GASES AND KINETIC-MOLECULAR THEORY

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CHAPTER 10

• GASES AND KINETIC-MOLECULAR THEORY

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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

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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

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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.

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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
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Pressure

• Pressure is force per unit area.
• lb/in2
• N/m2
• Gas pressure as most people think of it.

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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
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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

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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?

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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.

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Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale
absolute zero -273.15 0C
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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.

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Charles Law The Volume-Temperature
Relationship The Absolute Temperature Scale

• Mathematical form of Charles law.

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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

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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

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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.

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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?

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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!

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The Combined Gas Law Equation
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Avogadros Law and theStandard Molar Volume
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Avogadros Law and theStandard 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

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Avogadros Law and theStandard 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?

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Summary of Gas LawsThe 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.

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Summary of Gas LawsThe 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.

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Summary of Gas LawsThe 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.

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Summary of Gas LawsThe 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

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Summary of Gas LawsThe Ideal Gas Law
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Summary of Gas LawsThe 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!

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Summary of Gas LawsThe Ideal Gas Law
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Summary of Gas LawsThe 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!

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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.

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Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
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Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
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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!

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Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
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Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
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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 .....

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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?

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Daltons Law of Partial Pressures

• Vapor Pressure is the pressure exerted by a
  substances vapor over the substances liquid at
  equilibrium.

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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?

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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!

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Daltons Law of Partial Pressures
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Mass-Volume Relationships in Reactions Involving
Gases
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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

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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!

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Mass-Volume Relationships in Reactions Involving
Gases
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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.

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The Kinetic-Molecular Theory

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

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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.

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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.

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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.

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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.

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The Kinetic-Molecular Theory
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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.

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The Kinetic-Molecular Theory

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

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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!

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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?

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Diffusion and Effusion of Gases

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

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Diffusion and Effusion of Gases

• This is a demonstration of diffusion.

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Diffusion and Effusion of Gases

• The rate of effusion is inversely proportional to
  the square roots of the molecular weights or
  densities.

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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.

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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!

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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.

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Real GasesDeviations 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.

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Real GasesDeviations 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.

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Real GasesDeviations 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!

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Real GasesDeviations from Ideality

• Example 12-20 Solve Example 12-19 using the van
  der Waals equation.

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Real GasesDeviations from Ideality
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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?

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Synthesis Question
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Synthesis Question
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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?

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End of Chapter 12

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