The Gaseous Phase

1
The Gaseous Phase

• The three phases of matter, solids, liquids and
  gases, have different characteristics.
  
• A gas expands to fill any container it occupies
• A liquid has a fixed volume but takes the shape
  of the volume of the container it occupies
  
• A solid has both fixed volume and shape.
• These characteristics originate from the nature
  of the interactions between the atoms or
  molecules

2

• On a macroscopic scale, gases are distinguished
  from solids and liquids by their much smaller
  values of density.
  
• On the microscopic scale, the smaller values of
  density arise due to the much lower NUMBER
  DENSITY (number of molecules per cm3 of the
  sample) compared with liquids and solids.

3
Understanding the behavior of gases and how
reactions occur in the gas phase is of practical
importance CH4(g) O2(g) -- CO2(g) H2O(g) -
combustion of fuels N2(g) H2(g) -- NH3(g) -
production of ammonia for fertilizers
2NO(g) O2 (g) - 2NO2 (g) - responsible for ac
id rain
4

• Properties of Gases
• - A gas will fill the volume of the container
  which contains it.
  
• - The volume of the gas equals the volume of its
  container
  
• - Gases are highly compressible when pressure is
  applied to a gas, its volume readily decreases
  
• - Gases form homogenous mixtures with each other
  regardless of their identity or relative
  proportions
  
• These properties arise because the individual
  atoms/ molecules are relatively far apart

5

• Three properties of gases that are used to
  describe gases are pressure (P), volume (V) and
  temperature (T).
  
• The volume of a gas is defined by the volume of
  the container.
  
• Typical units for volume of gases is the liter, L.

6

• PRESSURE
• The force exerted by a gas on a unit area of the
  walls of its container is the pressure exerted by
  the gas.

SI Units for pressure Force is newton, N (kg m/s
2)
Area - m2 Pressure - N/m2 or pascal (Pa)
7

• Atmospheric Pressure
• Because of gravity, the earths atmosphere exerts
  a downward force and consequently a pressure on
  the earths surface.
  
• Atmospheric pressure pressure exerted by the
  atmosphere around us

A column of air 1m2 in cross section extending
through the atmosphere has a mass or roughly
10,000 kg.
8

• The acceleration produced by earths gravity is
  9.8 m/s2
  
• force mass x acceleration
• Force exerted by this air column is 1 x 105 N

1 x 105 Pa
More precisely, 1.01325 x 105 Pa 1 atmosphere
(atm)
9

• A barometer operates on the principle that the
  height of a liquid in a closed tube depends on
  the atmospheric pressure

Pressure g h d g 9.8 m/s h is the height of
the liquid in the sealed tube d is the density of
the liquid
10

• a) What is the height of a mercury column in a
  barometer at atmospheric pressure?
  
• b) What is the height of a water column in a
  barometer at atmospheric pressure?
  
• Explains why mercury is used in barometers and
  not water

11

• Units of pressure
• 1 atm 760 mm Hg 760 torr 1.01325 x 105 Pa
• There are other units of pressure (lbs/in2, bar)
  but we will typically deal with atm, mm of Hg or
  torr and Pa.

12

• The Gas Laws
• Through experimental observations, relationships
  have been established between the pressure (P),
  temperature (T) and volume (V) and number of
  moles (n) of gases.
• These relationships are called the GAS LAWS.
• Having defined P, V, T, and n for a gas, this
  information defines the physical condition or
  state of a gas.
  
• The relationships between P, V, T and n that will
  be discussed hold for IDEAL gases
  
• (or for low pressures ideal conditions)

13

• Relationship between Pressure and Volume Boyles
  Law

Boyle noted from the experiments he performed
that at a fixed temperature and for a fixed
amount of gas, as pressure on a gas increases,
the volume occupied by a gas decreases.
14

• Boyles Law
• The product of pressure and volume of a sample of
  gas is a constant, at constant temperature and
  for a fixed amount of gas.

15
Plot illustrating P-V relationship
16

• The conditions of 1.00 atm pressure and 0oC are
  called standard temperature and pressure (STP).
  
• Under STP conditions, the volume occupied by the
  gas in the J-tube is 22.4 L.

Since, PV constant P1V1 P2V2
17

• Temperature-Volume Relation - Charles Law
• The volume of a fixed quantity of gas at constant
  pressure increases linearly with temperature.
  
• V V0 a V0 t
• V0 is the volume of the gas at 0oC
• t is the temperature in oC
• a is the coefficient of thermal expansion

18
Volume
V V0 a V0 t
y mx b
19

• From a plot of V vs t we can determine V0 from
  the y-intercept.
  
• From the slope a V0, a can be determined

Volume
20

• Since gases expand by the same relative amount
  when heated between the same two temperatures (at
  low pressure) implies that a is the same for all
  ideal gases.
• For gases, at low pressure

For liquids and solids a varies from substance to
substance
21

• Re-writing the expression connecting V and t

Gas thermometer By measuring the volume of a gas
at 0oC and measuring the volume change as
temperature changes, the temperature can be
calculated
22

• Absolute temperature - Kelvin Scale

At t -273.15 oC volume of gas is zero
temperatures negative volume which
is physically impossible. Hence 273.15oC is the
lowest temperature that can be physically
attained and is the fundamental limit on
temperature.
23

• This temperature is called ABSOLUTE ZERO and is
  defined to be the zero point on the Kelvin scale
  (K)
  
• T (Kelvin) 273.15 t (Celsius)
• If we substitute the above expression in

and solve for V
24

• is a constant

Hence, V a T Charles Law In other words, o
n an absolute temperature scale, at a constant
pressure and for a fixed amount if gas, the
volume of the gas is proportional to the
temperature
Hence,
Note T is temperature in Kelvin
25

• Quantity-Volume relation - Avogadros Law

Volume is affected not just by pressure and
temperature, but also by the amount of gas.
Avogadros hypothesis - Equal volumes of gases at
the same temperature and pressure contain the
same number of molecules.
26
Avogadros law the volume of a gas maintained at
constant pressure and temperature is directly
proportional to the number of moles of gas.
V constant x n Hence, doubling the moles
of gas will cause the volume to double (as long
as T and P remain constant)
27

• The Ideal-Gas Equation

Boyles law V a P-1 (constant n, T)
Charles law V a T (constant n, P)

Avogadros law V a n (constant P, T)
Putting the three laws together
28

• P V n R T IDEAL GAS EQUATION
• An ideal gas is a gas whose pressure, volume and
  temperature behavior is completely described by
  this equation.
  
• R is called the universal gas constant since it
  is the same for all gases.
  

Note The ideal gas equation is just that -
ideal. The equation is valid for the most gases
at low pressures. Deviations from ideal
behavior are observed as pressure increases.
29

• The value and units of R depends on the units of
  P, V, n and T
  
• Temperature, T, MUST ALWAYS BE IN KELVIN
• n is expressed in MOLES
• P is often in atm and V in liters, but other
  units can be used.

Values of R Units Numerical value L-atm/(mol
-K) 0.08206 cal/(mol-K) 1.987 J/(mol-K)
8.314 m3-Pa/(mol-K) 8.314 L-torr/(mol-K) 62.
36
30

• Example
• Calcium carbonate, CaCO3(s), decomposes upon
  heating to give CO2(g) and CaO(s). A sample of
  CaCO3 is decomposed, and the CO2 collected in a
  250. mL flask. After the decomposition is
  complete, the gas has a pressure of 1.3 atm at a
  temperature of 31oC. How many moles of CO2 were
  generated?

Given Volume of CO2 250 mL 0.250 L Pressu
re of CO2 1.3 atm
temperature of CO2 31oC
First convert temperature to K
T 31 273 304 K
31

• To calculate n

Based on the units given for P and V, use
appropriate value for R R 0.08206 L-atm/(mol K
)
n 0.013 mol CO2
32

• Problem
• The gas pressure in a closed aerosol can is 1.5
  atm at 25oC. Assuming that the gas inside obeys
  the ideal-gas equation, what would the pressure
  be if the can was heated to 450oC?

Since, the can is sealed, both V and n stay
fixed. P (atm) t (oC) T(K) Initial 1.5 2
5 298
Final ? 450 723
33

• Molar Mass and Gas Density
• The ideal gas law, P V n R T can be used to
  determine the molar mass of gaseous compounds.

34
The ideal gas equation can be also be used to
determine the density of the gas
35

• Gas Stoichiometry
• If the conditions of pressure and temperature are
  known, then the ideal gas law can be used to
  convert between chemical amounts i.e. moles, and
  gas volume.
• Hence in dealing with chemical reactions
  involving gases, we can deal with volumes of
  gases instead of moles of gases, being that
  volume is usually an easier quantity to measure.

36

• Problem
• Dinitrogen monoxide, N2O, better known as nitrous
  oxide or laughing gas, is shipped in steel
  cylinders as a liquid at pressures of 10 MPa. It
  is produced as a gas in aluminium trays by the
  decomposition of ammonium nitrate at 200oC.
• NH4NO3(s) -- N2O(g) 2H2O(g)
• What volume of N2O(g) at 1.00 atm would be
  produced from 100.0g of NH4NO3(s) after
  separating out the H2O and cooling the N2O gas to
  273K. Assume a 100 yield in the production of
  N2O(g) .
• Assuming a 100 yield all the NH4NO3(s) is
  converted to N2O(g)
  
• Moles of NH4NO3(s) decomposed 100.0
  g/80.04g/mol
  
• 1.249 mol

37

• Hence, 1.249 mol of N2O(g) formed.
• To calculate the volume of N2O(g) produced, use
  the ideal gas equation.
  
• V n R T/ P
• V (1.249 mol) (0.08206 L-atm/(mol-K))
  (273)/(1.00 atm)
  
• 28.0 L N2O(g)

38

• Mixtures of Gases

Can we use the ideal gas equation to determine
the properties of gases in a mixture?
Dalton observed that the total pressure of a mix
ture of gases equals the sum of the pressures
that each would exert if each were present alone.
The partial pressure of a gas in a mixture of
gases is defined as the pressure it exerts if it
were present alone in the container.
39

• Daltons law states that the total pressure is
  the sum of the partial pressures of each gas in
  the mixture.
  
• For example, consider a mixture of two gases A
  and B in a closed container
  
• Assuming that the pressure is low enough, A and B
  obey the ideal gas equation.
  
• The fact that A and B behave as ideal gases
  implies that A and B do not interact with each
  other.
  

40
(No Transcript)
41

• From Daltons law
• Ptotal PA PB

Where ntotal nA nB is the total number of
moles
42
Mole Fraction
What is the fraction of the number of moles of A
in the mixture? To find this out, we need to divi
de the number of moles of A by the total number
of moles of gases in the mixture
Note mole fraction is unitless since it is a
fraction of two quantities with the same unit.
Also, sum of mole fractions of the components in
a mixture 1
43
Dividing PA/Ptotal
Hence, PA XA Ptotal
44

• A study of the effect of certain gases on plant
  growth requires a synthetic atmosphere composed
  of 1.5 mol percent of CO2, 18.0 mol percent of O2
  and 80.5 mol percent Ar.
• a) Calculate the partial pressure of O2 in the
  mixture if the total pressure of the atmosphere
  is to be 745 torr?
  
• b) If this atmosphere is to be held in a 120-L
  space at 295 K, how many moles of O2 are needed?
  

45

• Kinetic Theory of Gases
• The ideal gas equation describes how gases
  behave.
  
• In the 19th century, scientists applied Newtons
  laws of motion to develop a model to explain the
  behavior of gases.
  
• This model, called the kinetic theory of gases,
  assumes that the atoms or molecules in a gas
  behave like billiard balls.
  
• In the gas phase, atoms and molecules behave like
  hard spheres and do not interact with each
  other.
  

46

• Assumptions of Kinetic Theory of Gases
• 1) A gas consists of a large number of particles
  that are so small compared to the average
  distance separating them, that their own size can
  be considered negligible.
• 2) The particles of an ideal gas behave totally
  independent, neither attracting nor repelling
  each other.
  
• 3) Gas particles are in constant, rapid,
  straight-line motion, incessantly colliding with
  each other and with the walls of the container.
  All collisions between particles are elastic.
• 4) A collection of gas particles can be
  characterized by its average kinetic energy,
  which is proportional to the temperature on the
  absolute scale.

47

• Gas particles are constantly colliding with each
  other and the walls of the container.
  
• It is the collisions between the gas particles
  and the walls of the container that define the
  pressure of the gas.
  
• Every time a gas particle collides with the wall
  of the container, the gas particle imparts its
  momentum to the wall

momentum mass x velocity
48
The pressure exerted by the gas is proportional
to the momentum of the particle and the number of
collisions per unit time, the collision
frequency. Pressure a (momentum of particle) x
(rate of collisions with the wall)
The rate of collision is proportional to the num
ber of particles per unit volume (N/V) and the
speed of the particle (u).
49
P V a N m u2
The speed, u, is the average speed of the
particles, since not all the particles move with
the same speed.
50
Particles are moving in a 3-D space
Comparing this equation with the ideal gas
equation
P V n R T
This equation relates the speed of the gas
particles with the temperature of the gas
51

• n No N where No is Avogadros number

(m No) is the molar mass of the gas, M
The mean square speed depends on T and M
52
Temperature and Kinetic Energy
The average kinetic energy of a mole of particles
is
1 No m u2
2
53
Hence the average kinetic energy of the molecules
in a gas depends only on the temperature of the
gas and is independent of the mass or density of
the gas.
54
The kinetic theory of gases explains the observed
behavior of ideal gases i.e. it explains, the
three gas laws
The pressure exerted by a the gas is due to the
collisions between the particles and the walls of
the container. If the temperature stays the sam
e, then the average speed of the particles is the
same The pressure will depend on the number of c
ollisions per unit area of the wall per unit
time. Reducing the volume of the container, wil
l result in more frequent collisions and hence a
higher pressure.
55

• Charles Law

If the P and n are kept constant, the volume of
the container must increase with increasing T.
56

• Avogadros Law

At fixed pressure and temperature, the volume of
the gas is proportional to the number of
particles. This is explicit in the equation above
, for fixed P and T.
57

• Distribution of Molecular Speeds
• Atoms or molecules in a gas do not all travel at
  the same speed.
  
• There is a distribution of speeds, with some
  travelling slow, others fast, and the majority
  peaked about a value called the most probable
  speed.
• A plot of the number of molecules travelling at a
  given speed versus speed, at a fixed temperature
  is called the MAXWELL-BOLTZMANN distribution.

58
(No Transcript)
59
ump
urms
60
The root mean square speed, is the square root of
the square of the average speed
Root mean square speed sum the squares of the
speeds, divide by the number of particles and
then square root the resulting number.
Average speed divide the sum of the speed of all
particles by the number of particles.
1, 2, 3, 4, 5, 6 Average (123456)/6 3.5
RMS value ((149162536)/6)1/2 3.89 The
rms value is always slightly higher than the
average value.
61

• For the same molecule, as temperature increases
  the most probable speed shifts to higher values
  the distribution also broadens.
  
• For two gases at the same temperature, but
  different masses the rms speed for the lighter
  gas is higher than that for the heavier gas
  particles.

The average kinetic energy of all gases at the
same temperature is the same, regardless of
mass.

62
Temperature describes a system of gaseous
molecules only when their speed distribution is
represented by the Maxwell-Boltzmann
distribution. A collection of gas molecules who
se speed distribution can be represented by a
Maxwell-Boltzmann distribution is said to be at
thermal equilibrium.
63

• Motion of Gas Molecules- Diffusion Effusion
• Gas molecules do not travel in a straight line,
  but undergo a more random type motion.
  
• Each time a gas molecule collides with another
  its direction changes.
  
• The average distance covered by a gas molecule
  between two collisions is the mean free path.
  
• Lower the gas pressure, longer is the mean free
  path.
  

64

• If the pathway of a gas molecule from point A to
  B is tracked, its path would look like this

65
This type of irregular motion is called DIFFUSION
and is responsible for gases mixing like an the
odor filling up a room, The rate of diffusion
depends inversely on the mass of the molecule
heavier molecules diffuse more slowly than
lighter molecules.
66

• Effusion
• Effusion is the motion of gas molecules through a
  small hole.
  
• Within the container, each gas molecule undergoes
  the random motion, colliding with other gas
  molecules.
  
• During this process, if the gas molecule
  encounters the hole in the container it will
  emerge out of the container.
  

67

• Although each particle traces its own unique path
  to the hole, the faster the molecules move, the
  more quickly will they emerge from the hole.
  
• The rate of effusion is proportional to urms.
• For a mixture of two different gases, A and B, in
  the same container, and hence at the same
  temperature and pressure, the rate of effusion
  for each depends on urms of each.

68
Isotope separation by Diffusion
Rate of diffusion of a gas is inversely
proportional to the square root of the mass
Light atoms diffuse through a porous barrier fas
ter than heavier atoms. 235U is fissile, not 2
38U
69
Natural U is about 99.28 238U and 0.72 235U
For a nuclear reactor 10 235U For a weapons-
grade 90 235U
To enrich U in 235U one of the ways is to take
advantage of the different diffusion rates of
235U vs 238U
Need a gas use gaseous UF6
The enrichment factor, is theoretically 0.43,
but in practice only about 0.14
To produce 99 uranium-235 from natural uranium
4000 stages are required. The process requires
the use of thousands of miles of pipe, thousands
of pumps and motors, and intricate control
mechanisms.
70
The biggest obstacle was finding a suitable
material for the "porous barrier" that was able
to withstand the corrosive properties of the
uranium gas - one of the contributions of the
Manhattan Project at Columbia
Note other methods of enriching U with 235U were
also used
71
K-25, Oak Ridge National Lab
4 stories high and almost a half mile long
enclosed 2 million square feet of space, making
it the largest building in the world at the time.
The eventual cost of the K-25 complex 500
million.
72

• Real Gases Deviations from Ideal Behavior
• The fact that gases can liquefy at low
  temperatures or high pressures, indicate that
  gases do not behave ideally over all ranges of
  temperature and pressure.
• Gases liquefy because of interactions between
  molecules become important as the molecules come
  closer together.
  
• For Boyles law to hold a gas must never liquefy
  it must remain a gas at all pressures.
  
• This means that there must be no interactions
  between gas molecule.

73
Temperature at which He(g) condenses to He(l)
4 K
Ar(g) condenses to Ar(l) 87 K
74

• When a gas is compressed by an increase in
  pressure and corresponding decrease in volume,
  gas molecules are forced closer together.
  
• As the pressure increases, the amount by which
  the gas can be compressed decreases because of
  the finite volume occupied by each gas molecule.

For Boyles law to be valid over all ranges of
pressure means that gas molecules must have zero
volume
75

• The ideal gas equation is important in
  determining limiting values of pressure, volume,
  and is a useful description of the behavior of
  gases at low pressures and high temperatures.
• Deviations from ideal behavior can be quantified
  by a compressibility factor, Z
  

If Z 1, the gas behaves as an ideal gas
The further the compressibility factor is from 1,
the greater the deviation of the gas from an
ideal gas
76
(No Transcript)
77

• Equation of State for Real Gases van der Waals
  Equation
  
• Real gases
• 1) Particles of a real gas occupy space
• 2) Attractive and repulsive forces do exist
  between gas molecules.
  
• The van der Waals equation of state accounts for
  the real behavior of gases.
  
• The ideal gas equation P V n R T
• must be modified to account for the non-zero
  volume of each gas molecule, and the interactions
  between gas molecules.

78

• Accounting for Volume
• Because of the non-zero volume of each gas
  molecule, the volume available to a gas molecules
  is less than the volume of the container by V-
  nb
• b is the volume occupied by 1 mole of gas
  molecules (L/mol)
  
• n is the number of moles of gas (mol)
• V is the volume of the container.
• Accounting for Pressure
• Since real gas molecules interact with each
  other, the observed pressure is lower than the
  ideal gas pressure.

79

• Accounting for real gas behavior results in van
  der Waals equation of state

The van der Waals constants, a and b, are
different for different gases.
80
The constant b is related to the size of the gas
particle. Larger the value of b, larger is the
particle. Atom/Molecule b (L/mol) Ar 0.032
19 Cl2 0.05622 He 0.02370 H2 0.02661
81
The magnitude of the constant a is a measure of
the attractive forces between molecules.
Gases with larger a values liquefy or solidify m
ore easily than gases with smaller a values since
the attractive forces between molecules are
strong. Atom a (L2atm/(mol2)) boiling pt (K) A
r 1.345 87.3 He 0.03412 4.2 In gener
al atoms or molecules like He and H2 which have
small a and b values exhibit behavior fairly
close to ideal.