Chapter 11: Properties of Gases

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Chapter 11 Properties of Gases

• Gases have a number of properties that are very
  different from liquids and solids
• Gases are compressible
• Gases exert a pressure
• Gas pressure depends on the amount of confined
  gas
• Gases fill their container
• Gases mix freely with each other
• Gas pressure increases with temperature

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The qualitative observations of the properties of
gases leads to the conclusion that a gas is
comprised of widely spaced molecules in rapid
motion. Collisions of molecules with the walls
are responsible for the gas pressure.

• This simple model of gases is the basis of the
  kinetic-molecular theory (discussed in Section
  7.2)

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• Recall the pressure is a force per unit area
• The earth exerts a gravitational force on
  everything with mass near it
• What we call weight is the gravitational force
  acting on an object
• The pressure due to air molecules colliding with
  an object is called the atmospheric pressure

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Atmospheric pressure is measured with a
barometer. A Torricelli barometer consists of a
glass tube sealed at one end, about 80 cm in
length. The tube is filled with mercury, capped,
inverted, and the capped end immersed in a pool
of mercury. When the cap is removed the
atmosphere supports a the column of mercury about
760 mm high.
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• The height of the mercury column varies with
  altitude
• The average pressure at sea level or the standard
  atmosphere (atm) was defined as the pressure
  needed to support a column of mercury 760 mm high
  measures at 0oC
• The SI unit of pressure is the pascal (Pa)

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• You may encounter a number of pressure units
• The standard atmosphere is
• Chemical reactions often involve gases

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• A manometer is used to measure the pressure
  inside closed containers

Open-end manometer. (a) The pressure of the
trapped gas, Pgas equals the atmospheric
pressure, Patm. Trapped gas pressure (b) higher
and (c) lower than atmospheric pressure.
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A closed-end manometer for measuring gas
pressures less than 1 atm. When constructed (a)
the tube is fully evacuated and mercury is
allowed to enter and fill the closed arm
completely. (a) Mercury flows out of the closed
arm when the bulb contains gas at low pressure.
The difference in mercury levels, PHg, is the
pressure of the confined gas, Pgas.
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• Mercury is so dense (13.6 g mL-1) that small
  pressure changes are difficult to measure
• Other liquids can be used to make manometers

Columns of mercury and water that exert the same
pressure. Mercury is 13.6 times more dense than
water. Both columns have the same weight and
diameter, so they exert the same pressure.
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• Thus for a given difference in pressure, the
  difference in heights between the two levels is
  inversely proportional to the density of the
  liquid used in the manometer
• There are four variables that affect the
  properties of a gas pressure, volume,
  temperature, and the amount of the gas
• Simple experiments can be conducted that relate
  how these variables change
• The gas laws summarize these experiments

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Compressing a gas increases its pressure. A
molecular view of what happens when a gas is
squeezed into a smaller volume. The number of
collisions with a given area of the walls
increases which causes the pressure to rise.
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Robert Boyle studied how the volume of a fixed
amount of gas varies with pressure at constant
temperature. (a) Air trapped in a J-tube by
mercury. (b) As more mercury is added, the
pressure of the trapped gas increases and the
volume decreases.
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(a) A typical graph of volume versus pressure
showing volume decreasing as pressure increases.
(b) A straight line is obtained when volume is
plotted against (1/P), which shows that
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• The relationship between volume and pressure is
  called Boyles law or the pressure-volume law
• The volume of a given amount of gas held at
  constant temperature varies inversely with the
  applied pressure
• The proportionality can be removed by introducing
  a proportionally constant, C

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• Boyles law is remarkably successful, especially
  for common laboratory conditions
• However, no real gas obeys Boyles law exactly
  over a wide range of temperatures and pressures
• The hypothetical gas that does exactly obey
  Boyles law is called an ideal gas
• Real gases act more like ideal gases as their
  pressures decrease and temperatures increase

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• Jacques Alexander Charles studied how the volume
  of a gas sample varied with temperature

Charles law plots. Each line shows how the gas
volume changes with temperature for different
sized samples of the same gas.
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• When extrapolated to zero volume all the samples
  have the same temperature
• This temperature is called absolute zero and is
  the basis of the Kelvin temperature scale
• Charles law or the temperature-volume law can be
  expressed mathematically

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• Joseph Louis Gay-Lussac studied how the pressure
  and temperature of a fixed amount of gas at
  constant volume are related
• Gay-Lussacs law or the pressure-temperature law
  states
• The pressure of a fixed amount of gas held at
  constant volume is directly proportional to the
  Kelvin temperature
• Mathematically this is

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• The three gas laws are often used in a single
  equation called the combined gas law
• When using this equation the temperature must
  always be in kelvins
• Alternate forms of the previous gas laws result
  when certain variables cancel

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• Problems involving the gas laws are important

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• Example What will be the the final pressure of a
  sample of oxygen with a volume of 850 m3 at 655
  torr and 25.0oC if it is heated to 80.0oC and
  given a final volume of 1066 m3?
• ANALYSIS Use the combined gas law with
  temperature in kelvins.
• SOLUTION

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• The law of combining volume states
• When gases react at the same temperature and
  pressure, their combining volumes are in ratios
  of simple, whole numbers
• Example
• Amedeo Avogadro studied this and devised
  Avogadros principle
• When measured at the same temperature and
  pressure, equals volumes of gases contain equal
  number of moles

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• A corollary to Avogadros principle is
• The volume of a gas is directly proportional to
  its number of moles, n
• Thus, the volume of one mole of any gas at
  standard temperature and pressure (STP) or 0oC
  and 1 atm is 22.4 L (a constant for all ideal
  gases)
• This is called the standard molar volume of a gas

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• The combined gas law can be generalized to
  include changes in the number of moles of sample
• The ideal gas law is

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• The molecular mass is obtained by taking the
  ratio of mass to moles, which could be determined
  using the ideal gas law
• Gas densities (d), a ratio of gas mass to volume,
  can be calculated by taking the ratio of the
  molar mass to molar volume
• Example The molar mass of oxygen is 32.0 g/mol.
  What is the density of oxygen at STP?

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One mole of each gas occupies 22.4 at STP. Carbon
dioxide is more dense that oxygen due to molar
mass differences.

• We now need to consider mixtures of gases
• One useful way to describe a composition of a
  mixture is in terms of its mole fractions
• The mole fraction is the ratio of the number of
  moles of a given component to the total moles of
  all components

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• For a mixture of A, B, substances, the mole
  fraction of substance i (Xi) is
• This provides a convenient way to partition the
  total pressure of a mixture of gases
• Daltons law of partial pressures states the
  total pressure of a mixture of gases is the sum
  of their individual partial pressures

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• For a system of only gases, mole fractions and
  partial pressure partition the total pressure in
  the same fashion
• Gases are often collected over water in the
  laboratory
• These (collected) gases are saturated with water

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• The space above any liquid contains some of the
  liquids vapor
• The pressure this vapor exerts is called the
  vapor pressure

As the gas bubbles through the water, water vapor
gets into the gas so the total pressure inside
the bottle includes the partial pressure of the
water vapor.
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• The total pressure is the pressure of the gas
  plus the vapor pressure of water

Vapor pressure of water at various temperatures.
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• Example A sample of oxygen is collected over
  water at 20oC and a pressure of 738 torr. What is
  the partial pressure of oxygen?
• ANALYSIS The partial pressure of oxygen is less
  than the total pressure. Get the vapor pressure
  of water from table 11.2 (page 478).
• SOLUTION
• Partial pressures can be used to calculate mole
  fractions

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• This is possible because the number of moles of
  each gas is directly proportional to its partial
  pressure
• Using the ideal gas equation for each gas
• For a given mixture of gases, the volume and
  temperature is the same for all gases
• Using CV/RT gives

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• The partial pressure of a gas can be calculated
  using the total pressure and mole fraction

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• Gas volumes can be used in stoichiometry problems

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• Diffusion is the spontaneous intermingling of the
  molecules of one gas with another
• Effusion is the movement of gas molecules through
  a tiny hole into a vacuum
• The rates of both diffusion and effusion depend
  on the speed of the gas molecules
• The faster the molecules, the faster diffusion
  and effusion occur
• Thomas Graham studied effusion

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• He found that the effusion rate of a gas was
  inversely proportional to the square root of the
  density (d)
• This is known as Grahams law
• Where Mi is the molar mass of species i

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• The behavior of ideals gases can be explained

• Diffusion (b) Effusion

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• Postulates of the Kinetic Theory of Gases
• A gas consists of a large number of tiny
  particles that are in constant, random motion.
• The gas particles themselves occupy a net volume
  so small in relation to the volume of their
  container that their contribution to the total
  volume can be ignored.
• The collisions between particle and with the
  walls of the container are perfectly elastic.

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• Calculations show that PV is proportional to the
  average kinetic energy and the Kelvin
  temperature, thus
• The kinetic theory also explains the gas laws

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The kinetic theory and the pressure volume law
(Boyles law). When the gas volume is made
smaller going from (a) to (b), the frequency of
collisions per unit area of the containers wall
increases. Thus the pressure increases.
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The kinetic theory and the pressure-temperature
law (Gay-Lussacs law). The pressure increases
from (a) to (b) as measured by the amount of
mercury that must be added to maintain a constant
volume.
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The kinetic theory and the temperature-volume law
(Charles law). The pressure is the same in both
(a) and (b). At higher temperatures the volume
increases because the gas molecules have higher
velocities.
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• Grahams law requires that the rate of effusion
  for the different gases be compared at the same
  temperature and pressure
• When different gases have the same temperature,
  they have the same average kinetic energy
• The average kinetic energy can be expressed in
  terms of the average of the velocities squared or
  root mean square
• For the two gases labeled 1 and 2

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• Note that heavier gases move slower than lighter
  gases

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• The rate of effusion is proportional to the
  average molecular speed, thus
• The kinetic theory predicts that absolute zero is
  the temperature at which the average kinetic
  energy of an ideal gas is zero
• Real gases exhibit non-ideal behavior

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Deviation from the ideal gas law. A plot of PV/T
versus P for an ideal gas is a straight line. The
same plot for oxygen is not a straight line

• Deviations from ideal behavior occur because
• Gas molecules interact, and
• Gas molecules occupy a finite volume.

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• J. D. van der Waals corrected the ideal gas
  equation in a simple, but useful, way

(a) In an ideal gas, molecules would travel in
straight lines. (b) In a real gas, the paths
would curve due to the attractions between
molecules.
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• He did this by modifying the measured pressure
  and volume of a real gas so it fits the ideal gas
  equation
• The constants a and b are called the van der
  Waals constants

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• Table 11.3 (page 493) has a more complete set of
  van der Waals constants