Transcript Slide 1

chapter four
the statistical interpretation of
entropy
4-1Introduction
4.2 Entropy and disorder on the atomic scale
4.3 the concept of microstate
4- 4 Determination of the most probable
4.5 the effect of temperature
4.6 thermal equilibrium within asystem and the
boltzman equation
4.7 heat flow and the production of entropy
4.8 configurationally entropy and thermal
entropy
4.1. Introduction
the introduction of internal energy as a
state function was achieved as the result of
realization of the impossibility of the
creation of the perpetual motion machine
of the first kind ;
this is a direct result of the first law of the
thermodynamics which state that " though
energy can be transformed from one form
to another ;it cannot be created or
destroyed" .
*
the introduction of entropy as a state function was
achieved as the result of realization that there exist
possible and impossible processes ; and by examination
of the relationship which occur between the heat and
work effect of these processes .
* in spite of the fact that ; within the scope of classical
thermodynamics ; both internal energy (U); and entropy
(S); properties are simply mathematical functions of the
state of a system ;the physical significances of internal
energy is readily understandable and evidence by the
rapid acceptance of the first law of thermodynamics ;
the physical interpretation of entropy has to await the
development of statistical mechanics and the invention of
quantum theory
*
4.2 Entropy and disorder on the
atomic scale :
Gibbs described the entropy of a system as being a
measure of it's" degree of mixed -up-ness" when
this term is applied to the atomic –scale picture of
the systems ; i.e. the more
- Mixed up the constituent particles of the system the
higher the value its entropy
Entropy can thus be correlated with the atomic
scale randomness disorder of the system
Since the atomic disorder in the gaseous state
greatly exceeds that of the liquid state and the
atomic disorder of the liquid phase exceeds that of
the solid state as the entropy of the gaseous state
greatly exceeds that of the liquid state and that of
the liquid state and that of the solid state
This correlates greatly with macroscopic
phenomena , e . g of the melting temperature :
Hm
SL - Ss = t H > 0
mm
I.e. Sl > Ss , since
Hm is a positive value
Thus if the freezing occurs at the equilibrium freezing
temperature then the increase in the degree of order of
the freezing system exactly equals the decrease in the
degree of order of heat reservoir absorbing the heat of
solidification , and hence the total degree of the order in
the (system + reservoir ) is unchanged as a result of the
process ; I . e. the entropy has simply been transformed
the system to the heat reservoir .
The equilibrium melting or freezing temperature of
substances can thus be fired as being that temperature at
which no change in the degree of order of the (system +
heat path )
occurs as a result of the phase change ;I . e. only at this
temperature the solid phase is un equilibrium with the
liquid phase and hence only at this temperature the phase
change can occur reversibly .
The previous correlation cannot however , be irreversibly
applied because if a super cooled liquid spontaneously
freezes , it would appear that a decrease in the degree of
disorder is accompanies by an entropy increase due to the
irreversible freezing . in the degree of disorder in the
freezing system is lease than the increase in the degree of
order of the heat reservoir the heat of freezing and hence
the spontaneous process produces an overall increate in
the degree of disorder and thus an overall in crease in
crease in the entropy .
4-3 the concept of microstate
the development of a quantitative relationship between entropy
nd the “degree of mixed - up - ness ” secssitates the
uantification of the term “degree of mixed –up-ness” this can be
btained from the condition of “ elementary statistical mechanics”
Statistical mechanics is developed based on the assumption that
the equilibrium state of a system is simply the most probable of a
’s probable state .
*one of the major development in physical science which has led
considerable increase in in the understanding of the behavior of
matter is the “ a quantum theory “
A postulate of the quantum theory is that if a particle is confined
o move within a given fixed volume then its energy is quantized
.e. the energy values allowed to the particle is a specific discrete
nergy level separated from each other by “forbidden energy gaps
bonds )” .
* As
the volume allowed for movement increases the
spacing between the energy levels (energy gaps )
decreases , and when no restriction is placed on the
position of the particle ; energy allowed becomes continues
* The , based on the quantum theory , the energy level ,
available to particle in the solid are considerably more
widely spaced than are the levels available to a particle in
the gas .
* the effect of this quantization of energy can be illustrated
by examining the following hypothetical system. consider a
perfect crystal to contain these identical , and hence un
distinguishable , particles which are located on these
lattice sites :
A , B and C . suppose for simplicity that quantization in
such that the energy levels are equally spaced , with the
ground level Є0 being taken on zero , that energy value of
the ground value of the first level Є1 = u1 the second
level Є2 = 2 u , etc
Let the total energy of the system , U , equal 3u
this system can be realized , as shown in figure (1); in
hence different distribution as follow :
1)AU three particles in level (1) there is only one
arrangement of this distribution as shown in figure (2)
(figure 4 .2 in page 77 )
2) One particle in level (3) and the other 2 particles in
level zero ; there one thus these arrangements of this
distribution as shown in figure (2) .
3)One particle in level zero ; one particle in level 1;and one
particle in level (2) thus these are six distinguishable arrangement
of this distribution as shown in figure 2 .
* thus , altogether , there are 10 distinguishable way in which
these particles can be arranged among the energy levels , Є0 , Є1 ; Є2
and Є3 such that the total energy of the system U equal 3u .
* these distinguishable arrangement are called complexions or
microstates , and all of these 10 micro states correspond to a
single macro states .
4- 4 Determination of the most
probable microstate
In the above example ; the values of U ,V and N are
fixed , and hence; the macro state of the system on
fixed .
* With respect to the microstates , since the 10
microstates are contained in these distribution the
probability of the occurrence of the system in
distribution (1) is (1110) , the probability of the
system in distribution 2 is (3110) ; and the
probability of the system in distribution in 3 is (6110)
.
*
* the physical significance of these probability can be
viewed as follow : though from classical thermodynamics
points of view the macro states of the system is fixed ; the
microstate of the system is changeable in a way such that if
the microstate of the system were observed for a finite
length of time ; then the fraction of this time which the
system spent in each of the arrangement 1 , 2 and 3 would
be (1110) , ( 3110) and ( 6110).
•the number of arrangements within a given distribution
Ω is calculated as follow if N particles are distributed
among the energy level such that there are N0 in level Є0 ,
n1 in level n1 ; n2 in level Є2 ;….. nr in the highest level .
Of occupancy Єr , then the number of arrangement Ω is
i=r
given by :
n!
Ω=
=
n! / I = r( Π ni ! )
n0 ! n1 ! n2 ! ….. nr !
i=0
* the most probable distribution can be obtained by
determining the set
of numbers n0 , n1 , n2 , ….. nr which maximizes the value
of Ω . since for large value
of ni ;ln ni! = ni ln ni -ni
stirling's approximation , thus , Taking the logarithm of the
previous equation given :
i=r
ln Ω = n ln n – n - ∑
(ni ln ni - ni )
i=0
•Any infinitesimal interchange of particles among the
energy levels given :
δ ni
i=r
δ ln Ω = - ∑
i=r
i= 0
(δ ni ln ni ) –
ni
*
ni
i=r
= ∑ δ ni (Ln ni)
i= 0
of the set of ni ‘ S is such that Ω has its maximum value ,
then :
i=r
S Ln Ω = - ∑ Ln ni δ ni
(1)
i=0
* AS the microstate of the system is determined by the fixed
values of U ,V and n , any distribution of the system
particles among the energy levels must confirm with the
condition :
U = n0 Є0 + n1 Є1 + n2 Є 2 + ………..+ nr Єr = constant
i=r
= ∑ ni Єi
i =0
i=r
thus : du = ∑
Єi δ ni =0
(2)
i =0
* since N is constant , thus
i=n
N = ∑ ni = constant
i =0
i=r
Thus : δ n = ∑ δni = 0
i =0
(3)
the condition that Ω has it's maximum value for
the given macro state is that equation (1) ; (2) ; and
(3) are simultaneously satisfied .
* solution for the set of ni value corresponding to
the most probable distribution is achieved by means
of undetermined multiplies method .
thus ; multiply equation (2) by constant β ; where β
has the units of reciprocal of energy ; equation(3)
by the dimensionless constant α ; and adding the
resultant equations to equation (1) yields :
i=r
∑ ( Ln ni + α + β Єi ) δ ni = 0
(4)
i =0
* solution of equation {4} requires that each of
the bracketed terms be individually equal to zero
i.e.:
Ln ni + α + β Єi = 0
Or
-α
ni = e
-β Єi
e
i=r
i = r - β Єi
-α
Thus : n = ∑ ni = e ∑
e
i =0
i =0
i=r
- β Єi
n
-α
therefore e = n / ∑ e
=
P
i =0
i=r
where : P = ∑
- β Єi
e
is Known as the partition function and
i =0
hence ni =
n
P
e- β
Єi
(5)
* the distribution of N particles in the energy level; which
maximizes { i.e. the most probable distribution } is the one
in which the occupancy of the levels decreases
exponentially with incessancy energy as shown in figure {3}
[fig 4.3 page 81 ] ; examination of equation {5} indicate
that β must be positive quantity ; otherwise the level of
infinite energy would contain an infinite number of particles
.
* the experimentally obtained shape of figure {3} is
determined by the value of β which is shown to be
irreversibly proportional .
to the absolute temperature :
1
1
β α N or
β=
KT
T
where k is the boltyzman`s constant which is given by :
K = R / NA
where NA is the Avogadro's number
4.5 the effect of temperature
* as the temperature increase ; the upper levels
become relativity more populated ; and this
corresponds to an increase in the average energy of
the particles ; {i.e. an increase in the value of V \ N} ;
which for fixed value of V and N; U will increase the
value of V and N ;U will increase ; also as T increase
the value of β decrease and the shape of the
experiential distribution changes will be as shown in
figure (4) page 82 .
•as the macro state of system is fixed by fixing the values of U;V; AND N ;
then T as a dependent variable is fixed too
•*
when the number of particles in the system is very large ; then the
number of arrangements within the most probable distribution ; Ωmax ;is
the only term which makes a significant contribution to total number of
arrangement; Ωtotal which the system may have ; that isΩmax is
signifienceantly layer than the sum of all other arrangements ; hence :
• Ωtotal ~ Ωmax
Thus Ln Ωtotal = ln Ω max
i=r
= n ln n – n - ∑
( ni – ln ni – ni)
i=0
i=r
i=r
= n ln n – n - ∑ ni Ln ni + ∑ ni
i =0
i =0
i=r
= n Ln n – ∑ ni Ln ni
i=r
n
n
Єi
/
K
T
= n Ln n –∑
e
ln
e - Єi / K T
p
p
i=0
n
Єi
- Єi / K T
= n Ln n –
∑
(
Ln
n
–
Ln
P
)
e
p
kT
n
n
= n Ln n –
∑ ( Ln n– Ln P ) P +
∑ Єi e- Єi / K T
p
PkT
n
= n Ln P +
∑ Єi e- Єi / K T
PkT
n - Єi / K T
But U = ∑ ni , Єi = ∑ Єi x
e
p
n
= p ∑ Єi e- Єi / K T
Therefore :
UP ∑ Єi e- Єi / K T
N
Thus : Ln Ω = n Ln P +
n
UP
*
N
PkT
= n Ln P + U
kT
Or : U = K T Ln Ω + n K T ln P
n
4.6 thermal equilibrium within a system
and the boltzman equation
*consider
particles system at constant V in thermal
equilibrium at temperature T with aheat reservoir ; thus N
and Єi.s of the particles system are constant ; therefore P is
constant .
for a small exchange of energy between the particles system
And the heat reservoir ; we have :
d ln Ω = du
KT
*applying the first law thermodynamics given :
d Ln Ω =
δ q+ δ w
KT
since the particles system is at constant volume ; δ w = 0 , i.e.
δq
d Ln Ω =
KT
.*applying
the second law of thermodynamics given :
d δ ln Ω = ds
*as both S and Ω are state function ;; the above equation can be
written as :
S = K ln Ω
the equation is known as boatman's equation .
the previous equation is thus the required quantitative
relationship between the entropy of a system and the "degree of
mixed-up-ness" of a system" given as Ω "and defined as : the
number of ways which the available energy of the particles
system is shared among the particles of the system .
thus ; the equilibrium state of the system is that state while S is
maximum at the considered fixed volume of U ;V and n ;
this is based on the following "the most probable state of the
system is which Ω is maximum at the considered U;V ; and n of
the system .
therefore ; the batsman's equation provides a physical quality
to entropy
4.7 heat flow and the production of
entropy
*consider two closed system ; A and B ; let the energy of A to be
UA and the number of complexion of A to be Ω ; similarly ; let
the energy of B to be UB and the number of complexions to be
Ωb . if the two bodies are brought in contact with each other ;
the product Ωa Ωb will generally not have its maximum possible
value and heat will flow either from A to B or from B to A to
maximize the product Ωa Ωb .
if for example ; the heat flow from A to B this means that the
increase in the number of Ωb due to this heat exchance in larger
than the decrease in the value of ΩB
•when an amount of heat is transformed from A to B at
constant total energy ; then :
•d ln ΩA δ qA
KTA
δ qA
d ln ΩA
KTB
==-
δ qA
KTA
δ qA
KTB
thus δ Ln ΩA ΩB = δ Ln Ω A + δ Ln
δq
1 - 1
=
(
)
T
TA
B
K
ΩB
of course ; the flow will cease when ΩA ΩB will reach its
maximum value ; i.e. when ΩA ΩB = 0 and the condition for
that is TA = TB ;
And that condition for thermal equilibrium will prevail between
the two bodies .
*thus ;in the microscopic analysis ; an irreversible process is
one which takes the system from a lease probable to the most
probable state ; while in the corresponding macroscopic
analysis in irreversible process takes ; the system from a non
equilibrium to the equilibrium state .
* therefore ; what is considered in classical thermodynamics to
be an impossible process turn’s out as the result of microstate
examination to be an improbable process .
4.8 configurationally entropy and
thermal entropy
*
consider two crystal at the same temperature and
pressure ; one containing atoms of the element A and
the other containing atoms of element B ; when the two
crystals are placed in contact with one another ; the
spontaneous process which occurs in the diffusion of A
atoms into the crystal B lattice sites and diffusion of the
B atoms into the crystal A lattice .
*as this is a spontaneous process ; the entropy of the
system will increase and it might be predicted that
equilibrium will be reached "i.e. the entropy of the system
will reach a maximum value " when the concentration
gradients in the system have been eliminated " this is
similar to the case of heat flow under temperature gradient
; and this flow will be ceased and the thermal equilibrium
state will be reached when the temperature gradient in the
system in eliminated " .
* following the discussion of denbigh ; consider a crystal
containing of four atoms of A placed in contact with a
crystal containing four atoms of B as shown in figure 4.5
in page 87 .
•the mixing process in this system can be written as (A + B) un
At constant
mixed
(A + B) mixed
U , V , and N
•since the number of ways of arrangements in given by the
relation is given :
Ω conf
=
( n A + nB ) !
( nB !
Thus Ω conf
=
nA ! )
(4 +4)!
4!
4!
I . e number of ways = 70
8!
=
4! 4!
= 70
*now
; for the initial state where the arrangement in
crystal A 4:0 and cause quently the arrangement in 0:4 in
crystal B thus :
4!
4
!
Ω4:0 =
*
=1
4 ! 0!
0 ! 4!
when one atom of A is inter change with one atom of B
across x y , thus
4!
4!
=16
Ω1;3 =
*
3! 1!
1! 3!
•when tow atoms are interchanged , thus :
Ω2 : 2 = 4 ! * 4 !
=36
2!
2!
2! 2!
•when three atoms are changed , thus :
4!
4
!
=16
Ω1;3 =
3! 1!
1! 3!
•when the four atoms are interchanger , thus :4!
4!
=1
Ω0:4 =
*
4 ! 0!
0 ! 4!
•thus ∑ Ω = 1+ 16 + 36 + 16 + 1 = 70
Which equals = Ω 4 : 4
* the arrangement 2 : 2 thus us the most probable , 3 , 170,
therefore it corresponds , as expected , corresponds to the
elimination of the con cent ration gradient .
•Defining the configurationally entropy , S conf , as the entropy
component which causes from the number of distinguishable
ways in which the particles of the system can be mixed over
position in space , and the thermal entropy , Sth as component of
the entropy which arises from the number of ways in which the
entropy of the system can be shared among the particles , thus
S total + S th + S conf
* Thus , for the previous example
S conf = S conf (2) - S conf
= K ln Ω conf
(1)
(2) – K ln Ω conf
(1)
Ω conf (2)
= K ln
Ω conf (1)
In general : S tot = S th + S conf
= K ln Ωth + K ln Ω conf
= K ln Ωth Ω conf
Stot
= K ln
Ωth (2) Ω conf (2)
Ωth (1) Ω conf (1)
•Thus , for Tow closed system placed in Thermal contacts or for
two chemically identical open system , placed in thermal contact
Ω conf (2) = Ωconf (1)
Thus : Stot = K Ln
Ωth (2)
Sth
Ωth (1)
•Similarly , if particles of A are mixed with particles of B and if
this mixing process doesn’t effect the distribution of the particles
among the energy levels , i.e Ωth (2) = Ωth (1) , thus :
Stot = K Ln
Ω conf (2)
Ω conf (1)
S conf
* Such a situation corresponds to “ideal mixing ” and
requires that the quantization of energy within the initial
crystals are identical such a situation is the expectation rather
than the rules
•In general , when two or more pure elements are mixed at
constant U .V and n , Ω th (2) ≠ Ωth (1) complete spatial
randomization of the constituents particles does not occur . In
such case either of like particles (indicating difficulty in mixing
) or ordering indicating of tendency towards compound
formation occurs
•In all case , how ever , the equilibrium state of the mixed
system is that which at constant (U , V and N) maximized the
product Ωth Ω conf