114
1 The Gaseous State Chapter 10
1
The Gaseous State
Chapter 10
2
Objectives
1. Understand the definition of pressure. Use the definition to predict and measure pressure experimentally
2. Describe experiments that show relationships between pressure, temperature, volume, and moles of a gas sample
3. Use empirical gas laws to predict how change in one of the properties of a gas will affect the remaining properties.
4. Use empirical gas laws to estimate gas densities and molecular masses.
5. Use volume-to-mole relationships obtained using the empirical gas laws to solve stoichiometry problems involving gases.
3
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.
5
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
6
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.ohio-state.edu/betha/nealGasLaw/f
r1.1.html
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
8
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 other3. 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.
9
Properties of Gases
Observation HypothesisGases 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
10
Properties of Gases
Observation Hypothesis
Gases are easy to expand
Gas molecules do not strongly attract each other
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
11
Atmospheric Pressure
Intensive or Extensive Property?
12
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
13
Pressure
Unit Symbol Conversions
Pascal Pa 1 Pa = 1 N/m2
Psi lb/in2
Atmosphere Atm 1 atm = 101325 Pa = 14.7 lb/in2
Bar Bar 1 bar = 100000 Pa
Torr Torr 760 torr = 1 atm
Millimeter mercury
mm Hg 1 mm Hg = 1 torr
14
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.
15
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
16
Measuring Pressure
17
Pressure: Barometer
Barometer measures atmospheric pressure as a mercury column height.
18
Pressure: Open-Manometer
Manometer measures gas pressure as a difference in mercury column heights.
Two types: closed manometer
open manometer
19
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.
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?
21
The Gas Laws: State of Gas
Property Symbol Unit Property Type
Pressure P atm, torr, Pa
Intensive
Volume V L, cm3 Extensive
Temperature T K Intensive
Moles n mol extensive
22
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)
The Gas Laws
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=42
http://jersey.uoregon.edu/vlab/Piston/index.html
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)
Prentice Hall Simulations of Gas Laws
• http://cwx.prenhall.com/bookbind/pubbooks/hillchem3/chapter5/deluxe.html
• http://cwx.prenhall.com/bookbind/pubbooks/hillchem3/chapter5/deluxe.html
25
Boyle’s Law: ExperimentRelate volume to pressure when everything else is constant. Experiment: trapped air bubble at 298 K
Volume, mL Pressure, torr PV )mL torr)
10.0 760.0
20.0 379.6
30.0 253.2
40.0 191.0
Graphs?
26
Boyle’s Law: ExperimentRelate volume to pressure when everything else is constant. Experiment: trapped air bubble at 298 K
Volume, mL Pressure, torr PV (mL torr)
10.0 760.0 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?
27
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.
28
Boyle’s Law: Volume/Pressure Relationship
What happens to the volume of the gas as the pressure increases? Mathematical Relationship?
29
Plot of Boyle’s Law
V versus P V versus 1/P
Type of Graphs?
30
Boyle’s Law
P
k V
V
k
1
2
1
211 PPV
VP
P
nRTV
Boyle’s LawBoyle’s Law – the – the volumevolume of a fixed amount of gas of a fixed amount of gas at constant temperature and constant number of at constant temperature and constant number of moles is moles is inversely proportionalinversely proportional to the to the gas pressuregas pressure..
MOLECULAR VIEW
Boyle’s Law
V
P
k V
k RT
2 211 PPV
n
P
nRTV
Boyle’s LawBoyle’s Law – the – the volumevolume of a fixed amount of a fixed amount of gas at constant temperature and of gas at constant temperature and constant number of moles is constant number of moles is inversely inversely proportionalproportional to the to the gas pressure.gas pressure.
MOLECULAR VIEW:
Confining molecules to a smaller space increases the number (frequency) of collisions, and so increases the pressure
32
Charles' Law (V/T Relationships)
Relate volume to temperature, everything else is constant. Experiment: He bubble trapped at 1 atm.
V, mL T, ºC T, (K) V/T, mL/K
40.0 0.0 273.2
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
33
Charles' Law (V/T Relationships)
Relate volume to temperature, everything else is constant. Experiment: He bubble trapped at 1 atm.
V, mL T, ºC T, (K) V/T, mL/K
40.0 0.0 273.2 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
34
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
35
Charles’ Law
A plot of V versus T for a gas sample. What type of graph? Equation?
36
37
Charles' Law
Kinetic Interpretation of Charles's Law? Why higher pressure? Equation?
Frequency and force of collision…
Charles’ Law
2
T2
V
kT V
T
1
1T
V
P
nRk
P
nRV
The volume of the gas is directly proportional to its Kelvin temperature, when everything else is constant.
MOLECULAR VIEW
39
Charles’ Law
The volume of the gas is directly proportional to its Kelvin temperature, when everything else is constant.
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.
2T
2V
kT V
T
1
1T
V
P
nRk
P
nRV
40
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
41
Gay Lussac’s Law
2
2
T
P
kT P
nRT k
P
1
1
T
PT
Pk
V
nRT Molecular View;
The pressure of the gas is directly proportional to its Kelvin temperature, when everything else is constant.
42
Gay Lussac’s Law
2
2
T
P
kT P
nRT k
P
1
1
T
PT
Pk
V
nRT Molecular View;
The pressure of the gas is directly proportional to its Kelvin temperature, when everything else is constant.
Raising the temperature increases the number of collisions and the kinetic energy of the molecules. More collisions with greater energy (force) means higher pressure.
Combined Gas laws
2
2T
V
T
2P
1
T1
V1
P k
T
PV
n k nT PV
44
Avogadro’s Law: Relates n to Volume
45
Volume of Real Gases at STP
46
Avogadro’s Law: Relates n to Volume
nk V P
RT
P
nRT
A
Ak
V
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.
MOLECULAR VIEW
47
Avogadro’s Law: Relates n to Volume
nk V P
RT
P
nRT
A
Ak
V
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).
MOLECULAR VIEW
Type of gas does not influence distance between molecules too much.
48
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.
nk V
n
AV
49
Avogadro’s Law: Answer to Example 6
50
Example 7
Show the approximate level of the movable piston in drawings (a), (b), and (c ) after the indicated changes.
51
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]
52
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?
53
Ideal Gas Law
T toalproportiondirectly are V and Pboth nTPV
PV = nRT
Gas Constant R = 0.082057 (L atm)/(mol K)
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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)
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?
56
Gas Laws Summary
Changing variables
Variables held constant
Relationship 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
57
Gas Density and Molar Mass
dRT P(MM) P
DRT )
V
m
PV
mRT
RT
P(MM)
V
m D
MM
m VP
MM
m n nRT
P
RTMM
RTVP
)((
Purple M&M Do Red TooOr
Michael Mo do the right thing
58
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]
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Stoichiometry Involving Gases
Use regular Stoichiometry techniques, except that
for non STP conditions, and 22.4 L/mole for STP
conditions.
RT
PVn
60
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.
61
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.
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]
63
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)?
64
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?
65
Dalton’s Law of Partial Pressure
66
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?
67
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?
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Dalton’s Law of Partial Pressures
21total
22
1
P P P
n P n
V
RTVRTP
1
Ptotal = P1 + P2 + P3 +……. Pn
1 sample of fraction mole x n
n
n
n
11T
1
otalt
1
V
RTV
RT
P
P
total
1
P1 = x1PT
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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.
70
Dalton’s Law: Examples23. 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]
71
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]
72
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]
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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]
4.00 L CH4
1.50 L N2
3.50 L O2
.58 atm 2.70 atm .752 atm
Example 30Example 30
When these valves are opened, what is the partial pressureWhen these valves are opened, what is the partial pressureof each gas and the total pressure in the assembly?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]
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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 other3. 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.
76
Properties of Gases
Observation HypothesisGases 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
77
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
78
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.
79
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)
80
Maxwell’s Distribution Curves
http://jersey.uoregon.edu/vlab/Balloon/index.html
81
Gas Laws: Maxwell’s Distribution Curves
377 m/s
1500 m/s
900 m/s
compare
Molecules in a gas move at different speeds.
The Maxwell Distribution Curves show how many molecules are moving at a particular speed.
The distribution shifts to higher speeds at higher temperatures.
82
MKT of Gases: Equations
• KE = ½ m(urms)2
• Average KE = (3/2) RT• Maxwell equation for the root mean square
velocity:
• Urms = M
RT3
The Urms is not the same as the mean (average ) speed. The difference is small.
83
Average Molecular speed
Average molecular kinetic energy depends only on temperature for ideal gases.
Therefore: Higher temperature = higher root-mean-
square speed (RMS), rms
Higher molecular weight (molar mass) = lower urms speed (same temperature)
84
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)
M
RTrmsu
3
85
Average Speed of Some Molecules
86
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
87
Diffusion and Effusion
HCl and NH3: What will happen?
88
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.
1
2
2
1
M
M
r
r
Aof mass Molar
B of mass Molar
Aofdensity
B ofdensity
B of effusion of Rate
Aof effusion of Rate
89
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.
90
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.
91
1RT
PV
For 1 mole of gas:
PV = nRT equation when rearranged:
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
92
Factors that Affect Ideality
Deviation from ideal behavior as a function of temperature for nitrogen gas:
93
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.
94
Factors Affecting Ideality of Gases: low temperatures and small free volumes
Distance between molecules is related to gas concentration:
RT
P
V
n
•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)
95
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
96
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
2
2
V
naPP realideal
2
2
V
naPP realideal Correction for
lower pressure
97
Real Gases: Effect of Volume
(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.
Volume should go to zero, but it does not.
98
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
99
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
100
Van der Waals Constantsfor 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
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Ideal gas Real gas
Obey PV=nRT Always Only at low pressures
Molecular volume
Zero Small, but not zero
Molecular attraction
Zero Small
Molecular repulsion
Zero small
Real Gases: Comparison
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Real Gases
Large deviation form ideality:Large intermolecular attractive forces (IMF)Large Molar Mass (and subsequently volume)
Real Conditions:high pressureslow volumes
Ideal Conditions:Low pressures (atmospheric and up to ≈ 50 atmHigh temperatures
103
104
105
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.
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PV
P (at constant T)
CO2
O2
22.4
1 L
atm
PV versus P at Constant T (1 mole of Gas)
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• 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.
108
Boyle’s Law
109
Plots of Charles’ Law
A plot of V versus T for a gas sample. What type of graph?
110
Kinetic Molecular Theory of Gases1. 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.
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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.
112
Mercury Barometer
Open-ended Manometer
Pressure?
Measuring Pressure
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Maxwell- Boltzmann Velocity (energy) Distribution
Plot of Probability (fraction of molecules with given speed) versus root mean square velocity of the molecules.
114
Maxwell Distribution Curve
• Variation in particle speeds for hydrogen gas at 273K
The vertical line on the graph represents the root-mean-square-speed (urms).
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 temperature.
