Gases

1
Gases

• Gas Density and Molar Mass
• Density is mass per unit volume. Rearranging the
  ideal gas equation,

2
Gases

• Gas Density and Molar Mass
• Example In the Dumas-bulb technique for
  determining the molar mass of an unknown liquid,
  you vaporize the sample of a liquid that boils
  below 100 oC in a boiling-water bath and
  determine the mass of vapor required to fill the
  bulb. From the following data, calculate the
  molar mass of the unknown liquid mass of
  unknown vapor, 1.012 g volume of bulb, 354 cm3
  pressure, 742 torr temperature, 99 oC.

3
Gases

• Stoichiometry of Reactions Involving Gases
• Avogadros Hypothesis states that equal volumes
  of gases at the same temperature and pressure
  contain the same number of molecules.
• Application to stoichiometry the coefficients in
  balanced equations can represent volumes of
  gaseous substances as well as moles or molecules.
• Example N2(g) 3H2(g) 2NH3(g)
• At a given temperature and pressure, 1 L of N2(g)
  reacts with 3 L of H2(g) to produce 2 L NH3(g).
• Stoichiometric volume equivalences or ratios can
  be written
• 1 L N2(g) Û 3 L H2(g)
• 3 L H2(g) Û 2 L NH3(g) etc.

4
Gases

• Example What volume of NH3(g) is produced at
  1.00 atm and 0.00 oC from 10.0 L N2(g) at 150
  oC and 800 torr and 18.0 L H2(g) at 200 oC and
  350 torr?
• One way to solve this problem is to convert the V
  of N2(g) and H2(g) to a common T and P well
  use the T and P at which the product NH3(g) is
  formed.

Another way to solve this problem would be to
convert the volumes to moles, calculate the of
moles NH3 produced and convert to L NH3 at 1.00
atm and 0.00 oC.
5
Gases

• Example Calculate the volume of gas produced in
  a chemical reaction
• What volume of H2 at 25 oC and 610 torr is
  produced from 3.00 g Zn?
• Zn 2HCl ZnCl2(aq) H2(g)

6
Gases

• Gas Mixtures and Partial Pressures
• The partial pressure of a gas is the pressure
  that would be exerted by a single gas in a
  mixture of gases in the absence of the other
  gases.
• Daltons Law of Partial Pressures states that the
  total pressure exerted by a mixture of gases is
  the sum of the partial pressures of each gas in
  the mixture.
• PtotalP1 P2 P3
• This equation comes about because each ideal gas
  in a mixture behaves independently - each kind
  of molecule behaves like any of the other
  molecules.
• If each gas in the mixture behaves like an ideal
  gas

• In a mixture, all the gases are at the same T and
  are contained in the same V

7
Gases

• Gas Mixtures and Partial Pressures
• Example What is the total pressure exerted by a
  mixture of 2.00 g H2 and 8.00 g of N2 at 273 K
  in a 10.0 L container?

8
Gases

• Partial Pressures and Mole Fractions
• The ratio of the partial pressure of a gas to the
  total pressure of a mixture gives the ratio of
  the moles of a gas to the total moles of gas in
  the mixture.

• X1 is the mole fraction of gas 1 in the mixture
• The partial pressure of a gas is the mole
  fraction times the total pressure.
• Example Data from Voyager 7 give information
  about the composition of the atmosphere of
  Titan, Saturns largest moon. The total pressure
  is 1220 torr and the atmosphere consists of 82
  mol percent N2, 12 mol percent Ar and 6.0 mol
  percent CH4. What is the partial pressure of
  each gas?

9
Gases

• Collecting Gases Over Water a common laboratory
  method is to produce a gas by some chemical
  reaction and displacing water from a container.
• This allows the gas to be collected without
  mixing with air
• The volume of the water displaced allows
  determination of the volume of the gas produced
  in the reaction.
• However, the gas is saturated with water vapor
• If the gas is collected in such a way the
  pressure of the gas/water vapor mixture is the
  atmospheric pressure
• Patm Pgas Pwater
• Pgas Patm - Pwater
• Pwater is the vapor pressure of water at the
  temperature of the water and is tabulated in
  your text in Appendix G.

10
Gases

• Collecting Gases Over Water
• Example NH4NO2 can be decomposed to produce
  N2(g). A sample of NH4NO2 was decomposed and 511
  mL N2 collected over water at 26 oC and 745 atm.
  How many grams of NH4NO2 were decomposed?
• NH4NO2(s) N2(g) 2H2O(g)

11

• Gases
• Kinetic-Molecular Theory of Gases explains why
  gases follow the gas laws.
• Postulates of the theory
• Gases consist of a large number of molecules that
  are in constant, random motion.
• The volume of gas molecules is negligible
  compared to the total volume of their container.
• Attractive and repulsive forces between gas
  molecules are negligible
• Collisions between gas molecules are perfectly
  elastic energy can be exchanged between gas
  molecules by collision, but the average energy of
  the molecules in a gas is constant at constant
  temperature and does not change with time.
• The average kinetic energy of gas molecules is
  proportional to the absolute temperature and
  independent of the kind of gaseous molecules.
• Distribution of molecular speeds see Fig 12.18,
  p. 566.
• Not all the molecules have the same velocity -
  there is a distribution in the speeds of the
  molecules in a gas at constant temperature.
• If the temperature increases the distribution
  moves to higher velocities.
• The root-mean-square velocity is the velocity of
  a molecule having the average kinetic energy.

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13
Gases

• Kinetic Molecular Theory of Gases
• The rms speed is the square root of the sum of
  the squares of the speeds of the molecules in a
  sample of gas divided by the number of molecules
  in the sample.
• The average kinetic energy is calculated from the
  rms speed
• e mu2 (see Fig. 12.19, p. 567, for the
  effect of molecular mass on distribution of
  molecular velocity)
• m is constant for a given gas, so if energy is
  added to a gas, u must increase.
• Pressure is caused by molecular collisions with
  the walls of the container
• The faster the molecules move the greater the
  force of collision with the walls of a
  container.
• The faster the molecules move the more often the
  molecules will collide with the walls of a
  container.
• Temperature is a measure of the average kinetic
  energy of gas molecules.
• The higher the temperature the faster the
  molecules move

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15
Gases

• Kinetic Molecular Theory of Gases
• Explanation of Boyles Law - Pressure - Volume
  Relationship
• At constant temperature the gas molecules have a
  particular rms velocity.
• If the volume of the container increases, the
  distance molecules move before collision with
  the walls increases.
• The number of collisions per unit time decreases
  so the pressure drops if volume increases.
• Explanation of Gay-Lussacs Law - Temperature -
  Pressure Relationship
• At constant volume and temperature, the number of
  collisions per unit time with the walls of the
  container is constant so the pressure is
  constant.
• If the temperature increases at constant volume,
  the kinetic energy, and thus the rms velocity of
  the gas molecules increases.
• The number of collisions with the container walls
  per unit time increases and the force of the
  collisions increases so the pressure increases
  with increasing temperature.

16
Gases

• Kinetic Molecular Theory of Gases
• Explanation of Charless Law - Volume -
  Temperature Relationship
• Increasing the temperature of a gas at a given
  pressure increases the average kinetic energy of
  the gas molecules and thus their rms speed.
• This increases the number of collisions with the
  container walls per unit time and increases the
  force of the collisions.
• In order to maintain constant pressure the volume
  must increase to increase the distance
  molecules travel between collisions and reduce
  the number of collisions with the container walls
  per unit time.
• Thus, volume increases with temperature at
  constant pressure.
• Explanation of Avogadros Law Volume - Amount of
  Substance Law
• Increasing the number of molecules in a container
  at constant temperature will increase the number
  of collisions per unit time with the container
  walls.
• To maintain constant pressure, the volume of the
  container must increase in order to make the
  number of collisions with the wall constant.

17
Gases

• Kinetic Molecular Properties of Gases
• Because the average kinetic energy of gas
  molecules mu2
• two different gases with different m values at
  the same T, thus same e
• e1 m1u12 m2u22 e2
• since the molar mass is proportional to the
  molecular mass
• and it can be shown
• For a given temperature, the higher the molar
  mass, the smaller the rms velocity.
• See Fig. 12.9, p. 567, showing velocity
  distributions for molecules with different
  molar masses all at 25 oC.

18
Gases

• Kinetic Molecular Properties of Gases
• Grahams Law of Effusion Effusion is the passage
  of gas molecules through a small hole from a
  high pressure region into a vacuum.
• The higher the rms speed of molecules the higher
  the rate of effusion

• Example H2 effuses 2.9 times as fast as unknown
  gas at the same temperature. What is the molar
  mass of the unknown?

19
Gases

• Kinetic Molecular Properties of Gases
• Diffusion and Mean Free Path
• Diffusion is the process by which molecules move
  from an area of high concentration to an area
  of lower concentration.
• The rates of diffusion can be approximated by
  Grahams Law
• Diffusion is not instantaneous even though rms
  speeds are hundreds of meters/second because
  gas molecules under normal conditions undergo
  many collisions per second - perhaps 1010 per
  second at 1 atm and 25 oC.
• The average distance molecules travel between
  collisions under normal conditions is 10s of
  nm. This is the mean free path.
• Real Gases Deviations from Ideal Behavior
• should equal 1 for 1 mole of ideal
  gas at all pressures and temperatures.
• Generally deviations are significant only at
  pressures much greater than 1 atm.

20
Gases

• Real Gases Deviations from Ideal Behavior
• At high pressures, the gas molecules are close
  enough that their volume is not a small fraction
  of the volume occupied by the bulk sample.
• At high pressures, the gas molecules are close
  enough that attractive forces between gas
  molecules become important.
• The attraction between molecules at the walls of
  a container with nearby molecules reduces the
  force of impact of molecules with the wall and
  thus reduce the expected pressure.
• This reduces from its expected value of 1
• At very high pressures, the molecular volume
  increases above its expected value
  because gas molecules are not very compressible.
• At low temperatures, the gas molecules do not
  have as much kinetic energy as they have at high
  temperatures.
• Its harder for low temperature molecules to
  overcome attractive forces and
• is less than its expected value at low P.
  
• At high temperatures and pressures, is
  greater than its ideal value because the volume
  of the gas molecules becomes important.

21
Gases

• Real Gases Deviations from Ideal Behavior
• van der Waals Equation of State
• the ideal gas law
• Correct this equation for the effect of the
  molecular volume and the attraction between
  molecules

The volume in which the molecules are free to
move is reduced from the bulk volume by the
volume per mole of gas molecules (approximately
b) times n The pressure is reduced due to
attractions between molecules at high pressure.
For pairs of interacting molecules the effect
goes like the square of the molar density. The
value of a indicates how strongly molecules of
a gas are attracted to one another. Values for
the van der Waals constants a and b are given in
Table 12.3, p 572.
22

• Gases
• Real Gases Deviations from Ideal Behavior
• van der Waals Equation of State
• Example 1.000 mol of ideal gas at 22.41 L and
  0.00 oC would exert a pressure of 1.000 atm. Use
  the van der Waals equation to calculate the
  pressure of 1.000 mol of Cl2 gas at 22.41 L and
  0.00 oC.
• From Table 12.3, a6.49 L2-atm/mol2 and b0.0562
  L/mol

The 1.003 atm includes a correction to pressure
due to molecular volume. The 0.013 atm includes
a correction to pressure due to molecular
attraction. The molecular attraction between Cl2
molecules at 1 atm and 0.0 oC is the main
reason this gas deviates from ideality.