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chapter four the statistical interpretation of entropy

chapter four the statistical interpretation of entropy. أحمد محمد أحمد عبد الحميد ايمن محمد عوض محمد زين حسن حسن محمد أحمد فهمي زهران احمد جمال. 4-1Introduction 4.2 Entropy and disorder on the atomic scale 4.3 the concept of microstate 4- 4 Determination of the most probable

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chapter four the statistical interpretation of entropy

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  1. chapter four the statistical interpretation of entropy • أحمد محمد أحمد عبد الحميد • ايمن محمد عوض • محمد زين حسن حسن • محمد أحمد فهمي زهران • احمد جمال

  2. 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

  3. 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" .

  4. * 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

  5. 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

  6. 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 = > 0 H m tm

  7. 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 )

  8. 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 .

  9. 4-3 the concept of microstate • * the development of a quantitative relationship between entropy and the “degree of mixed - up - ness ” secssitates the quantification of the term “degree of mixed –up-ness” this can be obtained 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 all it’s probable state . • *one of the major development in physical science which has led to a 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 to move within a given fixed volume then its energy is quantized ,i.e. the energy values allowed to the particle is a specific discrete energy level separated from each other by “forbidden energy gaps (bonds )” .

  10. *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 :

  11. 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) .

  12. 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 ;Є2and 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 .

  13. 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) .

  14. * 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).

  15. 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 given by : • Ω = = n! / I = r( Π ni ! ) • n0 ! n1 ! n2 ! ….. nr ! n ! i=r i=0

  16. * 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

  17. Any infinitesimal interchange of particles among the energy levels given : • i = r • δ ln Ω = - ∑ (δ ni ln ni ) – ni* • i = r i= 0 • 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 δ ni ni

  18. * 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 ( 3 ) i =0

  19. 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

  20. *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 ni = e e i = r i = r - β Єi Thus : n = ∑ ni = e ∑ e i =0 i =0 i = r - β Єi therefore e = n / ∑ e = i =0 -β Єi - α - α n - α P

  21. i = r • where : P = ∑e is Known as the partition function and • i =0 • hence ni = e (5) - β Єi n - β Єi P

  22. * 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 .

  23. to the absolute temperature : β α N or β = where k is the boltyzman`s constant which is given by : K = R / NA where NA is the Avogadro's number 1 1 K T T

  24. 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 .

  25. 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 = 0

  26. i = r = n Ln n –∑ e - Єi / K T ln e- Єi / K T i = 0 = n Ln n – ∑ ( Ln n – Ln P - ) e- Єi / K T = n Ln n – ∑ ( Ln n– Ln P ) P + ∑ Єi e- Єi / K T = n Ln P +∑Єi e- Єi / K T = ∑ Єi e- Єi / K T n n p p n Єi p kT n n p PkT n PkT But U = ∑ ni , Єi = ∑ Єi x e- Єi / K T n p n p

  27. n Therefore : ∑ Єi e- Єi / K T Thus : Ln Ω = n Ln P + = n Ln P + Or : U = K T Ln Ω + n K T ln P UP N n UP * N PkT U kT

  28. 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 = *applying the first law thermodynamics given : d Ln Ω = du KT δ q+ δ w KT

  29. since the particles system is at constant volume ; δ w = 0 , i.e. d Ln Ω = .*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 . δ q KT

  30. 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

  31. 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 Ωbwill 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

  32. when an amount of heat is transformed from A to B at constant total energy ; then : • d ln ΩA • d ln ΩA • thus δ Ln ΩA ΩB = δ Ln Ω A + δ Ln ΩB • = δ qA δ qA = - KTA KTA δ qA δ qA = - KTB KTB δ q 1 1 - ( ) TB TA K

  33. 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 .

  34. 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 .

  35. *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 .

  36. the mixing process in this system can be written as (A + B) un mixed (A + B) mixed • since the number of ways of arrangements in given by the relation is given : • Ω conf = • Thus Ω conf= • I . e number of ways = 70 At constant U , V , and N ( nA + nB ) ! ( nB ! nA ! ) 8 ! ( 4 + 4 ) ! = 70 = 4! 4! 4! 4!

  37. *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 : • Ω4:0 = * • when one atom of A is inter change with one atom of B across x y , thus • Ω1;3 = * • when tow atoms are interchanged , thus : • Ω2 : 2 = * • when three atoms are changed , thus : • Ω1;3 = 4 ! 4 ! =1 4 ! 0! 0 ! 4! 4 ! 4 ! =16 3! 1! 1! 3! 4 ! 4 ! =36 2! 2! 2! 2! 4 ! 4 ! =16 3! 1! 1! 3!

  38. when the four atoms are interchanger , thus :- • Ω 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 4 ! 4 ! =1 4 ! 0! 0 ! 4!

  39. * Thus , for the previous example S conf = S conf (2) - S conf (1) = K ln Ω conf (2) – K ln Ω conf (1) = K ln In general : S tot = S th + S conf = K ln Ωth + K ln Ω conf = K ln Ωth Ω conf Stot = K ln Ω conf (2) Ω conf (1) Ωth (2) Ωconf (2) Ωth (1) Ωconf (1)

  40. 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 • 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 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 Ωth (2) Sth Ωth (1) Ω conf (2) Ω conf (1)

  41. 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

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