Transcript Chapter 3 Statistical thermodynamics

Chapter 3 Statistical thermodynamics
Content
3.1
3.2
3.3
3.4
3.5
3.6
Introduction
Boltzmann statistics
Partition function
Calculation of partition function
Contribution of Q to thermo_function
Calculation of ideal gas function
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 thermodynamics properties through the Stat. average.
According to some basic suppositions of the
substance structure,
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 substance’s thermodynamics properties.
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.
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.
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.
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 microcosmic state number is less than
the localized system.
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:
U  n11  n2 2     ni  i
i
3.1.7 Kinds of statistical system
Maxwell-Boltzmann statistics
usually called Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statistics
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
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
ε1，ε2，· · · , εi
one distributed form
N1，N2，· · · ，Ni
3.2.1.2
The microcosmic state number of this
partition is:
i  C  C
N1
N
N2
N  N1

N!
( N  N1 )!



N1 !( N  N1 )! N 2 !( N  N1  N 2 )!
N!
N!


N1 ! N 2 !  Ni !
i
(1)
3.2.1.3
There are many forms of partition,
the total microcosmic state number is:
N!
   i  
i
i  Ni !
i
(2)
No matter what partition, it has to
satisfy the following two conditions:
N  N
 N  U
i
(3)
i i
(4)
i
i
3.2.2 Most probable
distribution of localized system
The Ωi of every distribution is different
but there is a maximal value Ωmax
among them, in the macrocosmic system
which has enough particles, the whole
microcosmic number can approximately
be replaced by Ωmax, this is the most
probable distribution.
3.2.2.1
(
The problem is how to find out a
appropriate distribution Ni under two limit
conditions to make Ω the max one, in
mathematics, this is the question how to
work it out under the conditional limit of
Ni


formula (1). That is: i
i N i !
work out the extreme, make
N
i
i
 N ,  N i i  U
i
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 :

  i
Ni  e
The α and β in the formula is the non-fixed
gene which are brought in by the methods of
Lagrange multiply gene.
3.2.2.2 Most probable
distribution of localized system
Work it out by the mathematics methods:
N
1

e 



e
kT

i
i
or
  ln N  ln  e 
i
i
So the most probable distribution formula is:
  i / kT
e
Ni*  N
  i / kT
e

i
max
N!

*
N
 i!
i
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 microcosmic state number of energy level is called
the degeneration of that energy level, we use
gi to stand for it.
3.2.3.1
For example, the translation energy
formula of the gas molecule is:
2
h
2
2
2
i 
(nx  ny  nz )
3/ 2
8mV
The nx, ny and nz are the translation
quantum numbers which separately in
3.2.3.2
2
h
3
the x, y and z axis,  i 
3/ 2
8mV
so nx =1, ny =1 and nz =1, it only has
one probable state, so gi=1, it is nondegeneration.
3.2.3.3
When
h2
i 
6
3/ 2
8mV
nx
ny
nz
1
1
2
1
2
1
2
1
1
At this moment, under the situation
εi are the same, it has three different
microcosmic states, so gi=0.
3.2.4 Degeneration and
Microcosmic state number
Suppose one distribution of certain
localized system which has N particles:
Energy level
ε1， ε2，· · · ，εi
Every energy level degeneration g1，g2，· · · ，gi
One distributed form
N1，N2，· · · ，Ni
3.2.4.2
Choose N1 particles from N particles and then
put them in the energy level ε1, there are CN
selective methods;
But there are g1 different state in the ε1 energy
level, every particle in energy level ε1 has g1
methods, so it has gN11 methods;
Therefore, put N1 particles in energy level ε1,
it has gN11CN microcosmic number. Analogy in
turns, the microcosmic number of this distribution
methods is:
N1
N1
3.2.4.3
1  ( g  C )( g  C
N1
1
N1
N
N2
2
N2
N  N1
) 
( N  N1 )!
N!
N2
g 
 g2 

N !( N  N1 )!
N 2 !( N  N1  N 2 )!
N1
1
g
N1
1
g
N2
2
N!

N1 ! N 2 !   Ni !
giNi
 N !
i Ni !
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:
Ni
i
g
 (U ,V , N )   N !
i
i Ni !
The limit condition of sum still is:
 Ni  N  Ni  i  U
i
i
3.2.4.5
Use the most probable distribution
principle, ΣΩ1≈Ωmax , 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
N*i is:
3.2.4.6
 i / kT
g
e
Ni*  N i i / kT
 gi e
i
Compare with the most probable
distribution formula when we do not
consider degeneration , it has an
excessive item gi.
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!
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 nonlocalized system is:
Ni
i
g
1
 (U ,V , N ) 
N !

N! i
i Ni !
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 microcosmic state number is the maximal
one, the distribution form N*i (nonlocalized) is:
3.2.5.2 Most probable distribution of
non-localized system
  i / kT
gie
N N
 i / kT
 gie
*
i
i
It can be seen that the most probable
distribution formula of the localized and
non-localized system.
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:
Ni* gi e i / kT

*
  j / kT
N j g je
3.2.6.2 The other
form of Boltzmann formula
(2) Degeneration is not considered in the
classical mechanics, so the formula above is
i   j
N
ei / kT
  j / kT  exp(
)
kT
N
e
*
i
*
j
Suppose the lowest energy level is ε0,
εi - ε0 =Δεi , the particles in ε0 energy level
is N0, omit “*”, so the formula above can
be written as:
3.2.6.2 The other
form of Boltzmann formula
i / kT
Ni  N0e
p  p0e
 mgh / kT
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.
3.2.7 Entropy in Helmholz
free energy expression
According to the Boltzmann formula
which expose the essence of entropy
S  k ln   k ln max
N!
(1) for localized system, max 
*
N
non-degeneration
i
ln max  ln N !   ln Ni !
i
i
!
3.2.7.2 Entropy in Helmholz
free energy expression
Outspread of Stiring formula:
ln max  N ln N  N   Ni* ln Ni*   Ni*
 N ln N   N ln N
*
i
*
i
( N  N )
i
*
i
i
i
 N ln N   Ni* (   i )
i
 N ln N   N  U
 N ln  e
i
 i / kT
U

kT
i
(Ni*  e  i )
( Ni*  N ,
*
N
 i i  U )
i
i
(  ln N  ln  e
i
 i
1
,  )
kT
3.2.7.3 Entropy in Helmholz
free energy expression
ln max  N ln  e
i
 i / kT
U

kT
S (localized )  k ln max  kN ln e
 i / kT
U

T
F (localized )  U  TS   Nk ln  e
 / kT
i
i
3.2.7.4 Entropy in Helmholz
free energy expression
(2) for localized system, degeneration is gi
max
giNi
 N !
Ni !
i
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.
S (localized )  Nk ln  g i e  i / kT 
i
U
T
F (localized )   NkT ln  gi ei / kT
i
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:  ( g e ) 
  i / kT
S (non  localized )  k ln 


i
N
U
 T

i
N!
 ( gi e / kT ) N 

F (non  localized )  kT ln  i


N!


i
3.3 Partition function
3.3.1 definition
According to Boltzmann the most probable
distribution formula (omit mark “.”)
gi ei / kT
Ni  N
  i / kT
g
e
 i
i
Cause the sum item of the denominator is:
g e
i
i
i / kT
q
3.3 Partition function
q is called molecule partition function,
or partition function, its unit is 1. The
e-εi/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.
3.3.1.2 Definition
The comparison of any item in q;
The comparison of any two items
in q.
  i / kT
Ni gi e

N
q
  i / kT
Ni
gi e

N j g j e  j / kT
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 (εr), Vibration energy (εv), electron
energy (εe) and atom nucleus energy (εn), all of
the energy can be considered to be independent.
 t  r   v  e  n
3.3.2.2 Separation of partition function
The total energy of molecule is equal to
the summation of every energy
 i   i , t   i (inner)
  i ,t   i,t   i ,v   i ,e   i ,n
Every different energy has corresponding
degeneration, when the total energy is εi ,
the total degeneration is equal to the product
of every energy degeneration, that is:
gi  gi ,t  gi ,r  gi ,v  gi ,e  gi ,n
3.3.2.3 Separation of partition function
According to the definition of partition
function, put the expressions of εi and gi
into it, then we can get:
q   gi exp(
i
i
kT
)
  gi ,t gi ,r gi ,v gi ,e gi ,n exp(
i
 i ,t   i ,r   i ,v   i ,e   i ,n
kT
)
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:
3.3.2.4 Separation of partition function
q  [ gi ,t exp(
 i ,t
i
[ gi ,v exp(
i
[ gi ,n exp(
i
kT
 i ,v
kT
 i ,n
kT
)]  [ gi ,r exp(
 i ,r
i
)]  [ gi ,e exp(
i
kT
)] 
 i ,e
kT
)] 
)]
 qt  qr  qv  qe  qn
qt, qr, qv, qe and qn are separately called
Translation, Turn, Vibration, Electron, and
atomic nucleus partition functions.
3.3.2.5 Separation of partition function
Suppose the total particles is N
(1) Helmholz free energy F
F (non  localized)  kT ln[
( g i e
 / kT N
i
qN
 kT ln[ ]
N!
N!
i
)
]
3.3.3 Relation between Q
and thermodynamics function
F
(2) entropy S dF  SdT  pdV S  ( T )V , N
qN
 ln q
S (non  localized )  k ln[
]  NkT (
)V , N
N!
T
Or we can get the following formula directly
according to the entropy expression which was
get before.
 ( g e 
) 
q  U

 U

S (non  localized )  k ln
i
/ kT
N
i


i
N!



N
 k ln 

T
 N!  T
3.3.3.2 Relation between Q
and thermodynamics function
(3) thermodynamic energy U
U  F  TS
qN
qN
2  ln q
 kT ln[ ]  kT ln[ ]  NkT [
]V , N
N!
N!
T
 ln q
 NkT [
]V , N
T
Or the formula can be get from the comparison
of two expressions of S (non-localized)
2
3.3.3.3 Relation between Q
and thermodynamics function
(4) Gibbs free energy G
dF  SdT  pdV
 F 
 Inq 
p     NkT 

 V 
 V 
T, N
T, N
according to definition, G=F+pV, therefore:
qN 
 Inq 
G  kTIn   NkTV 

 V  r ,n
 N! 
3.3.3.4 Relation between Q
and thermodynamics function
(5) enthalpy H
H  U  pV
 ln q
 ln q
 NkT [
]V , N  NkTV (
)T , N
T
V
(6) heat capacity under constant volume
U

2  ln q
CV  (
)V 
[ NkT (
)V , N ]V
T
T
T
2
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 thermodynamics function can be worked
out.
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:
F (localized )  kTInqN
or
Inq
S (localized )  NkInq  NkT (
)V , N
T
U
S (loclized )  Nk ln q 
T
3.3.4.2 Relation between Q
and thermodynamics function
 ln q
U (localized )  NkT (
)V , N
T
2
 ln q
G (localized )   NkT ln q  NkTV (
)T , N
V
 ln q
 ln q
H (localized )  NkT (
)V , N  NkTV (
)T , N
T
V
2

2  ln q
CV (localized ) 
[ NkT (
)V , N ]V
T
T
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.
3.4 Calculation of partition function
Atomic nucleus partition function
Electron partition function
Translation partition function
Turn partition function
Vibration partition function
3.4.1 Partition function of atomic nucleus
qn  g n,0 exp(
 g n,0 exp(
 n,0
kT
 n,0
kT
)  g n,1 exp(
)[1 
g n,1
g n,0
 n,1
kT
exp(
)  
 n,1   n,0
kT
)  ]
The εn,0 εn,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.
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:
qn  g n,0 exp(
 n,0
kT
)
3.4.1.2 Partition function of electrons
If the energy of the nucleus ground
state energy level is chose as zero:
qn  gn,0  2sn  1
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.
3.4.2 Partition function of electrons
qe  ge,0 exp(
 ge,0 exp(
 e,0
kT
 e,0
kT
)  ge,1 exp(
)[1 
ge,1
ge,0
exp(
 e,1
kT
)  
 e,1   e,0
kT
)  ]
The electron energy interval is also very
large, ( εe,1-εe,0 )=400 kJ.mol-1 , except for F, Cl
minority elements, the second item in the
bracket is also be ignored.
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:
qe  ge,0 exp(
 e,0
kT
)
3.4.2.2 Partition function of electrons
qe  ge,0 exp(
 e,0
kT
)
If εe,0 is considered as zero, therefore
qe=g e,0=2j+1, j in the formula is electron
total momentum quantum number.Electron
total momentum distance which moves
around nucleus is also quantitative,
3.4.2.2 Partition function of electrons
the heft along certain chosen axis probably
has 2j+1 tropism.
Some freeness atom and steady ionic
j=0, g e,0 =1, are non-degeneration. If there
is a non-match electron, it probably has
two different spin, such as Na, its j=1/2,
g e,0 =2.
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:
2
2
x
2
n
2
y
2
2
z
2
h n
n
 i ,t 
(   )
8m a b
c
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, … , ∞ .
qt   gi ,t exp(
i
 i ,t
kT
)
3.4.3.2 Translation partition function
Put εi,t into it:



2
x
2
n y2
h n
nz2
qt     exp[
(  2  2 )]
8m a
b
c
nx 1 n y 1 nz 1

2
2
2
x
2

2
n y2
n
h
h
  exp(
 )   exp(
 2 )
8mkT a ny 1
8mkT b
nx 1

h2 nz2
exp(
 2 )  qt,x  qt,y  qt,z

8mkT c
nz 1
3.4.3.2 Translation partition function
Because for all quantum number
work out the summation from 0 ~ ∞ ,
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.
3.4.3.3 Translation partition function

h2 nx2
qt, x   exp(
 2)
8mkT a
nx 1

  exp( 2 n x2 ) (suppose
n x 1
h2
2

α
)
2
8m kTa
Because α2 is a very little value, the mark
of sum can be replaced by the mark of integral,
so:

qt,x   exp( 2 nx2 )dnx
0
3.4.3.4 Translation partition function

Cite the integral formula: 0 e
Then the formula turns to:
 x 2
1  12
dx  ( )
2 
1
1
2 mkT 1 2
2
qt,x 
( )  (
) a
2
2
h
qt,y and qt,z have the same expressions,
just a is turned to b or c, so:
2 mkT 3 2
2 mkT 3 2
qt  (
h
2
)
 a bc  (
h
2
)
V
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.
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:
 r  J ( J  1)
h2
8 I
2
J  0，
1，
2，

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.
m1m2
2
I (
)r
m1  m2
3.4.4.2 Turn partition function
The tropism of Turn angel momentum
is also quantitative , so the energy level
degeneration is g  2 J  1
i ,r
qr   gi ,r exp( 
i
 i ,r
kT
)
J ( J  1)h
  (2 J  1) exp(
)
2
8 IkT
J 0

2
3.4.4.2 Turn partition function
Make Qr 
h
2
8 2 Ik
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:
J ( J  1)Qr
qr   (2 J  1) exp(
)
T
J 0

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 T<<1,
therefore we use the mark of integral
instead of the mark of summation, and
make x=J(J+1), dx=(2J+1)dJ, put them
into it, then we can get:
3.4.4.3 Turn partition function
qr  

0

J ( J  1)Qr
(2 J  1) exp(
)dJ
T

0
xQr
exp( 
)dx
T
T 8 2 IkT


Qr
h2
3.4.4.4 Turn partition function
(2) The qr of some nucleus double
atoms and linearity multi-atom molecules
( σ is symmetry number, the microcosmic
state repeated time when it spins 360°)
8 IkT
qr 
2
h
2
3.4.4.4 Turn partition function
(3) The qr of non-linearity multi-atom
molecules
8 (2 kT )
qr 
3
h
2
3
2
(I x  I y  I z )
1
2
Ix, Iy and Iz separately are Turn
inertia in the three axes.
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,v=1,its vibration energy is
1
 v  (v  )h
2
v  0,1, 2,   
υ in the formula is Vibration quantum number,
when υ =0, εv,0 is called zero Vibration energy.
 v,0
1
 h
2
3.4.5.2 Vibration partition function
qv  
i
1
(v  )h

 i ,v
2
]
gi ,v exp(
)   exp[
kT
v 0
kT
Cause Qv=hv/k, Qv is called the Vibration
character temperature,it also has temperature
dimension, so:
Qv
3Qv
5Qv
qv  exp( )  exp(
)  exp(
)  
2T
2T
2T
Qv
Qv
2Qv
 exp( )  [1  exp( )  exp(
)  ]
2T
T
T
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 Qv=310K,
3.4.5.3 Vibration partition function
therefore the item υ =1 can not be ignored.
Under the condition of low temperature,
Qv
 1
T
Qv
exp( )  1
T
Cite the mathematic similar formula:
1
2
x  1 时， 1  x  x     
1 x
3.4.5.4 Vibration partition function
So the expression of qv is:
qv  exp(
v
2T
)
1
v
1  exp(
)
T
h
1
 exp(
)
2kT 1  exp( h )
kT
We regard the zero Vibration energy as
1
zero, that is εv,0=1/2hv=0, so: qv' 
h
1 exp(
kT
)
3.4.5.5 Vibration partition function
(2) qv of the multi-atom molecule
The Vibration liberty degree fv of multiatom molecule is: f v  3n  f t  fr
ft is Translation liberty degree, fr is Turn
liberty degree, n is total atom.
Therefore, the qv of the linearity multiatom molecule is:
3.4.5.5 Vibration partition function
hv
)
3 n 5
2kT
qv (linear )  
hv
i 1 1  exp( 
)
kT
exp( 
The qv of non-linearity multi-atom molecule
only need change (3n-5) to (3n-6).
3.5 Contribution of Q
to thermodynamic function
Contribution of atomic nucleus partition
function
Contribution of electron partition function
Contribution of Turn partition function
3.5.1 atomic nucleus’
Usually in the chemical reaction, nucleus is
always in ground state,

qn  g n,0 exp(
n,0
kT
)
If the ground state energy is chose as zero,
so:
q  g  2s  1
n
n,0
n
Sn is the nucleus spin quantum number, it
has nothing to do with the system temperature
and volume.
3.5.1.2 atomic nucleus’
qn has no contributions to thermodynamic energy, enthalpy and molar heat
capacity under constant volume, that is:
2  ln qn
U n  NkT (
)V , N  0
T
 ln qn
2  ln qn
H n  NkTV (
)T , N  NkT (
)V , N  0
V
T
U n
CV ,n  (
)V  0
T
3.5.1.3 atomic nucleus’
qn has little contributions to Fn, Sn
and Gn, that is:
Fn=-NkTInqn
Sn=NkInqn
Gn=-NkTInqn
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.
3.5.2 Electrons’
Usually electron is in ground state, and we
choose the ground energy as zero, so:
qe  ge,0  2 j  1
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:
Ue  He  Cv,e  0
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
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.
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.
2 mkT 3 2
) V
As it is known qt  (
2
h
For the non-localized system which has N
particles,calculate the contribution which is done
to thermodynamics function by qt.
3.5.3.2 Turn’
(1) Turn Helmholtz free energy
N
(qt )
Ft   kTIn
N!
2πm kT 3 / 2
  NkTIn(
) V  NkTInN  NkT
2
h
3.5.3.3 Turn’
(2) Turn entropy
Because
dF  SdT  pdV
Ft
St  ( ) V , N
T
2πm kT 3 / 2
2
 NK [ In(
)
h
q 5
 Nk[ In t  ]
N 2
5
V  InN  ]
2
This is called Sackur-Tetrode formula.
3.5.3.4 Turn’
(3) Turn thermodynamic energy
Ut  Ft  TSt
2  ln qt
 NkT (
)V , N
T
(4) Turn isometric heat capacity
CV ,t
U t
3
(
)V  Nk
T
2
3.5.3.5 Turn’
(5) Turn enthalpy and turn Gibbs free energy
Ht  Ut  pV
Gt  Ft  pV
F
p  ( )T , N
V
Put in the corresponding expressions Ut,
Ft then we can get turn enthalpy and turn
Gibbs free energy.
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:
3.6.1 Helmholtz free energy
F  Fn  Fe  Ft
N
t
q
  NkT ln qn  NkT ln qe  kT ln
N!
 n,0
  NkT [ g n,0 exp(
NkT [ g e,0 exp(
kT
 e,0
)] 
)] 
kT
2 mkT 3 2
NkT ln[(
) V ]  kT ln N !
2
h
3.6.1.2 Helmholtz free energy
 ( N  n,0  N  e,0 )  NkT ln g n,0 g e,0 
2 mkT 3 2
NkT ln[(
) V ]  NkT ln N  NkT
2
h
Both of the 1,2 items can be expurgated,
when ∆F is being calculation.
F
S  ( )V,N
T
3.6.2 S
2πmk 3 / 2
 Nk[ Ingn , 0 ge,0  In(
)

InV

InN
2
h
3
5
 InT  ]
2
2
This formula is also called Sachur-Tetrode formula
3.6.3 U
Because qn, qe are no useful for thermodynamics, only Turn energy has contribution
to it, so:
 ln qt
U  U t  NkT (
)V , N
T
3
 NkT
2
2
3.6.4 Cv
CV  CV ,t
U t
3
(
)V , N  Nk
T
2
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.
3.6.5 State equation of ideal gases
dF
p  ( ) T , N
dV
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.
3.6.5 State equation of ideal gases
InV
NkT
p  NkT[
]T , N 
V
V
The equation of ideal gas can be
educed by the stat. Thermodynamics
methods, this is the classical thermodynamics which it can not do.
ending