densities than liquids and solids
• Allotropes: O2 ↔O3
Kinetic Molecular Theory of Gases
1. Tiny particles in continuous motion ( the hotter
the gas, the faster the molecules are moving)
with negligible volume compared to volume of
container.
2. Molecules are far apart from each other
3. Do not attract or repel each other (?).
4. All collisions are elastic (gas does not lose
energy when left alone).
5. The energy is proportional to Kelvin
temperature. At a given temperature all gases
have the same average KE.
8
Properties of Gases
Observation
Hypothesis
Gases are easy to expand
Gases are easy to compress
Gases have densities that
are 1/1000 of solid or liquid
densities
Gases completely fill their
containers
Hot gases leak through
holes faster than cold gases
9
Properties of Gases
Observation
Gases are easy to
expand
Gases are easy to
compress
Gases have densities
that are 1/1000 of solid
or liquid densities
Hot gases leak through
holes faster than cold
gases
Hypothesis
Gas molecules do not
strongly attract each
other
Gas molecules don’t
strongly repel each other
Molecules are much
farther apart in gases
than in liquids and solids
Gas molecules are in
constant motion
10
Atmospheric Pressure
Intensive or Extensive
Property?
11
Pressure
• Pressure is due to collisions between gas
molecules and the walls of the container.
Magnitude determined by: force of collisions
and frequency.
• Pressure: force per unit area: P =F/A
• Standard temperature: 0ºC = 273.15 K
• Standard pressure: 1 atm in US; 1 bar
elsewhere
12
Pressure
Unit
Symbol
Conversions
1 Pa = 1 N/m2
Pascal
Pa
Psi
lb/in2
Atmosphere
Atm
Bar
Bar
1 atm = 101325 Pa =
14.7 lb/in2
1 bar = 100000 Pa
Torr
Torr
760 torr = 1 atm
Millimeter
mercury
mm Hg
1 mm Hg = 1 torr
13
Pressure: Examples
1. How much pressure does an elephant
with a mass of 2000 kg and total footprint
area of 5000 cm2 exert on the ground?
2. Estimate the total footprint area of a
tyrannosaur weighing 16 000 kg.
Assume it exerts the same pressure on
its feet that the elephant does.
14
Pressure
• Measuring pressure:
• Strategy:
– Relate pressure to fluid column heights
• You can’t draw water higher than 34 feet by
suction alone. Why?
• Hypothesis: atmospheric pressure
supports the fluid column
• Develop the equation
15
Measuring Pressure
16
Pressure: Barometer
Barometer measures atmospheric pressure as a mercury
column height.
17
Pressure: Open-Manometer
Manometer measures gas pressure as a difference in mercury
column heights.
Two types: closed manometer
open manometer
18
Measuring Gas Pressure
Closed-manometer :
the arm not connected
to the gas sample is
closed to the
atmosphere and is
under vacuum.
Explain how you can
read the gas pressure in
the bulb.
19
Pressure: Examples
3. Calculate the difference in pressure between the top
and the bottom of a vessel exactly 76 cm deep filled at
25 ºC with a) water; b) mercury (d = 13.6 g/cm3)
(7.43 x 103 Pa;100.9 x 103 Pa)
4. How high a column of air would be necessary to cause
the barometer to read 76 cm of mercury, if the
atmosphere were of uniform density 1.2 kg/m3?
dHg = 13.53 kg/m3
(8.6 km)
5. A Canadian weather report gives the atmospheric
pressure as 100.2 kPa. What is the pressure in
atmospheres? Torr? Mm Hg?
The Gas Laws: State of Gas
Property
Symbol Unit
Property
Type
atm, torr, Intensive
Pa
Pressure
P
Volume
V
L, cm3
Extensive
Temperature
T
K
Intensive
Moles
n
mol
extensive
21
The Gas Laws: State of Gas
• Any equation that relates P, V, T, and n for
a material is called an equation of state.
• Experiment shows PV = nRT is an
approximate equation of state for gases.
• R is the gas law constant
– Determined by measuring P, V, T, n and
computing R = PV/nT
– Value depends on units chosen for P, V, T
– Notice: 1 Joule = 1 N m = 1(Pa) (m3)
22
The Gas Laws
Gas laws deal with the MACROSCOPIC view of gases and we try
to explain the macroscopic properties by examining the
microscopic behaviors (many molecule behaviors)
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=42
http://jersey.uoregon.edu/vlab/Piston/index.html
Prentice Hall Simulations of Gas Laws
• http://cwx.prenhall.com/bookbind/pubbook
s/hillchem3/chapter5/deluxe.html
Boyle’s Law: Experiment
Relate volume to pressure when everything else is
constant. Experiment: trapped air bubble at 298 K
Volume, mL
10.0
Pressure, torr
760.0
20.0
379.6
30.0
253.2
40.0
191.0
Graphs?
PV )mL torr)
25
Boyle’s Law: Experiment
Relate volume to pressure when everything else is
constant. Experiment: trapped air bubble at 298 K
Volume, mL
10.0
Pressure, torr
760.0
PV (mL torr)
7.60 x 103
20.0
379.6
7.59 x 103
30.0
253.2
7.60 x 103
40.0
191.0
7.64 x 103
Graphs?
26
Boyle’s Law: Volume/Pressure Relationship
At constant n, and T, the volume of a gas decreases
proportionately as its pressure increases. If the pressure
is doubled, the volume is halved.
27
Boyle’s Law: Volume/Pressure
Relationship
What happens
to the volume of
the gas as the
pressure
increases?
Mathematical
Relationship?
28
Plot of Boyle’s Law
V versus P
V versus 1/P
Type of Graphs?
29
Boyle’s Law
Boyle’s Law – the volume of a fixed amount of gas
at constant temperature and constant number of
moles is inversely proportional to the gas pressure.
V
nRT
P
MOLECULAR VIEW
VP  k 1
V1 P1  V2 P2
V 
k1
P
30
Boyle’s Law
Boyle’s Law – the volume of a fixed amount of gas
at constant temperature and constant number of
moles is inversely proportional to the gas
pressure.
nRT
V
P
nRT  k
V 
k
P
V1 P1  V2 P2
MOLECULAR VIEW:
Confining molecules to a
smaller space increases
the number (frequency) of
collisions, and so
increases the pressure
Charles' Law (V/T Relationships)
Relate volume to temperature, everything else is constant. Experiment: He bubble
trapped at 1 atm.
V, mL
40.0
T, ºC
0.0
T, (K)
273.2
44.0
25.0
298.0
47.7
50.0
323.2
51.3
75.0
348.2
55.3
100.0
373.2
80.0
273.2
546.3
V/T, mL/K
32
Charles' Law (V/T Relationships)
Relate volume to temperature, everything else is constant. Experiment: He bubble
trapped at 1 atm.
V, mL
40.0
T, ºC
0.0
T, (K)
273.2
V/T, mL/K
0.146
44.0
25.0
298.0
0.148
47.7
50.0
323.2
0.148
51.3
75.0
348.2
0.147
55.3
100.0
373.2
0.148
80.0
273.2
546.3
0.146
33
Charles’ Law: Volume/Temperature
Relationships
At constant n and P, the volume of a gas increases
proportionately as its absolute temperature increases, If
the absolute temperature doubles, the volume is
doubled.
K = ºC + 273
34
Charles’ Law
A plot of V
versus T for a
gas sample.
What type of
graph?
Equation?
35
36
Charles' Law
Kinetic Interpretation of Charles's Law? Why higher
pressure? Equation?
Frequency and force of collision…
37
Charles’ Law
The volume of the gas is directly proportional to its
Kelvin temperature, when everything else is
constant.
nR
V 
T
P
V  kT
nR
k 
P
V
V
1  2
T
T
1
2
MOLECULAR VIEW
Charles’ Law
The volume of the gas is directly proportional to its Kelvin
temperature, when everything else is constant.
nR
V
T
P
V  kT
nR
k
P
V
V
1 2
T
T
1
2
MOLECULAR VIEW
Raising temperature increases the
number of collisions and force of
collisions (KE increases) with
container wall. If the walls are
flexible, they will be pushed back
and the gas expands.
39
Charles’ Law
Assume that you have a
sample of gas at 350 K in
a sealed container, as
represented in (a). Which
of the drawings (b) – (d)
represents the gas after
the temperature is lowered
from 350 K to 150 K
40
Gay Lussac’s Law
The pressure of the gas is directly proportional to its Kelvin
temperature, when everything else is constant.
nRT
P
V
k  nRT
Molecular View;
P  kT
P
k 
T
P1
P2

T1
T2
41
Gay Lussac’s Law
The pressure of the gas is directly proportional to its Kelvin
temperature, when everything else is constant.
nRT
P
V
k  nRT
P  kT
P
k 
T
P1
P2

T1
T2
Molecular View;
Raising the temperature increases
the number of collisions and the
kinetic energy of the molecules.
More collisions with greater
energy (force) means higher
pressure.
42
Combined Gas laws
PV
 nT
T
PV
k
T
k n
PV
P V
1 1 2 2
T
T
1
2
Avogadro’s Law: Relates n to Volume
44
Volume of Real Gases at STP
45
Avogadro’s Law: Relates n to Volume
At constant T and P, the volume of a gas is directly
proportional to moles of gas. Molar volume is almost
independent of the type of gas.
Samples of two gases with the same V, P, T contain the
same number of molecules.
nRT
V 
P
RT
kA 
P
V  k An
MOLECULAR VIEW
46
Avogadro’s Law: Relates n to Volume
At constant T and P, the volume of a gas is directly proportional
to moles of gas. Molar volume is almost independent of the type
of gas.
Samples of two gases with the same V, P, T contain the
same number of molecules (moles).
nRT
V
P
RT
kA 
P
V  k An
MOLECULAR VIEW
Type of gas does not influence
distance between molecules too
much.
47
Avogadro’s Law: Example 6
Show the approximate level of the movable piston in
drawings (a) and (b) after the indicated changes have
been made to the initial gas sample.
V n
V  kAn
48
Avogadro’s Law: Answer to Example 6
49
Example 7
Show the approximate level of the movable piston in
drawings (a), (b), and (c ) after the indicated changes.
50
Gas Laws: Examples
8.
A balloon indoors, where the temperature is 27.0 ºC,
has a volume of 2.00 L. What will be its volume
outdoors, where the temperature is -27.0 ºC?
(Assume no change in pressure)
[ 1.67 L]
9.
A sample of nitrogen occupies a volume of 2.50 L at 120 ºC and 1.00 atm. Pressure. To which of the
following approximate temperatures should the gas be
heated in order to double its volume while maintaining
a constant pressure?
-240 ºC
- 60.0 ºC
-12.0 ºC
30.0 ºC
[30.0 ºC]
51
Gas Laws: Examples
10. Calculate the volume occupied by 4.11 g
of methane gas at STP.
[5.74 x 103L]
11. What is the mass of propane, C3H8, in a
50.0 L container of the gas at STP?
52
Ideal Gas Law
PV  nT
both P and V are directly proportion al to T
PV = nRT
Gas Constant R = 0.082057 (L atm)/(mol K)
53
Examples
12. Sulfur hexafluoride, SF6 is a colorless, odorless,
very unreactive gas. Calculate the pressure (in
atm) exerted by 1.82 moles of the gas in a steel
container of volume 5.43 L at 69.5 ºC.
(9.42 atm)
13. Calculate the volume (in liters) occupied by 7.40 g
of CO2 at STP.
( 3.77 L)
54
Gas Laws: Examples
14. A gas initially at 4.0 L, 1.2 atm, and 66 º
undergoes a change so that its final volume an
temperature become 1.7 L and 42 º C. What is
its final pressure? Assume the number of moles
remains unchanged.
15. A certain container holds 6.00 g of CO2 at
150.0 ºC and 100. kPa pressure. How many
grams of CO2 will it hold at 30.0 ºC and the
same pressure?
Gas Laws Summary
Changing
variables
Variables
Relationship
held constant
Law
P, V
n, T
P1V1 = P2 =V2 Boyle’s Law
V, T
n, P
V/T = k
Charle’s Law
P, T
n, V
P/T = k
Gay-Lussac’s
n, V
P, T
V/n = k
Avogadro’s
P, V, T
n
PV/T = k
Combined
P, V, T, n
none
PV/(nT) = R
Ideal Gas Law
56
Gas Density and Molar Mass
m
n
MM
VP  nRT
D
m
VP 
RT
MM
m P(MM)

V
RT
mRT
m RT
DRT
MM 
 ( )(
)
PV
V P
P
P(MM)  dRT
Purple M&M Do Red Too
Or
Michael Mo do the right
thing
57
Density and Molar Mass: Examples
16. Calculate the density of methane gas, CH4, in grams per liter, at 25
ºC and 0.978 atm.
[0.641 g CH4/L]
17. Under what pressure must O2(g) be maintained at 25 ºC to have
density of 1.50 g/L?
[1.15 atm]
18. The density of a gaseous organic compound is 3.38 g/L at 40.0 ºC
and 1.97 atm. What is its molar mass?
[44.1 g/mol]
19. A gaseous compound is 78.14% boron, 21.86% hydrogen. At 27.0 º
C, 74.3 mL of the gas exerted a pressure of 1.12 atm. If the mass
of the gas was 0.0934 g, what is its molecular formula?
[B2H6]
58
Stoichiometry Involving Gases
Use regular Stoichiometry techniques, except that
PV
n
RT
for non STP conditions, and 22.4 L/mole for STP
conditions.
59
Stoichiometry: The Law of Combining
Volumes Involving Gases
When gases measured at the same temperature
and pressure are allowed to react, the volumes
of gaseous reactants and products are in small
whole-number ratios.
60
Stoichiometry: The Law of Combining Volumes
Involving Gases (Avogadro’s Explanation)
When the gases are measured at the same
temperature and pressure, each of the identical
flasks contains the same number of molecules.
61
Examples (Stoichiometry)
20. How many liters of O2(g) are consumed for every 10.0 L of CO2(g)
produced in the combustion of liquid pentane, C5H12, if each gas is
measured at STP?
[16.0 L O2]
21. Given the reaction
C6H12O6(s) + O2(g) → 6CO2(g) + 6H2O(g),
calculate the volume of CO2 produced at 37.0 ºC and 1.00 atm
when 5.60 g of glucose is used up in the reaction.
[4.75 L]
22. A 2.14 L- sample of hydrogen chloride gas at 2.61 atm and 28.0 ºC
is completely dissolved in 668 mL of water to form hydrochloric acid
solution. Calculate the molarity of the acid solution.
[0.338M]
Dalton’s Law of Partial Pressure
Assume that you have a mixture of He (4 amu) and Xe (
131 amu) at 300 K. Which of the drawings best represents
the mixture (blue= He; green = Xe)?
63
Dalton’s Law of Partial Pressure
1. What is the partial pressure of each gas – red, yellow, and green – if
the total pressure inside the following container is 600 mm Hg?
2. What is the volume of each gas inside the container, if the
total volume of this vessel is 1.0 L?
64
Dalton’s Law of Partial Pressure
65
Dalton’s Law of Partial Pressures
• Mole fraction: moles of component per mole
of mixture
• Avogadro’s Law: mole fraction = volume
fraction for ideal gas
Examples:
1. 2 L of He gas is mixed withy 3 L of Ne gas.
What is the mole fraction of each component?
2. Air is approximately 79% N2 and 21 %O2 by
mass. What is the mole fraction of O2 in the
air?
66
Dalton’s Law of Partial Pressures
Partial Pressure – the pressure of an individual gas
component in a mixture: PA
Examples:
1. One mole of air contains 0.79 moles of nitrogen and
0.21moles of oxygen. Compute the partial pressure of
these gases at a total pressure of 1.0 atm atm and at a
total pressure of 3.0 atm (about the pressure
experienced by a diver under 66 ft of seawater).
2. What is the mole fraction of water in the headspace of
a soda bottle, if the gas is at 2.0 atm and 25 ºC is
23.756 torr?
67
Dalton’s Law of Partial Pressures
Ptotal = P1 + P2 + P3 +……. Pn
P1  n1RT
V
P2 
n2 RT
V
Ptotal  P1  P2
n1 RT
P1
n
V

 1  1
Ptotal n t otal RT
nT
V
P1 = x1PT
x 1  mole fraction of sample 1
68
Dalton’s Law of Partial Pressures
Dalton’s Law: The total pressure of a mixture of
gases is just the sum of the pressures that each
gas would exert if it were present alone.
MOLECULAR VIEW
Molecules of a gas do not attract or repel each other.
The distances between particles are very large,
therefore each particular gas occupies the entire
container and adds its pressure to the total pressure in
the container.
69
Dalton’s Law: Examples
23. A mixture containing 0.538 mol He, 0.315 mol of Ne,
and 0.103 mol of Ar is confined in a 7.00 L vessel at 25
ºC.
A) Calculate the partial pressure of each of the gasses
in the mixture.
B) Calculate the total pressure of the mixture.
[P of He 1.88 atm; P of Ne 1.10 atm; P of Ar 0.360 atm; P total 3.34
atm]
24. The partial pressure of nitrogen in air is 592 torr. Air
pressure is 752 torr, what is the mole fraction of
nitrogen?
[7.87 x 10-1]
70
Dalton’s Laws: Examples
25. What is the partial pressure of nitrogen if the container
holding the air is compressed to 5.25 atm?
[4.13 atm]
26. Ca(s) + H2O(l) →Ca(OH)2 + H2(g)
H2(g) was collected over water. The volume of gas at
30.0 ºC and P= 988 mm Hg is 641 mL. What is the mass
(in grams) of the H2 gas obtained? The pressure of
water at 30.0 ºC is 31.82 mm Hg.
[0.0653 g]
71
Dalton’s Laws: Additional Problems
27. A gaseous mixture made from 6.00 g of oxygen and
9.00 g of methane is placed in a 15.0 – L vessel at
0.00°C What is the partial pressure of each gas, and
what is the total pressure in the vessel?
[0.281 atm O2; 0.841 CH4; 1.122 atm total]
28. A study of the effects of certain gases on plant growth
requires a synthetic atmosphere composed of 1.5 mol
percent of CO2, 18.0 mol percent O2; and 80.5 mol
percent of Ar. (a) calculate the partial pressure of O2 in
the mixture if the total pressure of the atmosphere is to
be 745 torr. (b) If this atmosphere is to be held in a 120
–L space at 295 K, how many moles of O2 are needed?
[PO2 = 134 torr; nO2 = 0.872 mol]
72
Dalton’s law of Partial Pressure
29. The apparatus shown consists of three bulbs
connected by stopcocks. What is the pressure inside the
system when the stopcocks are opened? Assume that
the lines connecting the bulbs have zero volume and that
the temperature remains constants.
[PCO2 = 0.710 atm; PH2 = 0.191 atm; P Ar = 0.511 atm; PT = 1.412 atm]
73
Example 30
4.00 L
CH4
1.50 L
N2
2.70 atm
.58 atm
3.50 L
O2
.752 atm
When these valves are opened, what is the partial pressure
of each gas and the total pressure in the assembly?
[P of CH4 = 1.2 atm; P of N2 = 0.097 atm;
P of O2 = 0.292 atm; P total : add all the pressures]
Kinetic Molecular Theory of Gases
1. Gas particles are in continuous motion ( the
hotter the gas, the faster the molecules are
moving) with negligible volume compared to
volume of container.
2. Molecules are far apart from each other
3. Do not attract or repulse each other (?).
4. All collisions are elastic (gas does not lose
energy when left alone).
5. The energy is proportional to Kelvin
temperature. At a given temperature all gases
have the same average KE.
75
Properties of Gases
Observation
Hypothesis
Gases are easy to expand
Gas molecules do not strongly
attract each other
Gases are compressible
Particles have small volumes
compared to continer. Lots of
empty space
Gases are easy to compress
Gas molecules don’t strongly repel
each other
Gases have densities that are
1/1000 of solid or liquid densities
Molecules are much farther apart
in gases than in liquids and solids
Hot gases leak through holes
faster than cold gases
Gas molecules are in constant
motion
76
Properties of Gases
Observation
Hypothesis
Gases undergo elastic
collisions: when gas is left
alone at constnat
temperature, it does not
liquefy or vaporize (no
energy exchange)
Gas molecules are like
billiard balls – do not stick to
each other (do not attract, do
not repel)
Hot gases leak through
holes faster than cold gases
Gas molecules are in
constant motion
77
Kinetic Molecular Theory of Gases
• Ideal gas limitations:
• Gases can be liquefied if cooled enough.
• Real gas molecules do attract one
another to some extent otherwise the
particles would not condense to form a
liquid.
78
Maxwell Distribution Curves
• Average Kinetic Energy at a given
temperature is constant for a gas
sample
• But, the speeds of the molecules vary
– (during to collisions with each other and with
the walls of the container)
• Physics: momentum is conserved
• (playing pool)
79
Maxwell’s Distribution Curves
http://jersey.uoreg
on.edu/vlab/Ballo
on/index.html
80
Gas Laws: Maxwell’s Distribution Curves
Molecules in a gas
move at different
speeds.
377 m/s
900 m/s
compare
1500 m/s
The Maxwell
Distribution Curves
show how many
molecules are
moving at a
particular speed.
The distribution
shifts to higher
speeds at higher
temperatures.
81
MKT of Gases: Equations
• KE = ½ m(urms)2
• Average KE = (3/2) RT
• Maxwell equation for the root mean square
velocity:
• Urms =
3RT
M
The Urms is not the same as the mean (average ) speed.
The difference is small.
82
Average Molecular speed
 Average molecular kinetic energy depends
only on temperature for ideal gases.
 Therefore:
 Higher temperature = higher root-meansquare speed (RMS), rms
 Higher molecular weight (molar mass) = lower
urms speed (same temperature)
83
Average Root Mean Square: Examples
31. Calculate the Urms speed, urms, of an N2 molecule at
25ºC.
(5.15 x 102 m/s)
32. Calculate the urms speed of helium atoms 25ºC.
(1.36 x 103 m/s)
33. Calculate the Urms speed of chlorine atoms at 25ºC.
(323 m/s)
u
rms

3RT
M
84
Average Speed of Some Molecules
85
Diffusion and Effusion
(a) Diffusion: mixing of gas molecules by random
motion under conditions where molecular collisions
occur.
(Ib) Effusion: the escape of a gas through a pinhole
without molecular collisions
86
Diffusion and Effusion
HCl and NH3: What
will happen?
87
Graham’s Law of Diffusion
• Under the same conditions of temperature
and pressure, the rate of diffusion of gas
molecules are inversely proportional to the
square root of their molecular masses.
r1
M2

r2
M1
Rate of effusion of A

Rate of effusion of B
density of B

density of A
Molar mass of B
Molar mass of A
88
Graham’s Law of Diffusion
34. It has taken 192 seconds for 1.4 L of an unknown gas
to effuse through a porous wall and 84 seconds for the
same volume of N2 gas to effuse at the same
temperature and pressure. What is the molar mass of
the unknown gas? (146 g/mol)
35. In a given period of time, 0.21 moles of a gas of MM =
26 gmol-1 effuses. How many moles of HCN would
effuse in the same period of time?
36. Calculate and compare the urms of Nitrogen gas at
35oC and 299K.
89
Real Gases
Problems with the Kinetic Molecular Theory
of "Ideal" Gases:
1. Gas particles have volume (they are not
point masses). The volume becomes
important under certain conditions.
2. When gas particles are close to each
other, they attract each other.
90
For 1 mole of gas:
PV = nRT equation when
rearranged:
PV
1
RT
Plot of (PV)/(RT) for
1 mole of gas
The value for the equation
is not always equal to 1
Corrections to the Ideal Gas
Equation is needed
91
Factors that Affect Ideality
Deviation from ideal
behavior as a function
of temperature for
nitrogen gas:
92
Factors that Affect Ideality of Gases
• Interactions between the molecules
(intermolecular forces): important at low
temperatures and small free volume
• Actual volumes of the molecules: important at
high pressures and small free volume.
• Free volume: the space in the container that
is not occupied by the molecules.
93
Factors Affecting Ideality of Gases: low
temperatures and small free volumes
Distance between molecules is related to gas concentration:
n
P

V RT
•At high concentration (high P, low V):
•Molecules are closer (higher concentration) = stronger
intermolecular attractions = deviation from ideality
•Repulsion make pressure higher than expected by decreasing
free volume
•Attractions make pressure lower than expected by breaking
molecular collisions (plastic collisions)
94
Effect of Intermolecular Attractions
Orange molecules
attract purple
molecules.
Therefore: purple
molecule exert less
force when it collides
with the wall.
No attractive forces =
more force
95
Real Gases: Effect of Pressure
At high pressures
• Intermolecular distances
between molecules
decrease
• Attractive forces start to
play a role
• Stickiness factor
• Measured pressure is
less than expected
Pideal  Preal
Pideal  Preal
n2
a 2
V
2
n
a 2
V
Correction for
lower pressure
96
Real Gases: Effect of Volume
Volume
should go
to zero,
but it does
not.
(a) At low pressure, the gas occupies the entire container
and its volume is insignificant compared to the volume of
the container.
(b) At high pressure, the volume of a real gas is somewhat
larger than the ideal value for an ideal gas as gas
molecules take up space.
97
Correction due to volume: (V
– nb)
V = volume of the container
n = number of moles
B = volume of a mole of particles
Correction for volume
98
Real Gases: Corrections
• Constant needed to correct intermolecular
attractive forces (make it larger)
• Constant needed to correct for volume of
individual gas molecules (make it smaller)
The constants are characteristic properties of the
substances: depend on the make-up and geometry of
the substance
99
Van der Waals Constants
for Common Gases
Compound
a (L2-atm/mol2)
b (L/mol)
He
0.03412
0.02370
Ne
0.2107
0.01709
H2
0.2444
0.02661
Ar
1.345
0.03219
O2
1.360
0.03803
N2
1.390
0.03913
CO
1.485
0.03985
CH4
2.253
0.04278
CO2
3.592
0.04267
NH3
4.170
0.03707
100
Real Gases: Comparison
Ideal gas
Real gas
Always
Only at low
pressures
Molecular
volume
Zero
Small, but not
zero
Molecular
attraction
Zero
Small
Molecular
repulsion
Zero
small
Obey PV=nRT
101
Real Gases
Large deviation form ideality:
Large intermolecular attractive forces (IMF)
Large Molar Mass (and subsequently volume)
Real Conditions:
high pressures
low volumes
Ideal Conditions:
Low pressures (atmospheric and up to ≈ 50 atm
High temperatures
102
103
104
Factors Affecting Ideality of Gases
– Tug-of-war between these two effects causes
the following:
• Repulsion win at very high pressure
• Attractions win at moderate pressure
• Neither attractions nor repulsions are important at
low pressure.
105
22.41 L atm
PV versus P at Constant T (1 mole of Gas)
O2
PV
CO2
P (at constant T)
106
• V of a real gas > V of an ideal gas
because V of gas molecules is significant
when P is high. Ideal Gas Equation
assumes that the individual gas molecules
have no volume.
107
Boyle’s Law
108
Plots of Charles’ Law
A plot of V versus T for a gas sample. What type of
graph?
109
Kinetic Molecular Theory of Gases
1. Gases are composed of tiny atoms or molecules (particles)
whose size is negligible compared to the average distance
between them. This means that the volume of the individual
particles in a gas can be assumed to be negligible (close to
zero).
2. The particles move randomly in straight lines in all directions
and at various speeds.
3. The forces of attraction or repulsion between two particles in
a gas are very weak or negligible (close to zero), except
when they collide.
4. When particles collide with one another, the collisions are
elastic (no kinetic energy is lost). The collisions with the
walls of the container create the gas pressure.
5. The average kinetic energy of a molecule is proportional to
the Kelvin temperature and all calculations should be carried
out with temperatures converted to K.
110
Notes: Kinetic Molecular Theory of Gases
a. The observation that gases are compressible agrees
with the assumption that gas particles have a small
volume compared to the container.
b. Elastic collisions agree with the observation that gases
when left alone in a container do not seem to lose
energy and do not spontaneously convert to the liquid.
c. The assumptions have limitations. For example, gases
can be liquefied if cooled enough. This means real gas
molecules do attract one another to some extent
otherwise the particles would never stick to one another
in order to condense to form a liquid.
111
Measuring Pressure
Open-ended
Manometer
Mercury Barometer
Pressure?
112
Maxwell- Boltzmann Velocity (energy)
Distribution
Plot of Probability (fraction of molecules with given speed)
versus root mean square velocity of the molecules.
113
Maxwell Distribution Curve
• Variation in particle speeds for hydrogen gas at 273K
urms
The vertical line
on the graph
represents the
root-meansquare-speed
(urms).
The root-mean-square-speed is the square root of the averages of the
squares of the speeds of all the particles in a gas sample at a particular
114
temperature.