Chapter 5: Gases

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Chapter 5 Gases

• 5.1 Measurements on Gases
• 5.2 The Ideal Gas Law
• 5.3 Gas Law Calculation
• 5.4 Stoichiometry of Gaseous Reactions
• 5.5 Gas Mixtures Partial Pressures and Mole
  Fractions
• 5.6 Kinetic Theory of Gases
• 5.7 Real Gases

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Measurements On Gases

• A gas is completely described by its volume,
  amount, temperature, and pressure.
• A gas expands uniformly to fill any container in
  which it is placed. The volume of a gas is the
  volume of its container. Volume can be expressed
  in liters, cubic centimeters or cubic meters.
• 1 L 103 cm3 10-3 m3

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Properties of a Gases

• The amount of matter in a gaseous sample is
  expressed in moles.
• The temperature of a gas is measured in degrees
  Celsius. In any calculation involving the
  physical behavior of gases, temperature must be
  expressed on the Kelvin scale.

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Properties (Contd)

• Pressure is defined as force per unit area (psi
  pound per square inch). Pressure is measured
  using a manometer.
• In a laboratory setting, pressure is expressed in
  millimeters of mercury (mm Hg) or atmospheres
  (atm).
• 1 atm 760mm Hg

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Pressure (Contd)

• In the International System, the standard unit of
  pressure is the pascal (Pa). A related unit is
  the bar (105 Pa).
• 1.013bar 1 atm 760mm Hg

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Example 1

• A balloon with a volume (V) of 2.36 x 104 m3
  contains 4.68 x 106g of He at 18ºC and 1.20bar.
  Express the volume of the balloon in liters, the
  amount in moles (n), the temperature (T) in K and
  the pressure (P) in both atmospheres and
  millimeters of mercury.
• (a) V 2.36 x 104m3 x 1L/10-3m3 2.36 x 107L
• (b) nHe 4.68 x 106g He x 1mole He/4.003g He
  1.17 x 106mol He

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Example (Contd)

• (c) T 18 273 291K
• (d) P 1.20bar x 1atm/1.013bar 1.18atm
• (e) P 1.20bar x 760mmHg/1.013bar 9.00 x 102
  mmHg

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Ideal Gas Law

• All gases are similar in their dependence of the
  volume on amount, temperature, and pressure.
• I. Volume is directly proportional to amount.
• V k1n
• Where k1 is a constant, independent of the values
  V or n.

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Ideal Gas Law (Contd)

• II. Volume is directly proportional to absolute
  temperature.
• V k2T
• Where k2 is a constant (Charles Law).
• III. Volume is inversely proportional to
  pressure.
• V k3/P
• Like k1 and k2, k3 is also a constant (Boyles
  Law).

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Ideal Gas Law

• Ideal Gas law as states is
• V constant x (n x T)/P
• The constants k1k2k3 are evaluated using
  Avogadros Law (Equal volumes of gases at the
  same T and P, contain the same number of moles).
  The new value is the constant R.
• PV nRT

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The Ideal Gas Constant

• The ideal gas constant, R, can be calculated from
  experimental values.
• Consider a gas at 0ºC and 1 atm (standard
  temperature and pressure, STP). At STP, 1 mole
  of a gas occupies a volume of 22.4L.
• Solving for R, we get 0.0821 L atm/(mol K).
• See Table 5.1 for R expressed in different units.

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Gas Law Calculations

• Can solve for
• (i) the final state of a gas, knowing its
  initial state and the changes in P, V, n, or T
  that occur.
• (ii) one of the four variables, P, V, n, or T,
  given the values of the other three.
• (iii) the molar mass or density of a gas.

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Final and Initial State Problems

• Determine the effect on the V, P, n, or T of a
  change on one or more of these variables.
• This involves a gas changing from an initial
  state to a final state.
• Ex. A sample of gas at 25ºC and 1 atm is heated
  to 95ºC. What is the new P?

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Example (Contd)

• Initial state P1V nRT1
• Final State P2V nRT2
• Dividing, the second equation by the first,
  cancels V, n, and R. Therefore
• P2/P1 T2/T1 (constant n, V)
• Solve for P2 P2 T2/T1 x P1
• P2 1atm x 368K/298K 1.23atm

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Calculation of P, V, n or T

• The ozone-friendly compound now used as a
  refrigerant in car air conditioners has the
  molecular formula C2F4H2. If 2.50g of this
  compound is introduced into an evacuated 500.0ml
  container at 10ºC, what pressure in atms is
  developed?

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Problem (Contd)

• First, convert V, n, and T into the proper units.
• V 500.0ml x 1L/1000ml 0.5000ml
• n 2.50g C2F4H2 x 1mole C2F4H2/102.04 C2F4H2
  0.0245mol
• P nRT/V
• 0.0245mol x 0.0821 x 283K/0.5000L
• 1.14atm

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Molar Mass and Density

• Ideal gas law can be used to determine
  experimentally the molar mass of a gas.
• Ex. Acetone is widely used as a solvent. A
  sample of acetone is placed in a 3.00L flask and
  vaporized by heating to 95ºC at 1.02atm. The
  vapor filling the flask at this temp. and
  pressure weighed 5.87g. Calculate the molar mass
  of acetone.

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Example (Contd)

• n PV/RT
• (1.02atm)(3.00L)/(0.0821)(368K)
• 0.101mol
• m M x n
• M m/n 5.87g/0.101mol 58.1g

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Density

• Density MP/RT
• Density of a gas is dependent on
• Pressure. Compressing a gas increases its
  density.
• Temperature. As temperature increases, density
  decreases.
• Molar Mass.

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Stoichiometry of Gaseous Reactions

• Law of Combining Volumes (Gay-Lussac) The volume
  ratio of any two gases in a reaction at constant
  temperature and pressure is the same as the
  reacting mole ratio.

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Example

• Octane, C8H18, is one of the hydrocarbons added
  to gasoline. On combustion, octane produces CO2
  and H2O. How many liters of oxygen, measured at
  0.974atm and 24ºC, are required to burn 1.00g of
  octane.

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Example (Contd)

• 2C8H18(l) 25O2(g) ? 16CO2(g) 18H2O(l)
• noxygen 1.00g C8H18 x 1mol C8H18/114.22g C8H18
  x 25mol O2/2mol C8H18 0.109mol O2
• Voxygen nRT/P
• 0.109mol x 0.0821 x 297K/0.974atm
• 2.73L

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Gas Mixtures Partial Pressures and Mole Fraction

• For a mixture of 2 gases A and B, the total
  pressure is given by the expression
• Ptot ntot(RT/V) (nA nB)RT/V
• Separating the terms on the right
• Ptot nA RT/V nB RT/V
• These two terms are, according to the ideal gas
  law, the pressures that A and B would exert if
  they were alone. These quantities are referred
  to as partial pressures.

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Partial Pressures (Contd)

• PA partial pressure A nART/V
• PB partial pressure B nBRT/V
• Ptot PA PB
• This relation was first derived by John Dalton in
  1801 and is referred to as Daltons Law of
  Partial Pressures.

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Partial Pressures (Contd)

• The total pressure of a gas mixture is the sum of
  the partial pressures of the components of the
  mixture.
• Consider a gaseous mixture of hydrogen and
  helium
• Phydrogen 2.46atm Phelium 3.69atm
• Ptot 2.46atm 3.69atm 6.15 atm

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Wet Gases Partial Pressures of Water

• When a gas such as hydrogen is collected by
  bubbling through water, it picks up water vapor.
  Therefore
• Ptot Pwater Phydrogen
• The partial pressure of water is equal to the
  vapor pressure of liquid water. It has a fixed
  value at a given temperature. The partial
  pressure of hydrogen can then be calculated by
  subtraction.

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Vapor Pressure (Contd)

• Vapor pressure, like density and solubility, is
  an intensive physical property that is
  characteristic of a particular substance.
• Water at 25ºC is 23.76 mm Hg
• 40ºC is 55.3 mm Hg
• 70ºC is 233.7 mm Hg
• 100ºC is 760.0 mm Hg

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Partial Pressure and Mole Fraction

• A mixture of containing gases A and B
• PA nART/V Ptot ntotRT/V
• PA/Ptot nA/ntot
• The fraction nA/ntot is referred to as the mole
  fraction of A in the mixture. It is the fraction
  of the total number of moles that is accounted
  for gas A. Using XA to represent the mole
  fraction,
• PA XAPtot
• The partial pressure of a gas in a mixture is
  equal to its mole fraction multiplied by the
  total pressure.

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Kinetic Theory of Gases

• Molecular Model
• I. Gases are mostly empty space total volume is
  negligible compared with that of the container.
• II. Gas molecules are in constant, chaotic
  motion collide frequently velocities constantly
  changing.
• III. Collisions are elastic no attractive
  forces to make molecules stick together.
• IV. Gas pressure is caused by collisions of
  millions of molecules with the walls of the
  container.

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Expression for Pressure, P

• According to laws of physics
• P Nmu2/3V
• Where V volume, N is the number of molecules, m
  is the mass, and u is the average speed.

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Average Kinetic Energy of Translational Motion
(Et)

• The kinetic energy, Et, of a gas molecule m
  moving at speed u is
• Et mu2/2
• From the previous equation for P, we see that mu2
  3PV/N, therefore
• Et 3PV/2N
• The ideal gas law states that PV nRT,
  therefore
• Et 3nRT/2N

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Kinetic Theory (Contd)

• At a given temperature, molecules of different
  gases must have the same average kinetic energy
  of translational motion.
• The average translational kinetic energy of a gas
  molecule is directly proportional to the Kelvin
  temperature.

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Real Gases

• The deviations from the ideal gas law become
  larger at high pressures and low temperatures.
• In general, the closer a gas is to the liquid
  state, the more it will deviate from the ideal
  gas law.
• Deviations from the ideal gas law occur because
  it neglects two forces
• 1. Attractive forces between gas particles.
• 2. The finite volume of gas particles.