Transcript trt 408 physical chemistry

TRT 401 PHYSICAL CHEMISTRY
PART 1: INTRODUCTION TO PHYSICAL
CHEMISTRY
What is physical chemistry?
Physical chemistry is a study of the physical basis of phenomena related to the
chemical composition and structure of substances.
Or
Physical chemistry is quantitative and theoretical study of the
properties and structure of matter, and their relation to the interaction
of matter with energy.
This course serves as an introduction to chemical thermodynamics, giving you an
understanding of basic principles, laws and theories of physical chemistry the are
necessary for chemistry, biochemistry, pre-medical, general science and engineering
students.
You will develop the ability to solve quantitative problems, and learn to use
original thought and logic in the solution of problems and derivation of equations.
You will learn to apply mathematics in chemistry in such a way that the equations
paint a clear picture of the physical phenomena
Physical chemistry includes numerous
disciplines:
Thermodynamics - relationship between energy interconversion
by materials, and the molecular properties
Kinetics - rates of chemical processes
Quantum Mechanics - phenomena at the molecular level
Statistical Mechanics - relationships between individual
molecules and bulk properties of matter
Spectroscopy - non-destructive interaction of light (energy) and
matter, in order to study chemical structure
Photochemistry - interaction of light and matter with the intent of
coherently altering molecular structure
Atoms and Molecules
Atoms are the submicroscopic particles that constitute the fundamental building
block of ordinary matter. They are most found in molecules, two or more atoms
joined in specific geometrical arrangement.
Carbon dioxide molecule
Carbon monoxide molecule
Oxygen
atom
Carbon
atom
In the study of chemistry, atoms are often portrayed as colored spheres, with each
color representing a different kind of atom.
For example, a black sphere represents a carbon atom, a red sphere represents
an oxygen atom.
Lanthanoids
Actinoids
For more interactive periodic table please refer http://www.ptable.com/
Chemical Bonding
cation
Sodium metal
Chlorine gas
anion
NaCl
Sodium Chloride
Oppositely charged ions are held together by ionic bonding,
forming a crystalline lattice.
Ionic compound
A compound that composed of cations and anions
bound together by electrostatic attraction
Covalent Compund
Compound that not contain a metallic element typically covalent compound
consisting of discrete molecules.
Single bond
covalent
 The shared pairs of electrons are bonding pairs.
 The unshared pairs of electrons are lone pairs or non bonding pairs.
Polar Covalent Bond
 In reality, fully ionic and covalent bonds represent the extremes of a spectrum
of bonding types.
 Most of covalent bonds are polar covalent bonds, in which the electrons are
shared unequally, but are nit fully transfered from one atom to the other.
 In polar bonds, one atom has a partial negative charge (δ-) and the other atom
has a partial positive charge(δ+).
Metallic Bonding
 Metallic bonding, occurs in metals. Since metals have low ionization energies, they tend
to electron easily.
 When metal atoms bonding together form a solid, each metal atom donates one or more
electron to an electron sea.
Example: sodium metal as an array of
positively charged Na+ ion immersed in
a sea of negatively charges electron (e-).
The table below summarize the three different types of bonding
Types of atom
Type of bond
Characteristic of bond
Metal & nonmetal
Ionic
Electron transferred
Nonmetal & nonmetal
Covalent
Electron shared
Metal & metal
Metallic
Electrons pooled
Electronegativity
The ability of an atom to attract electrons to itself in a chemical bond
(which results in polar bonds) is called electronegativity, χ (chi).
Fluorine is the most
Electronegativity element
Francium is the
least electronegative
element
Electronegativity generally increase as we move across a row in periodic table and
decrease as we move down a column.
Electronegativity and Bond Polarity
Bond polarity expressed numerically as dipole moment, μ which occurs when
there is a separation between a positive and negative charge.
μ = qr
μ: dipole moment; q: separating a proton and an electron; r: distance
Example:
q = 1.6 x 10-19 C
r = 130 pm (the approximate length of a short chemical bond)
μ= q x r
= (1.6 x 10-19 C)(130 x 10-12m)
= 2.1 x 10-29 . M
= 6.2 D
The debye (D) is a common unit used for reporting dipole moment (1D = 3.34 x 10-30 C.m)
Table 1 Dipole moments of several molecules in the gas phase
Molecule
∆ EN
Dipole Moment (D)
Cl2
0
0
ClF
1.0
0.88
HF
1.9
1.82
LiF
3.0
6.33
Matter
Matter: anything that occupies space and has mass.
Example: book, desk, pen, pencil even your body is are all compose of
matter.
Air also matter but it too occupies space and matter. These specific instance
of matter- such as air, sand, water- a substance.
Matter can be classify to its state- solid, liquid, or gas according to its
composition.
Example:
Table salt, ice &
diamond
Example:
Glass, plastic &
charcoal
Solid matter may be crystalline, in which case its atoms or molecules are
arranged in patterns with long range, repeating order.
Or
Its may be amorphous in which case its atoms or molecules do not have any
long range order.
In Gaseous Matter
Gases can be compressed-squeezed into a small volume because there is so much
Empty space between atoms or molecules in the gaseous state.
The Composition of Matter
Matter can be classified as either pure substances, which have fixed composition,
or mixtures, which have variable composition.
Pure substance (element and compounds) are unique materials with their own
chemical and physical properties, and are composed of only one type of atom or
molecule.
Element
A substance that cannot
be chemically broken down
Into simpler substance.
Compound
A pure substance that is composed
of atoms or two more different
elements.
Ionic
compound
Molecular
compound
The Composition of Matter
Mixture are simply random combinations of two or more different types of atoms of
molecules, and retain the properties of the individual substances. They can therefore
be separated (although sometime with difficulty) by physical means (such as boiling,
distillation, melting, crystallizing, magnetism, etc.)
Heterogenus Mixture
One in which the composition varies
from one region to another.
Homogenus Mixture
One with the same composition
Throughout.
Summary Composition of Matter
Variable composition?
YES
NO
Wet sand
NO
YES
Tea with sugar
YES
Pure water
NO
Helium gas
Quantifying Matter
Amount of substance(n): a measure of a number of specified entities (atoms,
molecules, or formula unit) present (unit; mole; mol).
1 mol of a substance contains as many entities as exactly 12 g of carbon-12 (ca.
6.02 x 1023 objects)
Avogadro’s Number: NA = 6.02 x 1023 mol-1
Extensive Property: Dependent upon the amount of matter in
the substance (e.g., mass & volume)
Intensive Property: Independent of the amount of matter in a
substance (e.g., mass density, pressure and temperature)
Molar Property: Xm, an extensive property divided by the
amount of substance, n: Xm = X/n
Molar Concentration:“Molarity” moles of solute dissolved in
litres of solvent: 1.0 M = 1.0 mol L-1
Units
 In science, the most commonly used set of units are those of the International
System of Units (the SI System, for Système International d’Unités).
 There are seven fundamental units in the SI system. The units for all other
quantities (e.g., area, volume, energy) are derived from these base units.
Table 2 Example list of units
for more info: http://physics.nist.gov/cuu/Units/
SI Prefix – Small & Large Unit
Table 3 SI Prefix Multipliers
SI Prefix – Large Unit
Table 4 SI Prefix – large unit
Energy
Energy is define as the ability to do work.
Work is done when a force is exerted through a distance.
Force through distance; work is done.
Energy is measured in Joules (J) or Calories (cal).
1 J = 1 kg m2 s-2
Energy may be converted from one to another, but it is neither created nor
destroyed (conversion of energy).
In generally, system tend to move from situations of high potential energy (less
stable) to situations having lower energy (more stable).
Energy is the capacity to supply heat or to do work. Energy can be exchanged
between objects by some combination of either heat or work:
Energy = heat + work
∆E= q + w
 work is done when a force is exerted through a distance
work = force x distance
 heat is the flow of energy caused by a temperature
difference.
Example of a billiard ball rolling across the table
and colliding straight on with a second,
stationary billiard ball.
Potential and Kinetic Energy
• Kinetic energy (EK) is the energy due to the motion of an object with mass m
and velocity v:
EK = ½ mv2
– Thermal energy, the energy associated with the temperature of an object, is
a form of kinetic energy, because it arises from the vibrations of the atoms
and molecules within the object.
• Potential energy (EP) is energy due to position, or any other form of “stored”
energy. There are several forms of potential energy:
– Gravitational potential energy
– Mechanical potential energy
– Chemical potential energy (stored in chemical bonds)
Potential energy increases when things that attract each other are separated or
when things that repel each other are moved closer.
• Potential energy decreases when things that attract each other are moved
closer, or when things that repel each other are separated.
• According to the law of conservation of energy, energy cannot be created or
destroyed, but kinetic and potential energy can be interconverted.
Example:
Energy transformation I
Energy transformation II
Water falling in a waterfall exchanges gravitational potential energy for
kinetic energy as it falls faster and faster, but the energy is never destroyed.
high EP
low EK
decreasing EP
increasing EK
low EP
high EP
EK converted to
thermal energy
and sound
Chemical Energy
The chemical potential energy of a substance results from the relative positions and
the attractions and repulsions among all its particles. Under some circumstances, this
energy can be released, and can be used to do work:
Using chemical energy to do work – The compound produced when gasoline burns have
Less chemical potential energy than the gasoline molecules.
Law of Conservation of Energy
• A law stating that energy can neither be created nor destroyed, only
converted from one form to another, and it can assume in different forms.
E.g.:
The cycle held the
gravitational energy
The energy transformed
into kinetic energy of
motion.
Contributions to Energy
Kinetic Energy, EK: Energy an object possesses as a result of its motion.
KE = ½mv2
Potential Energy, V: Energy an object possesses as a result of its position. Zero of
potential energy is relative:
1. Gravitational Potential Energy:
zero when object at surface (V = 0 when h = 0)
VG = mgh, m = mass, g = 9.81 m s-2, h = height
2. Electrical Potential Energy:
zero when 2 charged particles infinitely separated
qi = charge on particle i, r = distance
ε0 = 8.85 x 10-12 C2 J-1 m-1
(vacuum permittivity)
Equipartition of Energy
Molecules have a certain number of degrees of freedom: they can vibrate, rotate and
translate - many properties depend on these degrees of freedom:
Equipartition theorem:
All degrees of freedom have the same average energy at temperature T: total energy
is partitioned over all possible degrees of freedom
Quadratic energy terms:
½mvx2 + ½mvy2 + ½mvz 2
Average energy associated with each quadratic term is ½kT, where k = 1.38 x 10-23 J K-1
(Boltzmann constant), where k is related to the gas constant, R = 8.314 J K-1 mol-1 by
R = N Ak
However: this theorem is derived by classical physics, and can only be applied to
translational motion.
Relationship Between Molecular and Bulk Properties
The energy of a molecule, atom, or subatomic particle that is confined to a region of space
Is quantized, or districted to certain discrete values. These permitted energies are called
energy level.
Populations of States
At temperatures > 0, molecules are distributed over available energy levels according
to the Boltzmann Distribution, which gives the ratio of particles in each energy
state:
Boltzmann constant k=1.381 x 10-23 JK-1
At the lowest temperature T = 0, only the lowest energy state is occupied. At infinite
temperature, all states are equally occupied.
In real life, the population of states is described by an exponential function, with the
highest energy states being the least populated.
Degenerate states: States which have the same energy
These will be equally populated!
Boltzmann Distributions
Populations for (a) low
& (b) high temperatures
Boltzmann predicts an
exponential decrease in
population with
increasing temperature
At room T, only the
ground electronic state
is populated. However,
many rotational states
are populated, since the
energy levels are so
closely spaced.
More states are
significantly populated if
energy level spacing
are near kT!
The Electromagnetic Field
Light is a form of electromagnetic radiation. In classical physics, electromagnetic
radiation is understood in term of the electromagnetic field.
Electric field – charged particles (whether stationary or moving)
Magnetic field – acts only on moving charged particles.
Wavelength ,λ is the distance between the neighboring peaks of two wave, and is
frequency, v (nu) the number of times in a given interval at which its displacement
at a fixed point returns to its original value divided by the length of the time intervals.
Frequency is measure in Hertz, 1 Hz= 1 s-1.
Wavelength, λ = c/ v
Wavenumber, ṽ (nu tilde) the number of complete wavelengths in a given length.
Wavenumber, ṽ = v/c =1/λ
e.g.: A wave number of 5 cm-1 indicates there are 5 complete wavelength in 1 cm.
PART 2: INTRODUCTION TO PHYSICAL CHEMISTRY
Matter: Substance, intensive and extensive properties, molarity and molality
Substance
 A substance is a distinct, pure form of matter.
 The amount of a substance, n, in a sample is reported in terms of the unit called a mole (mol). In 1
mol are NA=6.0221023 objects (atoms, molecules, ions, or other specified entities). NA is the
Avogadro constant.
Extensive and intensive properties
 An extensive property is a property that depends on the amount of substance in the sample.
Examples: mass, volume…
 An intensive property is a property that is independent on the amount of substance in the sample.
Examples: temperature, pressure, mass density…
 A molar property Xm is the value of an extensive property X divided by the amount of substance, n:
Xm=X/n. A molar property is intensive. It is usually denoted by the index m, or by the use of small
letters. The one exemption of this notation is the molar mass, which is denoted simply M.
 A specific property Xs is the value of an extensive property X divided by the mass m of the
substance: Xs=X/m. A specific property is intensive, and usually denoted by the index s.
Measures of concentration: molarity and molality
 The molar concentration (‘molarity’) of a solute in a solution refers to the amount of substance of
the solute divided by the volume of the solution. Molar concentration is usually expressed in moles
per litre (mol L-1 or mol dm-3). A molar concentration of x mol L-1 is widely called ‘x molar’ and
denoted x M.
 The term molality refers to the amount of substance of the solute divided by the mass of the
solvent used to prepare the solution. Its units are typically moles of solute per kilogram of solvent
(mol kg-1).
Some fundamental terms:
System and surroundings:
For the purposes of Physical Chemistry, the universe is divided into two
parts, the system and its surroundings.
 The system is the part of the world, in which we have special interest.
 The surroundings is where we make our measurements.
The type of system depends on the characteristics of the boundary which
divides it from the surroundings:
(a) An open system can exchange matter and energy with its
(b) A closed system can exchange energy with its surroundings, but it
cannot exchange matter.
(c) An isolated system can exchange neither energy nor matter with its
surroundings.
Except for the open system, which has no walls at all, the walls in
the two other have certain characteristics, and are given special
names:
 An adiabatic (isolated) system is one that does not permit the
passage of energy as heat through its boundary even if there is a
temperature difference between the system and its
surroundings. It has adiabatic walls.
 A diathermic (closed) system is one that allows energy to escape as heat
through its boundary if there is a difference in temperature between the
system and its surroundings. It has diathermic walls.
Homogeneous system:
The macroscopic properties are identical
in all parts of the system.
Heterogeneous system:
The macroscopic properties jump at the
phase boundaries.
Phase:
Homogeneous part of a (possibly)
heterogeneous system.
Equilibrium condition:
 The macroscopic properties do not change without external influence.
 The system returns to equilibrium after a transient perturbation.
 In general exists only a single true equilibrium state.
Equilibrium in Mechanics:
Equilibrium in Thermodynamics:
H2O (water)
25°C
1 bar
stable
unstable
metastable
stable
H 2 + ½ O2
25°C
1 bar
metastable
The concept of “Temperature”:
 Temperature is a thermodynamic quantity, and not known in mechanics.
 The concept of temperature springs from the observation that a change in physical state (for
example, a change of volume) may occur when two objects are in contact with one another (as when
a red-hot metal is plunged into water):
A
+
B
A
B
If, upon contact of A and B, a change in any physical property of these systems is found, we know that
they have not been in thermal equilibrium.
The Zeroth Law of thermodynamics:
If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, than C is also
in thermal equilibrium with A. All these systems have a common property: the same
temperature.
Energy flows as heat from a region at a higher
temperature to one at a lower temperature if
the two are in contact through a diathermic
wall, as in (a) and (c). However, if the two
regions have identical temperatures, there is
no net transfer of energy as heat even though
the two regions are separated by a diathermic
wall (b). The latter condition corresponds to
the two regions being at thermal equilibrium.
The thermodynamic temperature scale:
In the early days of thermometry (and still in laboratory practice today), temperatures were related to
the length of a column of liquid (e.g. Mercury, Hg), and the difference in lengths shown when the
thermometer was first in contact with melting ice and then with boiling water was divided into 100
steps called ‘degrees’, the lower point being labelled 0. This procedure led to the Celsius scale of
temperature with the two reference points at 0 °C and 100 °C, respectively.
Assumption:
Linear relation between the Celsius temperature  and an
observable quantity x, like the length of a Hg column, the
pressure p of a gas at constant volume V, or the volume V of
the gas for constant pressure p:
(x)  a  x  b
x  x 0C
 (x)  100 C 
x100C  x 0C
For the pressure p, this transforms to:


p0C

p
 100  


C
p

p
p

p
100C
0C 
 100C 0C
Observation:
For all (ideal) gases one finds
100 
Left: The variation of the
volume of a fixed amount of
gas with the temperature
constant. Note that in each
case they extrapolate to zero
volume at -273.15 C.
Right: The pressure also
varies linearly with the
temperature, and
extrapolates to zero at
T= 0 (-273.15 C).
p0C
 273.15  0.01
p100C  p0C
 Introduction of the thermodynamic temperature scale (in
‘Kelvin’): T
p
 100 
K
p100C  p0C and
T


 273.15
K C
Work, heat, and energy:
 The fundamental physical property in thermodynamics is work: work is done when an object is
moved against an opposing force.
(Examples: change of the height of a weight, expansion of a gas that pushes a piston and raises the
weight, or a chemical reaction which e.g. drives an electrical current)
 The energy of a system is its capacity to do work. When work is done on an otherwise isolated
system (e.g. by compressing a gas or winding a spring), its energy is increased. When a system does
work (e.g. by moving a piston or unwinding the spring), its energy is reduced.
 When the energy of a system is changed as a consequence of a temperature difference between it
and the surroundings, the energy has been transferred as heat. When, for example, a heater is
immersed in a beaker with water (the system), the capacity of the water to do work increases
because hot water can be used to do more work than cold water.
Heat transfer requires diathermic walls.
 A process that releases energy as heat is called exothermic, a process that absorbs energy as heat
endothermic.
(a) When an endothermic process occurs in an
adiabatic system, the temperature falls; (b) if the
process is exothermic, then the temperature rises. (c)
When an endothermic process occurs in a diathermic
container, energy enters as heat from the
surroundings, and the system remains at the same
temperature; (d) if the process is exothermic, then
energy leaves as heat, and the process is isothermal.
Work, heat, and energy (continued):
Molecular interpretation
 In molecular terms, heat is the transfer of energy that makes use of chaotic molecular motion
(thermal motion).
 In contrast, work is the transfer of energy that makes use of organized motion.
 The distinction between work and heat is made in the surroundings.
When energy is
transferred to the
surroundings as heat,
the transfer stimulates
disordered motion of the
atoms in the
surroundings. Transfer of
energy from the
surroundings to the
system makes use of
disordered motion
(thermal motion) in the
surroundings.
When a system does
work, it stimulates
orderly motion in the
surroundings. For
instance, the atoms
shown here may be part
of a weight that is being
raised. The ordered
motion of the atoms in a
falling weight does work
on the system.
State functions and state variables
STATEMENT
 If only two intensive properties of a phase of a pure substance are known, all intensive
properties of this phase of the substance are known, or
 If three properties of a phase of a pure substance are known, all properties of this phase
of the substance are known.
example:
- p and T as independent variables means: Vm (=v) = f(p,T),
i.e. the resulting molar volume is pinned down, or
- p, T, n as independent variables means: V = f(p,T,n)
 The resulting function is termed a state function.
 The variables which describe the system state, are termed
- state variables, and are related to each other via the
- state functions.
The thermal equation of state and the perfect gas equation
Thermal equation of state:
 The thermal equation of state combines
volume V, temperature T, pressure p, and
the amount of substance n:
The “perfect gas” (or “ideal gas”):
 mass points without expansion
 no interactions between the particles
 a real gas, an actual gas, behaves
more and more like a perfect gas the
V = f(p,T,n) or Vm = v = f(p,T)
lower the pressure, and the higher the
Some empirical gas laws:
temperature
V = f(T) for p=const.:
p = f(T) for V=const.:
p = f(V) for T=const.:
“isobars”
1
V = const.  ( + 273.15°C)
= const.’  T
Charles’s Law
“isochores”
2
p = const.  ( + 273.15°C)
= const.’  T
“isotherms”
3
p  V = const.
Boyle’s Law
Combination of 1 and 3 for:




1 mol gas at
p0 = 1.013 bar
T0 = 273.15 K
v0 = 22.42 l
Step 1: Isobaric change
T0 ,p0 , v 0  v 
v0
T
T0
Step 2: Isothermal change
T,p0, v  p  v  p0  v
v
 p  v  0  p0  T
T0
}
= const. !
‘perfect gas equation’
A region of the p,V,T
surface of a fixed amount
of perfect gas. The points
forming the surface
represent the only states
of the gas that can exist.
pv=RT
pV=nRT
Sections through the surface
shown in the figure at constant
temperature give the isotherms
shown for the Boyle-Mariotte
law and the isobars shown for
the Gay-Lussac law.
R : ‘gas constant’
(= 8.31434 J K-1 mol-1)
Swap the changes:
a combination of 3 and 1 for




1 mol gas at
p0 = 1.013 bar
T0 = 273.15 K
v0 = 22.42 l
Step 1: Isothermal change
T0,p0,v0  p  v ''  p0  v 0
Step 2: Isobaric change
v ''
T0 ,p,v ''  v   T
T0
v
 p  v  0  p0  T
T0
}
= const. !

same result !!!

The change of a state variable is
independent of the path, on which the
change of the state has been made, as
long as initial and final state are
identical.
Some mathematical consequences:
(i) The change can be described as an ‘exact differential’,
i.e. the variables can be varied independently; e.g. for
z=f(x,y):
 z 
 z 
 z   z 
dz    dx    dy ;   ,   : partial differentials
 x  y
 x  y  y  x
 y  x
(ii) The mixed derivatives are identical (Schwarz’s
theorem):
  2z    2z 

 

 xy   yx 
(ii) Upon variation of x, y for z=const (Euler’s theorem):

z
 z 
x
 z 
 y 
dz  0    dx    dy     
z
 x  y
 x  z
 y  x
y

 
y
x
A more general approach to thermal expansion and compression:
 V = f(T) for p=const.:
(Gay-Lussac)
 p = f(T) for V=const.:
V = V0  (1 + )

1  p 
1  p 
    
p0    V p0  T  V
pV = const. and d(pV) = pdV + Vdp = 0

Exact differential of V=f(p,T):
 : (thermal) expansion coefficient
p = p0  (1 + )

 p = f(V) for T=const.:
(Boyle-Mariotte)
1  V 
1  V 



V0   p V0  T p
 : (isothermal) compressibility
1  V 

V  p  T
 V 
 V 
dV  
dT


 p  dp

T

p

T
generally valid!
   V  dT   V  dp
 V 
 V   T 
 
   p  (Euler)


 Due to  p T

T

p 
V


1 

p  generally valid!
Mixtures of gases: Partial pressure and mole fractions
Dalton’s law:
The pressure exerted by a mixture of perfect
gases is the sum of the partial pressures of the
gases.
The partial pressure of a gas is the
pressure that it would exert if it occupied
the container alone. If the partial pressure
of a gas A is pA, that of a perfect gas B is pB,
and so on, then the partial pressure when
all the gases occupy the same container at
the same temperature is
p  pA  pB  ...
where, for each substance J,
n R  T
pJ  J
V
The mole fraction, xJ, is the amount of J
expressed as a fraction of the total amount
of molecules, n, in the sample:
n
xJ  J
n  nA  nB  ...
n
When no J molecules are present, xJ=0;
when only J molecules are present, xJ=1.
Thus the partial pressure can be defined as:
pJ  x J  p
and
p A  pB  ... 
 (x A  xB  ...)  p  p
The partial pressures pA and pB
of a binary mixture of (real or
perfect) gases of total pressure p
as the composition changes
from pure A to pure B. The sum
of the partial pressures is equal
to the total pressure. If the gases
are perfect, then the partial
pressure is also the pressure that
each gas would exert if it were
present alone in the container.
Real gases: An introduction
Molecular interactions
Compression factor
Z
Real gases show deviations from the perfect gas
law because molecules (and atoms) interact with
each other: Repulsive forces (short-range
interactions) assist expansion, attractive forces
(operative at intermediate distances) assist
compression.
The variation of the
potential energy of two
molecules on their
separation. High positive
potential energy (at very
small separations)
indicates that the
interactions between them
are strongly repulsive at
these distances. At
intermediate separations,
where the potential energy
is negative, the attractive
interactions dominate. At
large separations (on the
right) the potential energy
is zero and there is no
interaction between the
molecules.
pv
RT
For a perfect gas, Z=1 under all conditions.
Deviation of Z from 1 is a measure of departure
from perfect behaviour.
At very low pressures, all the gases have Z1 and
behave nearly perfect. At high pressure, all gases
have Z>1, signifying that they are more difficult to
compress than a perfect gas, and repulsion is
dominant. At intermediate pressure, most gases
have Z<1, indicating that the attractive forces are
dominant and favor compression.
The variation of the
compression factor Z = pv/RT
with pressure for several
gases at 0C. A perfect gas
has Z = 1 at all pressures.
Notice that, although the
curves approach 1 as p  0,
they do so with different
slopes.
Real gases: The virial equation of state
Below, some experimental isotherms of carbon
dioxide are shown. At large molar volumes v and
high temperatures the real isotherms do not
differ greatly from ideal isotherms. The small
differences suggest an expansion in a series of
powers either of p or v, the so-called virial
equations of state:
Experimental
isotherms of
carbon dioxide at
several
temperatures.
The `critical
isotherm', the
isotherm at the
critical
temperature, is
at 31.04 C. The
critical point is
marked with a
star.
p  v  R  T  (1 Bp  Cp2  ...)
 B C

p  v  R  T   1  2  ... 
 v v

The third virial coefficient, C, is usually less
important than the second one, B, in the sense
that at typical molar volumes C/v2<<B/v. In
simple models, and for p  0, higher terms than
B are therefore often neglected.
The virial equation can be used to demonstrate
the point that, although the equation of state of
a real gas may coincide with the perfect gas law
as p  0, not all of its properties necessarily
coincide. For example, for a perfect gas dZ/dp = 0
(because Z=1 for all pressures), but for a real gas
dZ
 B  2pC  ...  B
dp
as p  0, and
dZ
B
d(1/ v)
as v  , corresponding to p  0.
Real gases: The Boyle temperature
Because the virial coefficients depend on the
temperature (see table above), there may be a
temperature at which Z1 with zero slope at low
pressure p or high molar volume v. At this
temperature, which is called the Boyle temperature,
TB, the properties of a real gas coinicide with those
of a perfect gas as p 0, and B=0. It then follows
that pvRTB over a more extended range of
pressures than at other temperatures.
The compression factor approaches 1 at low
pressures, but does so with different slopes.
For a perfect gas, the slope is zero, but real
gases may have either positive or negative
slopes, and the slope may vary with
temperature. At the Boyle temperature, the
slope is zero and the gas behaves perfectly
over a wider range of conditions than at other
temperatures.
Real gases: Condensation and critical point
Reconsider the experimental
isotherms of carbon dioxide.
What happens, when gas
initially in the state A is
compressed at constant
temperature (by pushing a
piston)?
• Near A, the pressure rises in approximate agreement with Boyle’s
law.
• Serious deviations from the law begin to appear when the
volume has been reduced to B.
• At C (about 60 bar for CO2), the piston suddenly slides in without
any further rise in pressure. Just to the left of C a liquid appears,
and there are two phases separated by a sharply defined surface.
• As the volume is decreased from C through D to E, the amount of
liquid increases. There is no additional resistance to the piston
because the gas can respond by condensation. The
corresponding pressure is the vapour pressure of the liquid at
this temperature.
• At E, the sample is entirely liquid and the piston rests on its
surface. Further reduction of volume requires the exertion of a
considerable amount of pressure, as indicated by the sharply
rising line from E to F. This is due to the low compressibility of
condensed phases.
The isotherm at the temperature Tc plays a special role :
• Isotherms below Tc behave as described above.
• If the compression takes place at Tc itself, a surface separating two phases does
not appear, and the volumes at each end of the horizontal part of the isotherm
have merged to a single point, the critical point of the gas. The corresponding
parameters are the critical temperature, Tc, critical pressure, pc, and critical
molar volume, vc, of the substance.
• The liquid phase of a substance does not form above Tc.
Real gases:
Critical constants, compression factors, Boyle temperatures,
and the supercritical phase
A gas can not be liquefied if the temperature
is above its critical temperature. To liquefy it
- to obtain a fluid phase which does not
occupy the entire volume - the temperature
must first be lowered to below Tc, and then
the gas compressed isothermally.
The single phase that fills the entire volume
at T> Tc may be much denser then is normally
considered typical of gases. It is often called
the supercritical phase, or a supercritical
fluid.
The van der Waals equation of gases: A model
Starting point: The perfect gas law pv = nRT
Correction 1:
Correction 2:
Attractive forces lower the
pressure
P= nRT – a n2
V- nb V2
P= RT
Vm- b
 replace p by (p+∏), where 
is the ‘internal pressure’. More
detailed analysis shows that
=a/v2.
Repulsive forces are taken into
account by supposing that the
molecules (atoms) behave as
small but impenetrable spheres
a, b: van der Waals coefficients
Comparison to the virial equation of state:
a
RT
– a
Vm2
Van der Waals equation
 replace v by (v-b), where b
is the ‘exclusion volume’. More
detailed analysis shows that b
is approximately the volume of
one mole of the particles.
Bb 
or
The surface of possible states
allowed by the van der Waals
equation.
Analysis of the van der Waals equation of gases
(3) The critical constants are related to the van
der Waals constants.
At the critical point the isotherm has a flat
inflexion. An inflexion of this type occurs if both
the first and second derivative are zero:
dp
RT
2a


0
2
3
dv
v
 v  b
Van der Waals isotherms at several values of T/Tc.
The van der Waals loops are normally replaced
by horizontal straight lines. The critical isotherm
is the isotherm for T/Tc = 1.
d2p
2RT
6a


0
dv 2  v  b 3
v4
at the critical point. The solution is
a
8a
v c  3b pc 
T

c
27b2
27Rb
(1) Perfect gas isotherms are obtained at high
enough temperatures and large molar volumes.
(2) Liquids and gases coexist when cohesive and
and the critical compression factor, Zc, is
dispersing effects are in balance. The ‘van der
Waals loops’ are unrealistic because they suggest predicted to be equal to
that under some conditions an increase in presure
pc v c 3
Z


c
results in an increase of volume. Therefore they are
RTc
8
replaced by horizontal lines
drawn so the loops define
for all gases.
equal areas above and below the
lines (‘Maxwell construction’)
Van der Waals constants of selected gases
The principle of corresponding states
Idea
b) Reduced melting temperature
Tm
Tc
If the critical constants are characteristic
properties of gases, than characteristic
points, like melting or boiling point, should
be unitary defined states. We therefore
introduce reduced variables
pr 
p
pc
vr 
v
vc
Tr 
c) Reduced boiling temperature
Tb
Tc
T
Tc
Examples
0.64 at 1.013 bar
d) Trouton’s rule (or: Pictet-Troutons’s rule)
A wide range of liquids gives
approximately the same standard entropy
of vaporization of
S  85 J K-1 mol-1
and obtain the reduced van der Waals
equation: pr  8Tr  32
3v r  1
0.44 at 1.013 bar
vr
a) Compression factors
But:

approximation!

works best for gases composed of spherical particles

fails, sometimes badly, when the particles are nonspherical or polar
The compression factors of four gases, plotted for three reduced
temperatures as a function of reduced pressure. The use of reduced
variables organizes the data on to single curves.
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