Gases

1
Gases

• Chapter 5

2
Gas Properties

• Four properties determine the physical behavior
  of any gas
• Amount of gas
• Gas pressure
• Gas volume
• Gas temperature

3
Gas pressure

• Gas molecules exert a force on the walls of their
  container when they collide with it

4
Gas pressure

• Gas pressure can support a column of liquid
• Pliquid ghd
• g acceleration due to the force of gravity
  (constant)
• h height of the liquid column
• d density of the liquid

5
Atmospheric pressure

• Torricelli barometer
• In the closed tube, the liquid falls until the
  pressure exerted by the column of liquid just
  balances the pressure exerted by the atmosphere.
• Patmosphere Pliquid ghd
• Patmosphere ? liquid height

Standard atmospheric pressure (1 atm) is 760 mm
Hg
6
Units for pressure

• In this course we usually convert to atm

7
Gas pressure

• Pliquid ghd
• Pressure exerted by a column of liquid is
  proportional to the height of the column and the
  density of the liquid
• Container shape and volume do not affect pressure

8
Example

• A barometer filled with perchloroethylene (d
  1.62 g/cm3) has a liquid height of 6.38 m. What
  is this pressure in mm Hg (d 13.6 g/cm3)?
• P ghd g hpce dpce g hHg dHg
• hpce dpce hHg dHg
• hHg hpce d pce (6.38 m)(1.62 g/cm3)
  0.760 m dHg 13.6 g/cm3
• hHg 760 mm Hg

9
Gas pressure

• A manometer compares the pressure of a gas in a
  container to the atmospheric pressure

10
Gas Laws Boyle

• In 1662, Robert Boyle discovered the first of the
  simple gas laws
• PV constant

For a fixed amount of gas at constant
temperature, gas pressure and gas volume are
inversely proportional
11
Examples 6-3A 6-3B

• A cylinder contains a gas at 5.25 atm pressure.
  When the gas is allowed to expand to a final
  volume of 12.5 L, the pressure drops to 1.85 atm.
  What was the original volume of the gas?
• 1.50 L of gas at 2.25 atm pressure expands to a
  final volume of 8.10 L. What is the final gas
  pressure in mm Hg?

12
Gas Laws Charles

• In 1787, Jacques Charles discovered a
  relationship between gas volume and gas
  temperature

relationship between volume and temperature
is always linear
all gases reach V 0 at same temperature,
273.15 C
volume (mL)
this temperature is ABSOLUTE ZERO
temperature (C)
13
A temperature scale for gasesthe Kelvin scale

• A new temperature scale was invented the Kelvin
  or absolute temperature scale
• K C 273.15
• Zero Kelvins absolute zero

14
Gas laws Charles

• Using the Kelvin scale, Charles results is
• For a fixed amount of gas at constant pressure,
  gas volume and gas temperature are directly
  proportional
• A similar relationship was found for pressure and
  temperature

15
Examples 6-4A 6-4B

• A gas at 25 C and 0.987 atm is heated under a
  piston. The volume expands from 0.250 L to 1.65
  L. What is the new temperature of the gas, if
  pressure has remained constant?
• If an aerosol can contains a gas at 1.82 atm at
  22 C, what will be the gas pressure in the can
  in an incinerator at 935 C?

16
Standard conditions for gases

• Certain conditions of pressure and temperature
  have been chosen as standard conditions for gases
• Standard temperature is 273.15 K (0 C)
• Standard pressure is exactly 1 atm (760 mm Hg)
• These conditions are referred to as STP
  (standard temperature and pressure)

17
Gas laws Avogadro

• In 1811, Avogadro proposed that equal volumes of
  gases at the same temperature and pressure
  contain equal numbers of particles.
• At constant temperature and pressure, gas volume
  is directly proportional to the number of moles
  of gas
• Standard molar volume at STP, one mole of gas
  occupies 22.4 L

18
Examples 6-6A 6-6B

• A small tank of propane is opened and releases
  30.0 L of gas at STP. What mass of propane was
  released?
• 128 g of dry ice sublimes into CO2 gas. What is
  the volume of this gas at STP?

19
Putting it all togetherIdeal Gas Equation

• Combining Boyles Law, Charles Law, and
  Avogadros Law give one equation that includes
  all four gas variables
• R is the ideal or universal gas constant
• R 0.08206 atm L/mol K

20
Using the Ideal Gas Equation

• Ideal gas equation may be expressed two ways
• One set of conditions ideal gas law
• Two sets of conditions general gas equation

21
Examples

• What is the volume occupied by 20.2 g NH3 gas at
  25 C and 752 mm Hg?
• How many moles of He gas are in a 5.00 L tank at
  10.5 atm pressure and 30.0 C?
• A 1.00 mL sample of N2 gas at 36.2 C and 2.14
  atm is heated to 37.8 C while the pressure is
  changed to 1.02 atm. What volume does the gas
  occupy at this temperature and pressure?

22
Ideal Gas Equation and molar mass

• Solving for molar mass (M)

23
Example

• A glass vessel weighs 40.1305 g when clean, dry,
  and evacuated. When filled with an unknown gas
  at 772 mm Hg and 22.4 C, the vessel weighs
  40.4868 g. What is the molar mass of the gas?
• 1.27 g of an oxide of nitrogen (believed to be
  either NO or N2O) occupies 1.07 L at 25 C and
  737 mm Hg. Which oxide is it?

24
Ideal Gas Equation and gas density
25
Gas density

• Gas density depends directly on pressure and
  inversely on temperature
• Gas density is directly proportional to molar mass

26
Examples

• What is the density of helium gas at 298 K and
  0.987 atm? Why can we say He is lighter than
  air?
• Hint what is the average molar mass of air,
  which is 78.08 N2, 20.95 O2, 0.93 Ar, and
  0.036 CO2?
• At what temperature will the density of O2 gas be
  1.00 g/L if the pressure is kept at 745 mm Hg?

27
Mixtures of Gases

• Ideal gas law applies to pure gases and to
  mixtures
• In a gas mixture, each gas occupies the entire
  container volume, at its own pressure
• The pressure contributed by a gas in a mixture is
  the partial pressure of that gas
• Ptotal PA PB (Daltons Law of Partial
  Pressures)

28
Mixtures of Gases

• When a gas is collected over water, it is always
  wet (mixed with water vapor).
• Ptotal Pbarometric Pgas Pwater vapor
• Example If 35.5 mL of H2 are collected over
  water at 26 C and a barometric pressure of 755
  mm Hg, what is the pressure of the H2 gas? The
  water vapor pressure at 26 C is 25.2 mm Hg.

29
Gas mixtures

• The mole fraction represents the contribution of
  each gas to the total number of moles.
• XA mole fraction of A

30
Gas mixtures

• The mole fraction represents the contribution of
  each gas to the total number of moles.
• XA mole fraction of A
• The pressure fraction is equal to the mole
  fraction

31
Gas mixtures

• The volume composition of a gas mixture is
• Avogadros hypothesis at constant T P, gas V
  is proportional to moles of gas
  
• The volume percent gives the mole fraction

32
Gas Mixtures

• For gas mixtures,

Each gas occupies the entire container. The
volume fraction describes the composition by
volume.
mole fraction equals pressure fraction equals volu
me fraction
33
Examples

• What is the total gas pressure in a mixture of
  1.0 g H2, 5.00 g He, and 12.5 g Ne, in a 5.0 L
  container at 55 C?
• A mixture of 0.197 mol CO2 and 0.00278 mol H2O
  gas are held at 30.0 C and 2.50 atm. What is
  the partial pressure of each gas in the mixture?
• Air is 78.08 N2, 20.95 O2, 0.93 Ar, and 0.036
  CO2 by volume. What is the partial pressure of
  each gas at a barometric pressure of 748 mm Hg?

34
Gases in Chemical Reactions

• To convert gas volume into moles for
  stoichiometry, use the ideal gas equation
• If both substances in the problem are gases, at
  the same T and P, gas volume ratios mole ratios.

35
Examples

• How many grams of NaN3 will produce 20.2 L N2 at
  30.0 C and 776 mm Hg? 2 NaN3 (s) ? 2 Na(l)
  3 N2 (g)
• What volume of O2 is consumed per liter of NO
  formed, at constant temperature and pressure
  4 NH3 (g) 5 O2 (g) ? 4 NO (g) 6 H2O (g)

36
A Model for Gas Behavior

• Gas laws describe what gases do, but not why.
• Kinetic Molecular Theory of Gases (KMT) is the
  model that explains gas behavior.
• developed by Maxwell Boltzmann in the mid-1800s
• based on the concept of an ideal or perfect gas

37
Ideal gas

• Composed of tiny particles in constant, random,
  straight-line motion
• Gas molecules are point masses, so gas volume is
  just the empty space between the molecules
• Molecules collide with each other and with the
  walls of their container
• The molecules are completely independent of each
  other, with no attractive or repulsive forces
  between them.
• Individual molecules may gain or lose energy
  during collisions, but the total energy of the
  gas sample depends only on the absolute
  temperature.

38
Molecular collisions and pressure

• Force of molecular collisions depends on
• collision frequency
• molecule kinetic energy, ek
• ek depends on molecule mass m and molecule speed
  u
• molecules move at various speeds in all directions

39
Molecular speed

• Molecules move at various speeds
• Imagine 3 cars going 40 mph, 50 mph, and 60 mph
• Mean speed u (40 50 60) 3 50
  mph
• Mean square speed (average of speeds squared) u2
  (402 502 602) 3 2567 m2/hr2
• Root mean square speed urms v2567 m2/hr2
  50.7 mph

40
Distribution of molecule speeds
41
The basic equation of KMT

• Combining collision frequency, molecule kinetic
  energy, and the distribution of molecule speeds
  gives the basic equation of KMT
• P gas pressure and V gas volume
• N number of molecules
• m molecule mass
• u2 mean square molecule speed (average of
  speeds squared)

42
Combine the Equations ofKMT and Ideal Gas
If n 1, N NA and PV RT
?Avogadros number
43
Combine the Equations ofKMT and Ideal Gas
NA x m (Avogadros number x mass of one
molecule) mass of one mole of molecules (molar
mass M)
44
Combine the Equations ofKMT and Ideal Gas
We can calculate the root mean square speed from
temperature and molar mass
45
Calculating root mean square speed

• To calculate root mean square speed from
  temperature and molar mass
• Units must agree!
• Speed is in m/s, so
• R must be 8.3145 J/mol K
• M must be in kg per mole, because Joule kg m2 /
  s2
• Speed is inversely related to molar mass light
  molecules are faster, heavy molecules are slower

46
Example 6-17A

• Which has the greater root mean square speed at
  25 C, NH3 gas or HCl gas? Calculate urms for
  the one with greater speed.

47
Interpreting temperature

• Combine the KMT and ideal gas equations again

Again assume n1, so N NA and PV RT
48
Interpreting temperature

• Absolute (Kelvin) temperature is directly
  proportional to average molecular kinetic energy
• At T 0, ek 0

49
Diffusion and Effusion

• Diffusion (a) is migration or mixing due to
  random molecular motion
• Effusion (b) is escape of gas molecules through a
  tiny hole

50
Rates of diffusion/effusion

• The rate of diffusion or effusion is directly
  proportional to molecular speed
• The rates of diffusion/effusion of two different
  gases are inversely proportional to the square
  roots of their molar masses (Grahams Law)

51
Using Grahams Law

• Grahams Law applies to relative rates, speeds,
  amounts of gas effused in a given time, or
  distances traveled in a given time.

52
Using Grahams Law with times

• Grahams law can be confusing when applied to
  times

rate amount of gas (n) time (t)
53
Use common sensewith Grahams Law

• When you compare two gases, the lighter gas
• escapes at a greater rate
• has a greater root mean square speed
• can effuse a larger amount in a given time
• can travel farther in a given time
• needs less time for a given amount to escape or
  travel
• Make sure your answer reflects this reality!

54
Examples

• 2.2 x 104 mol N2 effuses through a tiny hole in
  105 seconds. How much O2 effuses through the
  same hole in the same time?
• How long would it take for 2.2 x 104 mol H2 to
  effuse through the same hole as the 2.2 x 104
  mol N2, which effused in 105 seconds?

55
Examples

• A sample of Kr gas escapes through a tiny hole in
  87.3 sec. Under the same conditions, the same
  amount of unknown gas effuses in 131.3 sec. What
  is the molar mass of the unknown gas?
• How long would it take for the same amount of
  ethane gas to effuse, under the same conditions
  as the Kr gas in problem A?

56
Reality Check

• Ideal gas molecules Real gas molecules
• constant, random, same straight-line motion
• point masses are NOT points molecules
  have volume Vreal gas gt Videal gas
• independent of each other are NOT independent
  molecules are attracted to each other,
  so Preal gas lt Pideal gas
• gain / lose energy during same (some energy may
  be collisions, but total energy absorbed in
  molecular depends only on T (?ek) ???????????????
  ???)

57
Real gas corrections

• For a real gas,
• a corrects for attractions between gas molecules,
  which tend to decrease the force and/or frequency
  of collisions (so Preal lt Pideal)
• b corrects for the actual volume of each gas
  molecule, which increases the amount of space the
  gas occupies (so Vreal gt Videal)
• The values of a and b depend on the type of gas

58
An equation for real gasesthe van der Waals
equation
Add correction to Preal to make it equal to
Pideal, because intermolecular attractions
decrease real pressure
Subtract correction to Vreal to make it equal to
Videal, because molecular volume increases real
volume
59
When do I needthe van der Waals equation?

• Deviations from ideality become significant when
• molecules are close together (high pressure)
• molecules are slow (low temperature)
• At low pressure and high temperature, real gases
  tend to behave ideally
• At high pressure and low temperature, real gases
  do not tend to behave ideally

non-ideal conditions
60
Example 6-20A

• Calculate the pressure exerted by 1.00 mol CO2
  when confined to a volume of 2.00 L at 273 K.
  aCO2 3.59 L2atm/mol2 and bCO2 0.0427 L/mol.
  Which shows a greater departure from ideality,
  CO2 or Cl2 (aCl2 6.49 and bCl2 0.0562, same
  units)?

61
Examples 6-18 6-19A

• 2.2 x 104 mole of N2 gas effuses through a
  pinhole in 105 s. How much O2 effuses through
  the same hole in that amount of time?
• How long would it take for 2.2 x 104 mol H2 to
  effuse through the same hole?
• A sample of Kr gas escapes through a pinhole in
  87.3 s. Gas X requires 131.3 s for the same
  amount to escape. What is the molar mass of X?

62
Exercise 28

• A gas is collected over water when the barometric
  pressure is 756.2 mm Hg, but the water level
  inside the container of gas is 4.5 cm higher than
  outside. What is the total pressure of the gas
  inside the container, in mm Hg?

63
Exercise 29

• A 35.8-L cylinder of Ar gas is connected to an
  evacuated 1875-L tank. If the temperature is
  held constant and the final pressure is 721 mm
  Hg, what must have been the original gas pressure
  in the cylinder, in atm?

64
Exercise 31

• A fixed amount of gas held at a constant volume
  of 275 mL exerts a pressure of 798 mm Hg at 23.4
  C. At what temperature in C will the pressure
  of the gas become exactly 1.00 atm?

65
Exercise 33

• A 27.6 mL sample of PH3 gas is obtained at STP.
• a. What is the mass of this gas, in mg?
• b. How many molecules of PH3 are present?

66
Exercise 37

• A 12.8 L cylinder contains 35.8 g O2 at 46 C.
  What is the pressure of this gas, in atm?

67
Exercise 42

• A 10.0 g-sample of a gas has a volume of 5.25 L
  at 25 C and 762 mm Hg. If to this constant
  volume is added 2.5 g of the same gas, and the
  temperature raised to 62 C, what is the new gas
  pressure?

68
Exercise 44

• A 2.650 g sample of a gaseous compound occupies
  428 mL at 24.3 C and 742 mm Hg. The compound
  consists of 15.5 C, 23.0 Cl , and 61.5 F, by
  mass. What is its molecular formula?

69
Exercise 49

• In order for a gas-filled balloon to rise in air,
  the density of the gas in the balloon must be
  less than that of air.
• a. Consider air to have a molar mass of 28.96
  g/mol. Determine the density of air at 25 C and
  1 atm, in g/L.
• b. Show by calculation that a balloon filled
  with carbon dioxide at 25 C and 1 atm would not
  be expected to rise in air at 25 C.

70
Exercise 56

• A 3.57-g sample of a KCl-KClO3 mixture is
  decomposed by heating and produces 119 mL O2
  measured at 22.4 C and 738 mm Hg. What is the
  mass percent of KClO3 in the mixture? KClO3 (s)
  ? KCl (s) O2 (g)
• Hint the KCl is unchanged

71
Exercise 61

• A mixture of 4.0 g H2 and 10.0 g He gases in a
  4.3 L flask is maintained at 0 C.
• a. What is the total pressure in the container?
• b. What is the partial pressure of each gas?

72
Exercise 64

• A typical producer gas has this composition by
  volume 8.0 CO2, 23.2 CO, 17.7 H2, 1.1 CH4,
  and 50.5 N2.
• a. What is the density of this gas mixture at 23
  C and 763 mm Hg, in g/L?
• b. What is the partial pressure of CO in this
  mixture at STP?

73
Exercise 68

• A 1.072-g sample of He gas is found to occupy a
  volume of 8.446 L when collected over hexane at
  25.0 C and 738.6 mm Hg barometric pressure.
  Determine the vapor pressure of hexane at 25 C
  from these data.

74
Exercise 75

• If 0.00484 mol N2O gas effuses through an orifice
  in a certain period of time, how much NO2 gas
  would effuse in the same time under the same
  conditions?

75
Exercise 76

• A sample of N2 gas effuses through a tiny hole in
  38 s. What must be the molar mass of a gas that
  requires 64 s to effuse under identical
  conditions?

76
Exercise 79

• Use the van der Waals equation to calculate the
  pressure exerted by 1.50 mol SO2 gas when it is
  confined to a volume of 5.00 L at 298 K. For
  SO2, a 6.71 L2 atm/mol2 and b 0.0564 L/mol.
  Compare this real pressure to what you would
  predict assuming SO2 to be an ideal gas.

77
Exercise 87

• A 3.05-g sample of solid NH4NO3 is placed in an
  evacuated 2.18-L flask and heated to 250 C.
  What is the total gas pressure in atm in the
  flask at 250 C after the solid has completely
  decomposed NH4NO3 (s) ? N2O (g) 2 H2O (g)

78
Exercise 92

• What is the apparent molar mass of air, given
  that it is 78.08 N2, 20.95 O2, 0.93 Ar, and
  0.036 CO2 by volume?