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GASES AND KINETIC-MOLECULAR THEORY

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Title: GASES AND KINETIC-MOLECULAR THEORY


1
CHAPTER 10
  • GASES AND KINETIC-MOLECULAR THEORY

2
CHAPTER GOALS
  1. Comparison of Solids, Liquids, and Gases
  2. Composition of the Atmosphere and Some Common
    Properties of Gases
  3. Pressure
  4. Boyles Law The Volume-Pressure Relationship
  5. Charles Law The Volume-Temperature
    Relationship The Absolute Temperature Scale
  6. Standard Temperature and Pressure
  7. The Combined Gas Law Equation
  8. Avogadros Law and the Standard Molar Volume

3
CHAPTER GOALS
  • Summary of Gas Laws The Ideal Gas Equation
  • Determination of Molecular Weights and Molecular
    Formulas of Gaseous Substances
  • Daltons Law of Partial Pressures
  • Mass-Volume Relationships in Reactions Involving
    Gases
  • The Kinetic-Molecular Theory
  • Diffusion and Effusion of Gases
  • Real Gases Deviations from Ideality

4
Comparison of Solids, Liquids, and Gases
  • The density of gases is much less than that of
    solids or liquids.

Densities (g/mL) Solid Liquid Gas
H2O 0.917 0.998 0.000588
CCl4 1.70 1.59 0.00503
  • Gas molecules must be very far apart compared to
    liquids and solids.

5
Composition of the Atmosphere and Some Common
Properties of Gases
Composition of Dry Air
Gas by Volume
N2 78.09
O2 20.94
Ar 0.93
CO2 0.03
He, Ne, Kr, Xe 0.002
CH4 0.00015
H2 0.00005
6
Pressure
  • Pressure is force per unit area.
  • lb/in2
  • N/m2
  • Gas pressure as most people think of it.

7
Pressure
  • Atmospheric pressure is measured using a
    barometer.
  • Definitions of standard pressure
  • 76 cm Hg
  • 760 mm Hg
  • 760 torr
  • 1 atmosphere
  • 101.3 kPa

Hg density 13.6 g/mL
8
Boyles Law The Volume-Pressure Relationship
  • V ? 1/P or
  • V k (1/P) or PV k
  • P1V1 k1 for one sample of a gas.
  • P2V2 k2 for a second sample of a gas.
  • k1 k2 for the same sample of a gas at the same
    T.
  • Thus we can write Boyles Law mathematically as
    P1V1 P2V2

9
Boyles Law The Volume-Pressure Relationship
  • Example 12-1 At 25oC a sample of He has a volume
    of 4.00 x 102 mL under a pressure of 7.60 x 102
    torr. What volume would it occupy under a
    pressure of 2.00 atm at the same T?

10
Boyles Law The Volume-Pressure Relationship
  • Notice that in Boyles law we can use any
    pressure or volume units as long as we
    consistently use the same units for both P1 and
    P2 or V1 and V2.
  • Use your intuition to help you decide if the
    volume will go up or down as the pressure is
    changed and vice versa.

11
Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale
absolute zero -273.15 0C
12
Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale
  • Charless law states that the volume of a gas is
    directly proportional to the absolute temperature
    at constant pressure.
  • Gas laws must use the Kelvin scale to be correct.
  • Relationship between Kelvin and centigrade.

13
Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale
  • Mathematical form of Charles law.

14
Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale
  • Example 12-2 A sample of hydrogen, H2, occupies
    1.00 x 102 mL at 25.0oC and 1.00 atm. What
    volume would it occupy at 50.0oC under the same
    pressure?
  • T1 25 273 298
  • T2 50 273 323

15
Standard Temperature and Pressure
  • Standard temperature and pressure is given the
    symbol STP.
  • It is a reference point for some gas
    calculations.
  • Standard P ? 1.00000 atm or 101.3 kPa
  • Standard T ? 273.15 K or 0.00oC

16
The Combined Gas Law Equation
  • Boyles and Charles Laws combined into one
    statement is called the combined gas law
    equation.
  • Useful when the V, T, and P of a gas are changing.

17
The Combined Gas Law Equation
  • Example 12-3 A sample of nitrogen gas, N2,
    occupies 7.50 x 102 mL at 75.00C under a pressure
    of 8.10 x 102 torr. What volume would it occupy
    at STP?

18
The Combined Gas Law Equation
  • Example 12-4 A sample of methane, CH4, occupies
    2.60 x 102 mL at 32oC under a pressure of 0.500
    atm. At what temperature would it occupy 5.00 x
    102 mL under a pressure of 1.20 x 103 torr?
  • You do it!

19
The Combined Gas Law Equation
20
Avogadros Law and the Standard Molar Volume
21
Avogadros Law and the Standard Molar Volume
  • Avogadros Law states that at the same
    temperature and pressure, equal volumes of two
    gases contain the same number of molecules (or
    moles) of gas.
  • If we set the temperature and pressure for any
    gas to be STP, then one mole of that gas has a
    volume called the standard molar volume.
  • The standard molar volume is 22.4 L at STP.
  • This is another way to measure moles.
  • For gases, the volume is proportional to the
    number of moles.
  • 11.2 L of a gas at STP 0.500 mole
  • 44.8 L ? moles

22
Avogadros Law and the Standard Molar Volume
  • Example 12-5 One mole of a gas occupies 36.5 L
    and its density is 1.36 g/L at a given
    temperature and pressure. (a) What is its molar
    mass? (b) What is its density at STP?

23
Summary of Gas Laws The Ideal Gas Law
  • Boyles Law - V ? 1/P (at constant T n)
  • Charles Law V ? T (at constant P n)
  • Avogadros Law V ? n (at constant T P)
  • Combine these three laws into one statement
  • V ? nT/P
  • Convert the proportionality into an equality.
  • V nRT/P
  • This provides the Ideal Gas Law.
  • PV nRT
  • R is a proportionality constant called the
    universal gas constant.

24
Summary of Gas Laws The Ideal Gas Law
  • We must determine the value of R.
  • Recognize that for one mole of a gas at 1.00 atm,
    and 273 K (STP), the volume is 22.4 L.
  • Use these values in the ideal gas law.

25
Summary of Gas Laws The Ideal Gas Law
  • R has other values if the units are changed.
  • R 8.314 J/mol K
  • Use this value in thermodynamics.
  • R 8.314 kg m2/s2 K mol
  • Use this later in this chapter for gas
    velocities.
  • R 8.314 dm3 kPa/K mol
  • This is R in all metric units.
  • R 1.987 cal/K mol
  • This the value of R in calories rather than J.

26
Summary of Gas Laws The Ideal Gas Law
  • Example 12-6 What volume would 50.0 g of ethane,
    C2H6, occupy at 1.40 x 102 oC under a pressure of
    1.82 x 103 torr?
  • To use the ideal gas law correctly, it is very
    important that all of your values be in the
    correct units!
  • T 140 273 413 K
  • P 1820 torr (1 atm/760 torr) 2.39 atm
  • 50 g (1 mol/30 g) 1.67 mol

27
Summary of Gas Laws The Ideal Gas Law
28
Summary of Gas Laws The Ideal Gas Law
  • Example 12-7 Calculate the number of moles in,
    and the mass of, an 8.96 L sample of methane,
    CH4, measured at standard conditions.
  • You do it!

29
Summary of Gas Laws The Ideal Gas Law
30
Summary of Gas Laws The Ideal Gas Law
  • Example 12-8 Calculate the pressure exerted by
    50.0 g of ethane, C2H6, in a 25.0 L container at
    25.0oC.
  • You do it!

31
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
  • Example 12-9 A compound that contains only
    carbon and hydrogen is 80.0 carbon and 20.0
    hydrogen by mass. At STP, 546 mL of the gas has
    a mass of 0.732 g . What is the molecular (true)
    formula for the compound?
  • 100 g of compound contains 80 g of C and 20 g of
    H.

32
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
33
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
34
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
  • Example 12-10 A 1.74 g sample of a compound that
    contains only carbon and hydrogen contains 1.44 g
    of carbon and 0.300 g of hydrogen. At STP 101 mL
    of the gas has a mass of 0.262 gram. What is its
    molecular formula?
  • You do it!

35
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
36
Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
37
Daltons Law of Partial Pressures
  • Daltons law states that the pressure exerted by
    a mixture of gases is the sum of the partial
    pressures of the individual gases.
  • Ptotal PA PB PC .....

38
Daltons Law of Partial Pressures
  • Example 12-11 If 1.00 x 102 mL of hydrogen,
    measured at 25.0 oC and 3.00 atm pressure, and
    1.00 x 102 mL of oxygen, measured at 25.0 oC and
    2.00 atm pressure, were forced into one of the
    containers at 25.0 oC, what would be the pressure
    of the mixture of gases?

39
Daltons Law of Partial Pressures
  • Vapor Pressure is the pressure exerted by a
    substances vapor over the substances liquid at
    equilibrium.

40
Daltons Law of Partial Pressures
  • Example 12-12 A sample of hydrogen was collected
    by displacement of water at 25.0 oC. The
    atmospheric pressure was 748 torr. What pressure
    would the dry hydrogen exert in the same
    container?

41
Daltons Law of Partial Pressures
  • Example 12-13 A sample of oxygen was collected
    by displacement of water. The oxygen occupied
    742 mL at 27.0 oC. The barometric pressure was
    753 torr. What volume would the dry oxygen
    occupy at STP?
  • You do it!

42
Daltons Law of Partial Pressures
43
Mass-Volume Relationships in Reactions Involving
Gases
44
Mass-Volume Relationships in Reactions Involving
Gases
  • In this section we are looking at reaction
    stoichiometry, like in Chapter 3, just including
    gases in the calculations.
  • 2 mol KClO3 yields 2 mol KCl and 3 mol O2
  • 2(122.6g) yields 2 (74.6g) and 3
    (32.0g)
  • Those 3 moles of O2 can also be thought of as
  • 3(22.4L) or 67.2 L at STP

45
Mass-Volume Relationships in Reactions Involving
Gases
  • Example 12-14 What volume of oxygen measured at
    STP, can be produced by the thermal decomposition
    of 120.0 g of KClO3?
  • You do it!

46
Mass-Volume Relationships in Reactions Involving
Gases
47
The Kinetic-Molecular Theory
  • The basic assumptions of kinetic-molecular theory
    are
  • Postulate 1
  • Gases consist of discrete molecules that are
    relatively far apart.
  • Gases have few intermolecular attractions.
  • The volume of individual molecules is very small
    compared to the gass volume.
  • Proof - Gases are easily compressible.

48
The Kinetic-Molecular Theory
  • Postulate 2
  • Gas molecules are in constant, random, straight
    line motion with varying velocities.
  • Proof - Brownian motion displays molecular motion.

49
The Kinetic-Molecular Theory
  • Postulate 3
  • Gas molecules have elastic collisions with
    themselves and the container.
  • Total energy is conserved during a collision.
  • Proof - A sealed, confined gas exhibits no
    pressure drop over time.

50
The Kinetic-Molecular Theory
  • Postulate 4
  • The kinetic energy of the molecules is
    proportional to the absolute temperature.
  • The average kinetic energies of molecules of
    different gases are equal at a given temperature.
  • Proof - Brownian motion increases as temperature
    increases.

51
The Kinetic-Molecular Theory
  • The kinetic energy of the molecules is
    proportional to the absolute temperature. The
    kinetic energy of the molecules is proportional
    to the absolute temperature.
  • Displayed in a Maxwellian distribution.

52
The Kinetic-Molecular Theory
  • The gas laws that we have looked at earlier in
    this chapter are proofs that kinetic-molecular
    theory is the basis of gaseous behavior.
  • Boyles Law
  • P ? 1/V
  • As the V increases the molecular collisions with
    container walls decrease and the P decreases.
  • Daltons Law
  • Ptotal PA PB PC .....
  • Because gases have few intermolecular
    attractions, their pressures are independent of
    other gases in the container.
  • Charles Law
  • V ? T
  • An increase in temperature raises the molecular
    velocities, thus the V increases to keep the P
    constant.

53
The Kinetic-Molecular Theory
54
The Kinetic-Molecular Theory
  • The root-mean square velocity of gases is a very
    close approximation to the average gas velocity.
  • Calculating the root-mean square velocity is
    simple
  • To calculate this correctly
  • The value of R 8.314 kg m2/s2 K mol
  • And M must be in kg/mol.

55
The Kinetic-Molecular Theory
  • Example 12-17 What is the root mean square
    velocity of N2 molecules at room T, 25.0oC?

56
The Kinetic-Molecular Theory
  • Example 12-18 What is the root mean square
    velocity of He atoms at room T, 25.0oC?
  • You do it!

57
The Kinetic-Molecular Theory
  • Can you think of a physical situation that proves
    He molecules have a velocity that is so much
    greater than N2 molecules?
  • What happens to your voice when you breathe He?

58
Diffusion and Effusion of Gases
  • Diffusion is the intermingling of gases.
  • Effusion is the escape of gases through tiny
    holes.

59
Diffusion and Effusion of Gases
  • This is a demonstration of diffusion.

60
Diffusion and Effusion of Gases
  • The rate of effusion is inversely proportional to
    the square roots of the molecular weights or
    densities.

61
Diffusion and Effusion of Gases
  • Example 12-15 Calculate the ratio of the rate of
    effusion of He to that of sulfur dioxide, SO2, at
    the same temperature and pressure.

62
Diffusion and Effusion of Gases
  • Example 12-16 A sample of hydrogen, H2, was
    found to effuse through a pinhole 5.2 times as
    rapidly as the same volume of unknown gas (at the
    same temperature and pressure). What is the
    molecular weight of the unknown gas?
  • You do it!

63
Real Gases Deviations from Ideality
  • Real gases behave ideally at ordinary
    temperatures and pressures.
  • At low temperatures and high pressures real gases
    do not behave ideally.
  • The reasons for the deviations from ideality are
  • The molecules are very close to one another, thus
    their volume is important.
  • The molecular interactions also become important.

64
Real Gases Deviations from Ideality
  • van der Waals equation accounts for the behavior
    of real gases at low temperatures and high
    pressures.
  • The van der Waals constants a and b take into
    account two things
  • a accounts for intermolecular attraction
  • b accounts for volume of gas molecules
  • At large volumes a and b are relatively small and
    van der Waals equation reduces to ideal gas law
    at high temperatures and low pressures.

65
Real Gases Deviations from Ideality
  • What are the intermolecular forces in gases that
    cause them to deviate from ideality?
  • For nonpolar gases the attractive forces are
    London Forces
  • For polar gases the attractive forces are
    dipole-dipole attractions or hydrogen bonds.

66
Real Gases Deviations from Ideality
  • Example 12-19 Calculate the pressure exerted by
    84.0 g of ammonia, NH3, in a 5.00 L container at
    200. oC using the ideal gas law.
  • You do it!

67
Real Gases Deviations from Ideality
  • Example 12-20 Solve Example 12-19 using the van
    der Waals equation.

68
Real Gases Deviations from Ideality
69
Synthesis Question
  • The lethal dose for hydrogen sulfide is 6.0 ppm.
    In other words, if in 1 million molecules of air
    there are six hydrogen sulfide molecules then
    that air would be deadly to breathe. How many
    hydrogen sulfide molecules would be required to
    reach the lethal dose in a room that is 77 feet
    long, 62 feet wide and 50. feet tall at 1.0 atm
    and 25.0 oC?

70
Synthesis Question
71
Synthesis Question
72
Group Question
  • Tires on a car are typically filled to a pressure
    of 35 psi at 3.00 x 102 K. A tire is 16 inches
    in radius and 8.0 inches in thickness. The wheel
    that the tire is mounted on is 6.0 inches in
    radius. What is the mass of the air in the tire?

73
End of Chapter 12
  • Gases are the simplest state of matter.
  • Liquids and solids are more complex.
  • They are the subject of Chapter 13.
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