Chapter 3 Statistical thermodynamics

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Chapter 3 Statistical thermodynamics
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Content

• 3.1 Introduction
• 3.2 Boltzmann statistics
• 3.3 Partition function
• 3.4 Calculation of partition function
• 3.5 Contribution of Q to thermo_function
• 3.6 Calculation of ideal gas function

3
3.1 Introduction

• 3.1.1 Method and target
• According to the Stat. Unit
  mechanic properties (such as rate, momentum,
  vibration) are related with the system
  microcosmic and macrocosmic properties, work out
  the thermo-dynamics properties through the
  Stat. average. According to some basic
  suppositions of the substance structure,

4

• and the spectrum data which get from
  the experiments, we can get some basic
  constant of the substance structure, such
  as the space between the nucleus, bond
  angle, vibration frequency and so on to
  work out the molecule partition function.
  And then according to the partition function we
  can work out the substances thermo-dynamics
  properties.

5
3.1.2 Advantage

• Related with the system microcosmic
  and macrocosmic properties, it is satisfied
  for some results we get from the simple
  molecule. No need to carry out the
  complicated low temperature measured heat
  experiment, then we can get the quite
  exact entropy.

6
3.1.3 Disadvantage

• The structure model must be supposed
  when calculating, certain approximate
  properties exist for large complicated
  molecules and the agglomerated system, it still
  has some difficulties in calculating.

7
3.1.4 Localized system

• Particles can be distinguished from
  each other. For example, in the crystal,
  particles vibrate in the local crystal
  position, every position can be imagined
  to have different numbers to be
  distinguished, so the micro-cosmic state
  number of localized system is very large.

8
3.1.5 Non-localized system

• Basic particles can not be
  distinguished from each other. Such as, the
  gas molecule can not be distinguished from
  each other. When the particles are the
  same,its micro- cosmic state number is
  less than the localized system.

9
3.1.6 Assembly of independent particles

• The reciprocity of the particles
  is very faint, therefore it can be
  ignored, the total energy of the system is
  equal to the summation of every
  particles energy, that is

10
3.1.7 Kinds of statistical system

• Maxwell-Boltzmann statistics
• usually called Boltzmann statistics
• Bose-Einstein statistics
• Fermi-Dirac statistics

11
3.2 Boltzmann Statistics

• Microcosmic state number of localized system
• Most probable distribution of localized
  system
• Degeneration
• Degeneration and Microcosmic state number
• Most probable distribution of non-localized
  system
• The other form of Boltzmann formula
• Entropy in Helmholz free energy expression

12
3.2.1 Microcosmic state number of localized
system

• One macrocosmic system which
  is consisted by N independent particles which
  can be extinguished, it has many different
  partition forms in the quantitative
  level. Suppose one of the
  partition forms is

energy level e1,e2, ,
ei one distributed form N1,N2, ,Ni
13
3.2.1.2

• The microcosmic state number of this
  partition is

14
3.2.1.3

• There are many forms of partition,
  the total microcosmic state number is

No matter what partition, it has to
satisfy the following two conditions
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3.2.2 Most probable distribution of
localized system

• The Oi of every distribution is
  different but there is a maximal value
  Omax among them, in the macrocosmic system
  which has enough particles, the whole
  microcosmic number can approximately be
  replaced by Omax, this is the most
  probable distribution.

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3.2.2.1

• The problem is how to find
  out a appropriate distribution Ni under two
  limit conditions to make O the max
  one, in mathematics, this is the question
  how to work it out under the conditional
  limit of formula (1). That is

(
work out the extreme, make
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3.2.2.2

• Firstly, outspread the factorial by
  the String formula, then use the methods
  of Lagrange multiply gene, the most
  probable distribution we get

• The a and ß in the formula is the
  non-fixed gene which are brought in by
  the methods of Lagrange multiply gene.

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3.2.2.2 Most probable distribution of
localized system

• Work it out by the mathematics methods

or
So the most probable distribution formula
is
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3.2.3 Degeneration

• Energy is quantitative, but probably
  several different quanta state exist in
  every energy level, the reflection on
  spectrum is that the spectrum line of
  certain energy level usually consisted by
  several very contiguous exact spectrum line
  .
• In the quanta mechanics,the probable
  micro- cosmic state number of energy level
  is called the degeneration of that energy
  level, we use gi to stand for it.

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3.2.3.1

• For example, the translation energy
  formula of the gas molecule is

The nx, ny and nz are the translation
quantum numbers which separately in
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3.2.3.2

• the x, y and z axis,

• so nx 1, ny 1 and nz 1, it only
  has
• one probable state, so gi1, it is
  non-degeneration.

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3.2.3.3
When

• At this moment, under the situation
  ei are the same, it has three different
  microcosmic states, so gi0.

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3.2.4 Degeneration and Microcosmic state
number

• Suppose one distribution of certain
• localized system which has N particles

Energy level e1,
e2, ,ei Every energy level degeneration
g1,g2, ,gi One distributed form
N1,N2, ,Ni
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3.2.4.2

• Choose N1 particles from N particles
  and then put them in the energy level e1,
  there are CNN1
• selective methods
• But there are g1 different state in
  the e1 energy level, every particle in
  energy level e1 has g1 methods, so it
  has gN11 methods
• Therefore, put N1 particles in
  energy level e1, it has gN11CNN1
  microcosmic number. Analogy in turns, the
  microcosmic number of this distribution
  methods is

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3.2.4.3

26
3.2.4.4

• Because there are many
  distribution forms, under the situation
  which U, V and N are definite, the total
  microcosmic state numbers are

The limit condition of sum still is
27
3.2.4.5

• Use the most probable distribution
  principle, SO1Omax , use the Stiring formula
  and Lagrange multiply gene method to work
  out condition limit, when the microcosmic
  state number is the maximal one, the
  distribution form Ni is

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3.2.4.6

• Compare with the most probable
  distribution formula when we do not consider
  degeneration , it has an excessive
  item gi.

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3.2.5 Most probable distribution of
non-localized system

• Because particles can not be
  distinguished in the non-localized system,
  the microcosmic number which distribute in
  the energy level is less than the
  localized system, so amend the equal
  particle of the localized system microcosmic
  state number formula, that is the
  calculation formula divides N!

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3.2.5 Most probable distribution of
non-localized system

• Therefore, under the condition that
  U, V and N are the same, the total
  microcosmic state number of the non-localized
  system is

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3.2.5.2 Most probable distribution of
non-localized system

• Use the most probable distribution
  principle, use the Stiring formula and Lagrang
  multiply gene method to work out the condition
  limit,when the micro-cosmic state number is
  the maximal one, the distribution form Ni
  (non-localized) is

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3.2.5.2 Most probable distribution of
non-localized system

• It can be seen that the most
  probable distribution formula of the
  localized and non-localized system.

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3.2.6 The other form of Boltzmann formula

• (1) Compare the particles of energy
  level i to j, use the most probable
  distribution formula to compare, expurgated
  the same items, then we can get

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3.2.6.2 The other form of Boltzmann
formula

• (2) Degeneration is not considered
  in the classical mechanics, so the formula
  above is

Suppose the lowest energy level is
e0, ei - e0 ?ei , the particles in e0
energy level is N0, omit , so the
formula above can be written as
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3.2.6.2 The other form of Boltzmann
formula

• This formula can be used
  conveniently, such as when we discuss the
  distribution of pressure in the gravity
  field, suppose the temperature is the same
  though altitude changes in the range from 0 to h,
  then we can get it.

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3.2.7 Entropy in Helmholz free energy
expression

• According to the Boltzmann formula which
  expose the essence of entropy

(1) for localized system,
non-degeneration
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3.2.7.2 Entropy in Helmholz free
energy expression

• Outspread of Stiring formula

38
3.2.7.3 Entropy in Helmholz free
energy expression

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3.2.7.4 Entropy in Helmholz free
energy expression

• (2) for localized system, degeneration
  is gi

The deduce methods is similar with the
previous one,among the results we get, the
only excessive item than the result of (1)
is item gi.
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3.2.7.5 Entropy in Helmholz free
energy expression

• (3) for the non-localized system
• Because the particles can not be
  distinguished, it need to equally amended,
  divide N! in the corresponding localized
  system formula, so

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3.3 Partition function

• 3.3.1 definition
• According to Boltzmann the most
  probable distribution formula (omit mark
  .)

Cause the sum item of the denominator
is
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3.3 Partition function

• q is called molecule partition
  function, or partition function, its unit
  is 1. The e-ei/kT in the sum item is called
  Boltzmann gene.The partition function q is
  the sum of every probable state Bolzmann
  gene of one particle in the system, so q is
  also called state summation.

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3.3.1.2 Definition

• The comparison of any item in q
• The comparison of any two items in q.

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3.3.2 Separation of partition function

• The energy of one molecule is
  considered as the summation of the
  Translation energy of whole particles motion
  and the inner motion energy of the molecule.
• The inner energy concludes the
  Translation energy (er), Vibration energy
  (ev), electron energy (ee) and atom
  nucleus energy (en), all of the energy can
  be considered to be independent.

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3.3.2.2 Separation of partition function

• The total energy of molecule is equal to
  the summation of every energy
• Every different energy has
  corresponding degeneration, when the total
  energy is ei , the total degeneration is
  equal to the product of every energy
  degeneration, that is

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3.3.2.3 Separation of partition function

• According to the definition of
  partition function, put the expressions of
  ei and gi into it, then we can get

47
3.3.2.3 Separation of partition function

• It can be proved in the
  mathematics, the product summation of
  several independent variables is equal to
  the separate product summation, so the
  formula above can be written as

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3.3.2.4 Separation of partition function

• qt, qr, qv, qe and qn are
  separately called Translation, Turn,
  Vibration, Electron, and
• atomic nucleus partition functions.

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3.3.2.5 Separation of partition function

• Suppose the total particles is N
• (1) Helmholz free energy F

50
3.3.3 Relation between Q and thermodynamics
function

• (2) entropy S

Or we can get the following formula
directly according to the entropy expression
which was get before.
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3.3.3.2 Relation between Q and
thermodynamics function

• (3) thermodynamic energy U

Or the formula can be get from the
comparison of two expressions of S
(non-localized)
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3.3.3.3 Relation between Q and
thermodynamics function

• (4) Gibbs free energy G

according to definition, GFpV, therefore
53
3.3.3.4 Relation between Q and
thermodynamics function

• (5) enthalpy H

(6) heat capacity under constant volume
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3.3.3.4 Relation between Q and
thermodynamics function

• According to the expressions
  above, only if the partition function is
  known, the value of the thermo- dynamics
  function can be worked out.

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3.3.4 Relation between Q and thermodynamics
function

• According to the method which
  the relationship of non-localized system and
  thermodynamics function is the same, we
  can get

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3.3.4.2 Relation between Q and
thermodynamics function

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3.3.4.3 Relation between Q and
thermodynamics function

• It can be seen from the formulas
  above U, H and the expression of Cv are
  the same in the localized and non-localized
  system
• However, in the expressions of F,
  S and G, compared with the localized system,
  it lacks the relational 1/N! constant,
  but it can be expurgated each other
  when we calculate the change of the
  functions. This chapter mainly discusses
  non-localized.

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3.4 Calculation of partition function

• Atomic nucleus partition function
• Electron partition function
• Translation partition function
• Turn partition function
• Vibration partition function

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3.4.1 Partition function of atomic nucleus

• The en,0 en,1 in the formula
  separately stand for the atom nucleus energy
  which is in the ground and the first
  excited state, gn,0 gn,1 separately
  stand for the degeneration of the
  corresponding level.

60
3.4.1.2 Partition function of electrons

• Because in the chemical reaction,
  nucleus is always in the ground state,
  otherwise the energy level interval between
  the ground and the first excited state is very
  large,so commonly all the items after the second
  one in the bracket are ignored, so

61
3.4.1.2 Partition function of electrons

• If the energy of the nucleus
  ground state energy level is chose as
  zero

That is the atom nucleus partition
function is equal to the ground state
degeneration, it comes from the nucleus
spin effect. Sn in the formula is
the nucleus spin quantum number.
62
3.4.2 Partition function of electrons

• The electron energy interval is also
  very large, ( ee,1-ee,0 )400 kJ.mol-1 ,
  except for F, Cl minority elements, the
  second item in the bracket is also be
  ignored.

63
3.4.2 Partition function of electrons

• Though the temperature is very
  high, the electron is also probably be
  excited, but usually the electron is not
  excited, the molecule has been
  decomposed.Therefore, usually the electron
  is always in the ground state, so

64
3.4.2.2 Partition function of electrons

• If ee,0 is considered as zero,
  therefore qeg e,02j1, j in the formula is
  electron total momentum quantum number.Electron
  total momentum distance which moves around
  nucleus is also quantitative,

65
3.4.2.2 Partition function of electrons

• the heft along certain chosen axis probably
  has 2j1 tropism.
• Some freeness atom and steady
  ionic j0, g e,0 1, are non-degeneration. If
  there is a non-match electron, it probably
  has two different spin, such as Na, its
  j1/2, g e,0 2.

66
3.4.3 Translation partition function

• Suppose the particle which quality
  is m moves in the cubic system which
  volume is a.b.c, according to the
  Translation energy expression which is get
  from the fluctuation equation

67
3.4.3 Translation partition function

• h in the formula is plank
  constant, nx, ny, nz is the Translation
  quantum number which are in the x, y, z
  axis, its value is positive integer 1, 2,
  , 8 .

68
3.4.3.2 Translation partition function

• Put ei,t into it

69
3.4.3.2 Translation partition function

• Because for all quantum number
  work out the summation from 0 8 , it
  concludes all of the states, the item gi,t
  will not appear in the formula. The
  Translation partition function in the three
  axes is analogous, here we just explain one
  qt,x of them, others can be analogy.

70
3.4.3.3 Translation partition function

• Because a2 is a very little value,
  the mark of sum can be replaced by the
  mark of integral, so

71
3.4.3.4 Translation partition function

• Cite the integral formula

Then the formula turns to
qt,y and qt,z have the same
expressions, just a is turned to b or c,
so
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3.4.4 Turn partition function

• The Turn partition function of
  single atom molecule is zero, qr of
  different nucleus double atoms molecules,
  the same nucleus double atoms molecules
  and linearity multi-atom molecules have
  analogous forms, but the qr expression of
  non-linearity multi-atom molecules is more
  complicated.

73
3.4.4 Turn partition function

• (1) The qr of different nucleus
  double atoms molecule, suppose it is a rigid
  rotor and turns around the centroid, its
  energy level formula is

74
3.4.4 Turn partition function

• J in the formula is the Turn
  energy level quantum number, I is the
  Turn inertia,suppose the double atoms quantity
  are m1, m2, r is nucleus interval.

75
3.4.4.2 Turn partition function

• The tropism of Turn angel momentum
  is also quantitative , so the energy level
  degeneration is

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3.4.4.2 Turn partition function

• Qr is called Turn character
  temperature, because the right side of the
  formula has the dimension of temperature.
  Put Qr into qr expression, then we can
  get

Make
77
3.4.4.3 Turn partition function

• Work out the Qr from the Turn
  inertia I. Except H2, the Qr of most
  molecules is very small, Qr Tltlt1,
  therefore we use the mark of integral
  instead of the mark of summation, and make
  xJ(J1), dx(2J1)dJ, put them into it,
  then we can get

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3.4.4.3 Turn partition function

79
3.4.4.4 Turn partition function

• (2) The qr of some nucleus
  double atoms and linearity multi-atom molecules
  ( s is symmetry number, the microcosmic state
  repeated time when it spins 360)

80
3.4.4.4 Turn partition function

• (3) The qr of non-linearity
  multi-atom molecules

Ix, Iy and Iz separately are Turn
inertia in the three axes.
81
3.4.5 Vibration partition function

• (1) The qv of double atoms molecule
• suppose the molecule only does
  one kind of simple Vibration which rate is
  V, the Vibration is non-degeneration, g
  i,v1,its vibration energy is

? in the formula is Vibration quantum
number, when ? 0, ev,0 is called zero
Vibration energy.
82
3.4.5.2 Vibration partition function

• Cause Qvhv/k, Qv is called the
  Vibration character temperature,it also has
  temperature dimension, so

83
3.4.5.3 Vibration partition function

• Vibration character temperature is
  one of the important properties, the higher
  Qv is, the smaller percentage of the
  excited state is, the second item and the
  items after it in the qv expression can be
  ignored.
• The Qv of some molecule are
  lower, such as iodine Qv310K,

84
3.4.5.3 Vibration partition function

• therefore the item ? 1 can not be
  ignored.
• Under the condition of low temperature,

Cite the mathematic similar formula
85
3.4.5.4 Vibration partition function

• So the expression of qv is

We regard the zero Vibration energy as
zero, that is ev,01/2hv0, so
86
3.4.5.5 Vibration partition function

• (2) qv of the multi-atom molecule
• The Vibration liberty degree fv of
  multi-atom molecule is

• ft is Translation liberty degree, fr
  is Turn liberty degree, n is total atom.
• Therefore, the qv of the linearity
  multi-atom molecule is

87
3.4.5.5 Vibration partition function

• The qv of non-linearity multi-atom
  molecule
• only need change (3n-5) to (3n-6).

88
3.5 Contribution of Q to thermodynamic
function

• Contribution of atomic nucleus partition
  function
• Contribution of electron partition
  function
• Contribution of Turn partition function

89
3.5.1 atomic nucleus

• Usually in the chemical reaction,
  nucleus is always in ground state,

If the ground state energy is chose as
zero, so
Sn is the nucleus spin quantum number,
it has nothing to do with the system
temperature and volume.
90
3.5.1.2 atomic nucleus

• qn has no contributions to
  thermo-dynamic energy, enthalpy and molar heat
  capacity under constant volume, that is

91
3.5.1.3 atomic nucleus

• qn has little contributions to Fn,
  Sn and Gn, that is
• Fn-NkTInqn
• SnNkInqn
• Gn-NkTInqn

92
3.5.1.3 atomic nucleus

• When we calculate the changing
  value, this item will be expurgated, so
  we will ignore the contribution of qn.
  Only when we calculate the prescribed
  entropy, the contribution of qn has to
  be considered.

93
3.5.2 Electrons

• Usually electron is in ground
  state, and we choose the ground energy as
  zero, so

Because the total angle momentum quantum
number j of electron has nothing to do
with temperature and volume, qe has no
contribution to thermodynamics, enthalpy and
isometric heat capacity, that is
94
3.5.2.2 Election

• qe has little contributions to Fe,
  Se , Ge, that is
• Fe (non-localized)-NkTInqe
• Se (non-localized)NkInqe
• Ge (non-localized)-NkTInqe

95
3.5.2.2 Election

• Except for Se, when we
  calculate the changing value of Fe and
  Ge, this item also can be expurgated
  commonly if the first excited state of
  election can not be ignored and the ground
  state is not equal to zero,so the whole
  expression of qe must be put it into to
  calculate.

96
3.5.3 Turn

• Because the interval of Turn
  energy level is very little, Turn partition
  function has great contributions to
  thermodynamics function, such as entropy and
  so on.
• As it is known

• For the non-localized system which
  has N particles,calculate the contribution
  which is done to thermodynamics function by
  qt.

97
3.5.3.2 Turn

• (1) Turn Helmholtz free energy

98
3.5.3.3 Turn

• (2) Turn entropy
• Because

This is called Sackur-Tetrode formula.
99
3.5.3.4 Turn

• (3) Turn thermodynamic energy

• (4) Turn isometric heat capacity

100
3.5.3.5 Turn

• (5) Turn enthalpy and turn Gibbs free
  energy

Put in the corresponding expressions Ut,
Ft then we can get turn enthalpy and
turn Gibbs free energy.
101
3.6 Calculation of thermodynamic function
for Single atom ideal gas

• Because the inter motion of single
  atom molecule has no Translation and
  Vibration, only the atom nucleus, electron
  and outer Turn have contributions to
  thermodynamics.
• Ideal gas is localized system, so a
  series of its thermodynamics are showed by
  the partition function calculation formulas
  as following

102
3.6.1 Helmholtz free energy

103
3.6.1.2 Helmholtz free energy

• Both of the 1,2 items can be
  expurgated, when ?F is being calculation.

104
3.6.2 S

• This formula is also called
  Sachur-Tetrode formula

105
3.6.3 U

• Because qn, qe are no useful for
  thermo-dynamics, only Turn energy has
  contribution to it, so

106
3.6.4 Cv

• The conclusion is the same with
  the result of the classical energy share
  theory, single atom molecule only has three
  translation liberty degree, every liberty
  degree contribute 1/2k, then N particles
  total have 3/2Nk.

107
3.6.5 State equation of ideal gases

• Put the expression of F into it,
  because other items have nothing to do with
  volume only one item has relationship
  with V in translation item, put it
  in, and then we can get the state
  equation of ideal gas.

108
3.6.5 State equation of ideal gases

• The equation of ideal gas can be
  educed by the stat. Thermodynamics methods,
  this is the classical thermo- dynamics which
  it can not do.

ending