Unit 8: Gas Laws

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Unit 8 Gas Laws
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Comparison of Solids, Liquids, Gases

• The density of gases is much less than that of
  solids or liquids.
• Gases are easily compressed and they completely
  fill any container in which they occupy
• Tells us that gas molecules are far apart
  interactions among them are weak
• Gases exert pressure on their surroundings in
  turn, pressure must be exerted to confine a gas
• Gases diffuse into one another (i.e. they are
  miscible, unless they react with one another)
• Mix completely e.g. air is a mixture of gases
• Conversely, different gases in a mixture do not
  separate on standing

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Pressure

• Pressure is force per unit area.
• lb/in2 commonly known as psi
• Atmospheric pressure is measured using a
  barometer.
• Definitions of standard pressure
• 1 atmosphere
• 760 mm Hg
• 76 cm Hg
• 760 torr
• 101.3 kPa

Mercury barometer the air pressure is measured
in terms of the height of the mercury column i.e.
the vertical distance between the surface of the
mercury in the open dish and that inside the
closed tube. The pressure exerted by the
atmosphere is the pressure exerted by the column
of mercury
Hg density 13.6 g/mL
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Boyles Law Inverse relationship of pressure
volume
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Boyles Law The Volume-Pressure Relationship
Graphical Representation of Boyles Law

• At constant temp and with equal mols, volume of
  a gas varies inversely with pressure-
• V a 1 ? V 1 k1 ? PV
  k1
• P P
• The product of volume (V) and pressure (P) is a
  constant
• This allows us to equate initial and final
  conditions to solve practical problems.

Fig. 12-4, p. 405
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Boyles Law The Volume-Pressure Relationship

• Used to calculate-
• Volume resulting from pressure change
• Pressure resulting from volume change
• 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 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
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Charles Law The Volume-Temperature Relationship

• 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

• Volume of a gas varies directly with temperature
  if the pressure and of mols of gas remain
  constant
• V a T ? V Tk2 ? V k2
• T
  
• Ratio of volume (V) and temperature (T) is a
  constant
• This allows us to equate initial and final
  conditions to solve practical problems

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Charles Law The Volume-Temperature Relationship

• Example 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

• Have seen that both temperature pressure affect
  the volume of a gas
• Often convenient to choose some standard
  temperature and pressure as a reference point for
  discussing gasses.
• Standard temperature and pressure is given the
  symbol STP.
• Standard P ? 1 atm or 101.3 kPa
• Standard T ? 273 K or 0 oC

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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 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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Avogadros Law

• Mathematically Avogadros law can be stated as
  At constant temperature pressure, the volume,
  V, occupies by a gas is directly proportional to
  the number of mols, n, of gas
• V a n or V kn or V
  k (constant temp pressure)
• 
  n
• For 2 samples of a gas at the same temperature
  pressure, the relation between volumes and number
  of moles can be represented as
• V1 V2
  (constant T , P)
• n1 n2

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Using Avogadros Law

• Example 4 If 5.50 mol of CO occupy 20.6 L, how
  many liters will 16.5 mol of CO occupy at the
  same temperature and pressure?
• What do we know?
• n1 5.50 mol n2 16.5 mol
• V1 20.6 L V2 ? L
• V2 V1n2 (20.6 L)(16.5 mol)
• n1 (5.50 mol)
• 61.8 L CO

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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

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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 0.0821 L . atm / mol . K
• It is a proportionality constant called the
  universal gas constant.

An ideal gas is one that exactly obeys these gas
laws. Many gases show slight deviations from
ideality, but at normal temperatures, pressures
the deviations are usually small enough to ignore
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Summary of Gas Laws The Ideal Gas Law

• Example 5 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 Laws The Ideal Gas Law
PV nRT ?
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Daltons Law of Partial Pressures
An illustration of Daltons law. When the 2 gases
A and B are mixed in the same container at the
same temperature, they exert a total pressure
equal to the sum of their partial pressures.
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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 ....
• Where
• PA nA RT PB nBRT
  PC nCRT etc
• V
  V V
• The pressure that each gas exerts in a mixture is
  called its partial pressure. No way has been
  devised to measure the pressure of an individual
  gas in a mixture it must be calculated from
  other quantities.

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Daltons Law of Partial Pressures

• Example 6 If 100 mL of hydrogen, measured at
  25.0 oC and 3.00 atm pressure, and 100 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

• Example 7 A 10.0 L flask contains 0.200 mol of
  methane, 0.300 mol of hydrogen and 0.400 mol of
  nitrogen at 250C
• (a) What is the pressure (in atm) inside the
  flask?
• (b) What is the partial pressure of each
  component of the mixture of gases?

• (a) Solution
• Given mols of each component. Can use ideal gas
  equation to find total pressure from mols
• ntotal 0.200 0.300 0.400 0.900 mols
• V 10.0L T 25 273 298K
• Ptotal nRT (0.900mol)(0.0821L.atm / mol.K)
  (298K) 2.20atm
• V
  10.0L

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Daltons Law of Partial Pressures

• (b) Solution
• Now we find partial pressures for each component
• PCH4 (nCH4)RT (0.200)(0.0821)(298)
  0.489 atm
• V 10.0
• Similar calculations for N2 and H2 give
• PH2 0.734 atm PN2 0.979 atm
• As a check, the sum of all the partial pressures
  should be equal to the total pressure (Daltons
  Law)
• 0.489 0.743 0.979 2.20 atm

• Problem solving tip Sometimes the amount of gas
  is expressed in other units that can be converted
  to mols. E.g. molar mass. Can then convert mass ?
  mols

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Mole Fraction and Partial Pressure

• Can describe the composition of any gas mixture
  in terms of the mole fraction of each component .
• The mole fraction, XA, of component A in a
  mixture is defined as

• Like any other fraction, mole fraction is
  dimensionless.
• Can relate the mole fraction of each component to
  its partial pressure

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Mole Fraction and Partial Pressure

• XA PA similarly, XB PB and so
  on
• Ptotal
  Ptotal

• We can rearrange these equations to give another
  statement of Daltons Law of Partial Pressures

PA XA Ptotal PB XB Ptotal and so
on The partial pressure of each gas is equal to
the mole fraction in the gas mixture times the
toal pressure of the mixture
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Mole Fraction and Partial Pressure

• Example 14 Find the mole fractions of the gases
  in example 13.
• A 10.0 L flask contains 0.200 mol of methane,
  0.300 mol of hydrogen and 0.400 mol of nitrogen
  at 250C
• Solution, using the mols given
• X methane nmethane / ntotal 0.200 /0.900
  0.222
• X hydrogen nhydrogen / ntotal 0.300/ 0.900
  0.333
• X nitrogen nnitrogen / ntotal 0.400/
  0.900 0.444
• Alternatively, could use partial total pressure
  in Example 13 (b)
• X methane Pmethane / P total 0.489 atm
  / 2.20 atm 0.222
• X hydrogen Phydrogen / P total 0.734 atm
  / 2.20atm 0.334
• X nitrogen Pnitrogen / P total 0.979
  atm / 2.20 atm 0.445

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The Kinetic-Molecular Theory

• The basic assumptions of the kinetic-molecular
  theory for an ideal gas (one that obeys the gas
  laws)-
• Gases consist of discrete molecules which are
  small and very far apart relative to their own
  size, and occupy no volume (they can be
  considered as points)
• The observation that gases can be easily
  compressed supports this.

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The Kinetic-Molecular Theory

• The gas molecules are in continuous random,
  straight-line motion with varying speeds.
• The collisions between gas molecules and with the
  walls of the container are elastic, (that is, no
  energy is gained or lost during the collision).
• At any given instant only a small fraction of the
  gas molecules are involved in collisions.

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The Kinetic-Molecular Theory

• There are no attractive or repulsive forces
  between molecules.
• NOTE at HIGH pressures and LOW temperatures
  (conditions under which a gas liquefies
  attractions and repulsions between gas molecules
  become significant and the gas behaves
  non-ideally.
• A real gas is one that does not behave as an
  ideal gas due to interactions between gas
  molecules.