System with Variable Energy, Volume and Number of

ISSN 00360244, Russian Journal of Physical Chemistry A, 2011, Vol. 85, No. 13, pp. 13–19. © Pleiades Publishing, Ltd., 2011.

CHEMICAL THERMODYNAMICS AND THERMOCHEMISTR

System with Variable Energy, Volume and Number of Particles: Evaluation of Partition Function and Thermodynamic Quantities1 I. M. Stankovi c , V. M. Markovi c , and L. Z. KolarAni c  Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12–16, 11158 Belgrade, P.O. Box 47, Serbia email: Received December 20, 2010

Abstract—Partition function and thermodynamic parameters of the system open with respect to energy E, volume V and number of particles N, whereas temperature T, chemical potential µ and pressure P were kept constant, were considered by the procedure proposed by Gibbs. The mean values of variable energy, volume and number of particles were obtained by universal expression for such values. The entropy was determined by its general statistical definition. The thermodynamic potential and other thermodynamic quantities were also turned out naturally. Keywords: System with variable energy, volume and number of particles, partition function, thermodynamic quantities, entropy. DOI: 10.1134/S0036024411130280 1 INTRODUCTION

particles N0 are constant (figure):

The aim of this paper is to derive the partition func tion and thermodynamic quantities for a system that exchanges energy E, volume V and number of particles N with the reservoir, the system which we shall call henceforth totally open system. The analysed system is in thermal, chemical and mechanical equilibrium with the environment. That means to take temperature T, chemical potential μ and pressure P constant. In the literature available to the authors, the consideration of such system from statistical thermodynamic point of view is not found, whereas the analogue analysis of a system open with respect to energy, energy and vol ume, or, energy and number of particles is widespread [1–30].

0

E = E i + E 'i = const, 0

V = V j + V 'j = const,

(1)

0

N = N k + N 'k . In other words, the whole system (reservoir + con sidered system) is isolated from the surrounding and in a state of mutual equilibrium. Therefore, if g(Ei, Vj, Nk) is the number of different states of the considered system with energy Ei, the volume Vj, and the number of particles Nk, and g(E 'i , V 'j , N 'k ) is the number of dif ferent states of the reservoir with the energy E i' , the

RESULTS

volume V 'j and the number of particles N 'k , the proba

The Partition Function of an Imaginary Totally Open System To define the expression for the partition function of the system, we shall start from the expression for the probability that the analyzed system has the energy Ei, the volume Vj, and the number of particles Nk, W(Ei, Vj, Nk). This system is in equilibrium with the reservoir with the energy E 'i , the volume V 'j and the number of particles N 'k . The total energy of the reservoir and the system E0, the total volume V0 and the total number of

Ei, Vj, Nk

E'i, V'j, N'k

The system in the thermodynamic equilibrium with the reservoir.

1 The article is published in the original.

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bility W(Ei, Vj, Nk) is directly proportional to their product: W ( E i, V j, N k ) = Ag ( E 'i, V 'j , N 'k )g ( E i, V j, N k ).

V j > V j' ,

N k > N k' .

and

(3)

Hence, we can expand the function lng(E 'i , V 'j , N 'k ) in a Taylor series in a neighbourhood of the point (E 'i = E0 – Ei ≈ E0, V j' = V0 – Vj ≈ V0, N k' = N0 – Nk ≈ N0) and lock up ourselves to the linear terms of the expansion. In the obtained expression: 0

0

we can introduce indefinite multipliers α, β and γ: 0

0

0

0

0

0

0

0

0

0

0

∂ ln g (E , V , N )/∂E = β, ∂ ln g (E , V , N )/∂V = γ,

(5)

∂ ln g (E , V , N )/∂N = α and easily obtain W(Ei, Vj, Nk) in the form: 0

0

0

W ( E i, V j, N k ) = Ag ( E i, V j, N k )g (E , V , N ) × exp ( – βE i – γV j – αN k ).

(6)

Since, each partial derivative of any quantity with respect to one parameter implies that all other parameters of which the quantity depends on are constant, we can drop the indication what is constant in expressions (4) and (5), for brevity. The value of the constant A can be determined from the normal ization condition:

∑ ∑ ∑ W ( E , V , N ) = 1. i

i

j

j

k

μN k – E i – PV j⎞ g ( E i, V j, N k ) exp ⎛   ⎝ ⎠ kB T = . μN k – E i – PV j⎞  g ( E i, V j, N k ) exp ⎛  ⎝ ⎠ kB T

(7)

i

j

W ( E i, V j, N k )

Π ( T, P, μ ) =

μN k – E i – PV j

⎞ . ∑ ∑ ∑ g ( E , V , N ) exp ⎛⎝  ⎠ k T i

i

j

k

Taking into account physical meaning of the indefinite multipliers α, β and γ (Appendix) the probability that

(10)

k

B

k

Thermodynamic Functions The evaluation of thermodynamic quantities for totally open system has been done in analogy with the deriving of corresponding expressions for the thermo dynamic quantities of other types of systems that are common in literature which exchange less than three quantities with the environment: isolated, closed, open and isothermalisobaric system. @Mean values of energy, volume and number of particles.@ Knowing the probability distribution W(Ei, Vj, Nk) we can find the mean values of energy, volume and number of particles by definition for mean values of any quantity. Thus, E=

∑ ∑ ∑ E W(E , V , N ) i

i

= 1 Π

j

∑∑∑ i

V=

∑∑∑ j

j

The summation in the denominator goes over all the states accessible to the system. However, the probability to find a system in all states with the energy Ei, the volume Vj, and the number of particles Nk differs from the probability that the system will be in one state with energy Ei, the volume Vj, and the number of particles Nk for the statistical wait g.

g ( E i, V j, N k ) exp ( – βE i – γV j – αN k ) =   . (8) g ( E i, V j, N k ) exp ( – βE i – γV j – αN k ) i

k

Here, the number of different states with same energy, volume and number of particles g(Ei, Nk, Vj) is the well known degeneracy or the statistical weight of this sys tem [6]. The expression in the denominator (9) is the corresponding partition function or statistical sum, i.e. the total number of different states of the system under consideration:

k

The probability distribution of the states of a system with variable energy, volume and number of particles can finally be written in the form:

(9)

∑∑∑

0

0 0 0 ∂ ln g (E , V , N ) ln g (E 'i, V 'j , N 'k) ≈ ln g (E , V , N ) – E i  0 ∂E (4) 0 0 0 0 0 0 ∂ ln g ( E , V , N ) ∂ ln g ( E , V , N ) – V j   – N k   = α; 0 0 ∂V ∂N

0

W ( E i, V j, N k )

(2)

The system is much smaller than the reservoir, its energy, volume and the number of particles contained in it is small in comparison with the energy volume and number of particles in the reservoir: E i > E i' ,

the system has the energy Ei, the volume Vj, and the number of particles Nk is:

j

k

k

μN k – E i – PV j⎞ E i g exp ⎛   , ⎝ ⎠ kB T j

j

∑∑∑ i

j

(11)

∑ ∑ ∑ V W(E , V , N ) i

1 =  Π

i

k

j

k

i

j

k

k

μN k – E i – PV j⎞ V j g exp ⎛   , ⎝ ⎠ kB T

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SYSTEM WITH VARIABLE ENERGY, VOLUME AND NUMBER OF PARTICLES

N=

∑ ∑ ∑ N W(E , V , N ) k

i

1 =  Π

j

∑∑∑ i

j

k

i

j

k

k

μN k – E i – PV j⎞ N k g exp ⎛   . ⎝ ⎠ kB T

(13)

∑∑∑ i

j

k

μN k – E i – PV j⎞  N k g exp ⎛  ⎝ ⎠ kB T

μN k – E i – PV j

j

j

B

k

ΠV = – , kB T and the expression for the mean volume of the system is: (17)

The mean energy of this system in terms of the partition function and the parameters of the system (T, μ, P = const) can be obtained by differentiating the partition function with respect to the tempera ture: 1 ⎛ ∂Π = –  ⎞ 2 ⎝ ∂T ⎠ P, µ kB T

j

k

(18)

i

j

(19)

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k

(21)

(22)

we can easily obtain expression: S = –kB

∑ ∑ ∑ W(E , V , N ) i

i

j

j

k

k

(23)

μN E PV × ⎛ k – i – j – ln Π⎞ , ⎝ kB T kB T kB T ⎠ which directly gives the relation: μN E PV S = –  +  +  + k B ln Π. T T T

(24)

Thermodynamic potential. From the last equation we can see that E – TS = μN – PV – k B T ln Π.

(25)

The expression E – TS is equal to the Helmoltz free energy A. The total differential of the Helmholtz free energy A: dA = μdN + Ndμ – PdV – VdP – k B d ( T ln Π ) (26) can be compared with the corresponding thermody namic expression where V = V and N = N : (27)

resulting in: – k B d ( T ln Π ) = – SdT + VdP – Ndμ.

Thus, the mean energy of the system is: 2 ln Π⎞ . E = μN – PV + k B T ⎛ ∂ ⎝ ∂T ⎠ µ, P

k

dA = – SdT – PdV + μdN

B

Π × ( μN k – E i – PV j ) = – 2 ( μN – E – PV ). kB T

j

k

μN k – E i – PV j⎞ W ( E i, V j, N k ) = exp ⎛   /Π, ⎝ ⎠ kB T

μN k – E i – PV j

⎞ ∑ ∑ ∑ g exp ⎛⎝  ⎠ k T i

i

j

and the probability that system will be in a state with the energy Ei, the volume Vj, and the number of parti cles Nk:

(16)

∂ ln Π V = – k B T ⎛ ⎞ . ⎝ ∂P ⎠ µ, T

∑ ∑ ∑ W ( E , V , N ) ln W ( E , V , N ) = – k B 〈 ln W〉 ,

(15)

⎞ ( – V ) ∑ ∑ ∑ g exp ⎛⎝  ⎠ k T i

Entropy. Using the general definition of entropy given by Gibbs [6]:

(14)

The expression for the mean volume of the system in terms of the partition function and the parameters of the system (T, μ, P = const) can be obtained by dif ferentiating the partition function with respect to the pressure: 1 ⎛ ∂Π ⎞ =  ⎝ ∂P ⎠ T, µ k B T

+ k B T  . ⎝ ∂T ⎠ µ, P

i

Thus, the expression for the mean number of parti cles is:

(20)

2 ⎛ ∂ ln Π⎞

S = –k

ΠN = . kB T

N = k B T ( ∂ ln Π/∂μ ) T, P .

Using the expressions for the mean values of the num ber of particles (15) and the volume (17) we can easily obtain: ∂ ln Π ∂ ln Π E = μk B T ⎛ ⎞ + Pk B T ⎛ ⎞ ⎝ ∂μ ⎠ P, T ⎝ ∂P ⎠ µ, T

Let us start with the expression for the mean num ber of particles in terms of the partition function and parameters of the system (T, μ, P = const). It can be easily obtained by differentiating the partition func tion with respect to chemical potential only: 1 ⎛ ∂Π ⎞ =  ⎝ ∂μ ⎠ T, P k B T

15

(28)

In general, the thermodynamic potential is defined as X = –kBTlnY, where Y is the partition function of any considered system. In this case, X = –kBTlnΠ such that: dX = – SdT + VdP – Ndμ. Vol. 85

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The partition functions, thermodynamic potential and thermodynamic functions derived under different conditions Isolated system, Microcanonical ensemble

( ) ( ) ( )

( ) ( ) ( )

∂ ln g 1 = ∂S =k = kβ T ∂ E N ,V ∂ E N ,V ∂ ln g P = ∂S =k = kγ T ∂V N , E ∂V N , E µ ∂ ln g = − ∂S = −k = −k α T ∂ N E ,V ∂ N E ,V

g( E,V, N ) S = k ln g µ dS = 1 dE + P dV − dN T T T

Closed system, open system with respect to E, canonical ensemble

( ) ( ) ( )

Z=

(

S = − ∂A = k ln Z + kT ∂ ln Z ∂T V , N ∂T ∂ A ∂ ln Z P =− = kT ∂V T , N ∂V T , N ∂ A ∂ ln Z μ= = −kT ∂ N V ,T ∂ N V ,T

Z( T,V, N )

∑ g(E i )e −E / kT i

i

A = − kT ln Z

dA = −SdT − PdV − µ Nd

(

(

)

)

)

V,N

Open system with respect to E and N, grand canonical ensemble

(∂∂φT ) ∂φ P = −( ) ∂V

i

(

)

= k ln Ξ + kT ∂ ln Ξ ∂T V , µ V ,µ = kT ∂ ln Ξ = kT ln Ξ ∂V T , µ V T,µ ⎛ ∂φ ⎞ ⎛ ∂ ln Ξ ⎞ N = −⎜ ⎟ = kT ⎜ ⎟ ⎝ ∂μ ⎠V ,T ⎝ ∂μ ⎠V ,T

S =−

Ξ(T ,V , µ) Ξ = ∑ ∑ g(E i , N k ) exp (μ N k − E i / kT ) k

Φ = − PV = − kT ln Ξ

d Φ = −SdT − PdV − Nd µ

(

)

Open system with respect to E and V, isothermalisobaric canonical ensemble

( ) ( ) ( )

Δ(T , P, N )

Δ = ∑∑ g(Ei ,V j i

(

)

N, P

(

)

µ,P

S = − ∂G = k ln Δ + kT ∂ ln Δ ∂T N , P ∂T ∂ G ∂ ln Δ V = = −kT ∂P T , N ∂P T , N ∂ G ∂ ln Ξ μ= = −kT ∂N T ,P ∂N T ,P

−Ei −PV j )e kT

j

G = − k T ln Δ

d G = −SdT + Vd P + µ dN

( ) ( )

Open system with respect to E, V and N

Π(T , P, µ) Π = ∑∑∑ g(E i , N k ,V j ) exp (µN k − Ei − PV j / kT ) i

j

k

X = − kT ln Π

dX = −SdT + VdP − Nd µ

From the last expression directly follows: ∂X ∂ S = – ⎛ ⎞ = –  ( – k B T ln Π ) ⎝ ∂T⎠ µ, P ∂T ∂ ln Π⎞ , = k B ln Π + k B T ⎛  ⎝ ∂T ⎠ µ, P

(30)

( ) ( )

S = − ∂X = k ln Π + kT ∂ ln Π ∂T µ,P ∂T ∂ X ∂ ln Π V = = −kT ∂P T ,µ ∂P T ,µ ⎛ ∂X ⎞ ⎛ ∂ ln Π ⎞ N = −⎜ ⎟ = kT ⎜ ⎟ ⎝ ∂µ ⎠T ,P ⎝ ∂µ ⎠T ,P

(

)

∂X ∂ V = ⎛ ⎞ =  ( – k B T ln Π ) ⎝ ∂P⎠ T, µ ∂P ∂ ln Π = k B T ⎛ ⎞ , ⎝ ∂P ⎠ T, µ

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SYSTEM WITH VARIABLE ENERGY, VOLUME AND NUMBER OF PARTICLES

∂X⎞ ∂ ( – k T ln Π ) N = ⎛  = –  B ⎝ ∂μ⎠ T, P ∂μ

(32)

ln Π⎞ . = k B T ⎛ ∂ ⎝ ∂μ ⎠ T, P

Other thermodynamic functions of the system, as the Helmholtz free energy, the Gibbs free energy, enthalpy and heat capacity, can be expressed simply by substituting the obtained expressions for the entropy, the mean number of particles and the mean volume into the classical thermodynamic expressions. Thus, we get for the Gibbs energy: ln Π⎞ – k T ln Π; G = μk B T ⎛ ∂ B ⎝ ∂μ ⎠ P, T

(33)

is that the partition function has different value for dif ferent systems. If the partition function is labelled as Y, the entropy for every system with variable energy is in the following form: ln Y⎞ . S = k B ln Y + k B T ⎛ ∂ ⎝ ∂T ⎠ µ, P

(39)

For systems with variable number of particles, the mean number of particles is: ∂ ln Y N = k B T ⎛ ⎞ , ⎝ ∂μ ⎠ P, T

(40)

while for systems with variable volume, the mean vol ume is: ∂ ln Y⎞ . V = – k B T ⎛  ⎝ ∂P ⎠ µ, T

for the Helmholtz free energy: ln Π⎞ + Pk T ⎛ ∂ ln Π⎞ A = μk B T ⎛ ∂ B ⎝ ⎝ ∂μ ⎠ P, T ∂P ⎠ µ, T

17

(41)

(34)

– k B T ln Π;

CONCLUSION

for enthalpy: ln Π⎞ + k T 2 ⎛  ∂ ln Π⎞ ; H = μk B T ⎛ ∂ B ⎝ ∂μ ⎠ P, T ⎝ ∂T ⎠ µ, P

(35)

for isochoric heat capacity: 2 ∂ ln Π⎞ ∂ ln Π C V = 2k B T ⎛ ⎞ + k B T ⎛   ; ⎝ ∂T ⎠ µ, P ⎝ ∂T 2 ⎠ µ, P

(36)

for isobaric heat capacity: 2

ln Π⎞ + μk T ⎛  ∂ ln Π⎞ C P = μk B ⎛ ∂ B ⎝ ⎝ ∂μ ⎠ P, T ∂μ∂T ⎠ P 2

∂ ln Π⎞ + k T 2 ⎛ ∂ ln Π⎞ + 2k B T ⎛  B ⎝ ∂T ⎠ µ, P ⎝ ∂T 2 ⎠

(37)

;

APPENDIX 1. THE PHYSICAL MEANING OF PARAMETERS Α, Β AND Γ

µ, P

and for the difference of the isochoric and isobaric heat capacities: 2

ln Π⎞ + μk T ⎛  ∂ ln Π⎞ . C P – C V = μk ⎛ ∂ B ⎝ ⎝ ∂μ ⎠ P, T ∂μ∂T ⎠ P

Thermodynamic quantities of the totally open sys tem in terms of the partition function Π and parame ters of the system T, μ and P are derived in analogy with other systems in literature. For this purpose, ther modynamic potential of the form X = –kBTlnΠ is defined. As considered system is open with respect to energy, volume and number of particles, their mean values are also obtained directly by universal expres sion for such values. Moreover, the entropy is deter mined by its general statistical definition. Thus, it is shown that the hypothetic system with variable energy, volume and number of particles is the best one for nat ural evaluation of all thermodynamic quantities.

(38)

DISCUSSION The obtained thermodynamic functions are com pared with the corresponding ones evaluated when the system is isolated or open with respect to energy, energy and volume, or, energy and number of particles (table). In table are given common names of ensem bles, their partition functions with the constant parameters denoted in brackets and expression for corresponding thermodynamic potential as the basic relation for calculation of thermodynamic functions. All derived thermodynamic quantities are the same functions of the partition function; the only difference RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A

There are different manners to find the physical meaning of parameters α, β and γ, that is, to express them in function of well known quantities in classical thermodynamics. With aim to complete the derivation of probability distribution of the states of a system with variable energy, volume and number of particles, we shall briefly present one of them [9]. Since the partition function of isolated system known as statistical weight or degeneracy g(E, V, N) is a function of energy, volume and number of particles, the total differential of lng can be expressed as: ∂ ln g ( E, V, N)⎞ dE d ln g ( E, V, N ) = ⎛  ⎝ ⎠ V, N ∂E ln g ( E, V, N)⎞ dV + ⎛  ∂ ln g ( E, V, N)⎞ dN (A1) + ⎛ ∂ ⎝ ⎠ E, N ⎝ ⎠ E, V ∂V ∂N = βdE + γdV + αdN, Vol. 85

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18

where parameters α, β and γ are equal to the ones defined in equation (5). Therefore, the last expression can be rewritten in the form: dE = 1d ln g – γdV – α  dN. β β β

(A2)

and compared to the classical one which correlate dE, dV and dN dE = TdS – PdV + μdN.

(A3)

If S = kB lng, it follows that the parameters α, β and γ are equal to: μ α = – , kB T

1 β = , kB T

P γ = . kB T

and

(A4)

With aim to show that S = kB lng, we need to con sider a system A0, made of two subsystems such that A0 = A1 + A2, each of which in equilibrium, and apply on them one by one statistical and classical thermody namic knowledge. From statistical point of view, the number of microstates for the total system g0(E1, E2) is equal to the product of the numbers of microstates for the two initial systems g1(E1) and g2(E2): g0(E1, E2) = g1(E1)g2(E2).

(A5)

If they are in a thermal contact exchanging the energy, the equilibrium will be achieved when the number of microstates for the total system is maximal: 0

∂g ( E 1 )/∂E 1 = 0.

(A6)

Since E0 = E1 + E2 = const, we can easily obtain the relation:

Since the entropy of the total system is maximal in equilibrium, we obtain: ∂S 1 ( E 1 ) ∂S 2 ( E 2 ) ∂S  =   =  = const. ∂E ∂E 1 ∂E 2

(A12)

The entropy of the system with a constant volume and number of particles, in classical thermodynamics is: dS = dE/T,

(A13)

( ∂S/∂E ) N, V = 1/T.

(A14)

such that: It follows that the systems are in thermal equilibrium when their temperatures are equal and constant. Since the parameters β have same characteristics for the sys tems in thermal equilibrium (A10), it follows that there must be direct relation between T and β. Consid ering (A9) and (A14) it follows that: d ln g/dS = βT.

(A15)

Since both β and T are constants, their product will also be a constant: βT = 1/T,

(A16)

where k is the Boltzmann’s constant. It follows that the parameter β is equal to: β = 1/k B T.

(A17)

ACKNOWLEDGMENTS This work is supported by Ministry of Science and Technological Development of Serbia, project no. 172015.

0

∂g ( E 1, E 2 ) ∂g 1 ( E 1 )   = g 2 ( E 2 ) ∂E 1 ∂E 1 ∂g 2 ( E 2 ) ∂E 2 + g 1 ( E 1 )   , ∂E 2 ∂E 1

REFERENCES (A7)

which ensues: ∂ ln g 1 ( E 1 )/∂E 1 = ∂ ln g 2 ( E 2 )/∂E 2 .

(A8)

Taking into account expression:

ˆ

∂ ln g ( E )/∂E = β,

(A9)

we obtain that all subsystems in thermal equilibrium must satisfy the following condition: β 1 = β 2 = β = const.

(A10)

Furthermore, considering the entropy of the com bined subsystem A0, one comes to the physical mean ing of parameter β. Its entropy is the sum of entropies of two subsystems: 0

S ( E 1, E 2 ) = S 1 ( E 1 ) + S 2 ( E 2 ).

1. T. L. Hill, An Introduction to Statistical Thermodynam ics, 2nd ed. (AddisonWesley, Reading, MA, 1962). 2. H. Eyring, D. Henderson, B. Stover, and E. Eyring, Statistical Mechanics and Dynamics (Wiley, New York, 1964). 3. R. H. Fowler, Statistical Mechanics (Cambridge Univ. Press, Cambridge, 1966). 4. A. Münster, Statistical Thermodynamics (Springer, Ber lin, 1969), Vol. 1. 5. B. Mili c, Statistical Physics (Naucna knjiga, Belgrade, 1970) [in Serbocroat]. 6. Ya. P. Terletskii, Statistical Physics (NorthHolland, Amsterdam, 1971). 7. B. G. Levich, Theoretical Physics, Vol. 2: Statistical Physics, Electromagnetic Processes in Matter (North Holland, Amsterdam, 1971). 8. H. Schilling, Statistische Physik in Beispielen (VEB Fachbuchverlag, Leipzig, 1972; Mir, Moscow, 1976). 9. R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972). 10. A. Isihara, Statistical Physics (Academic Press, New York 1971; Mir, Moscow, 1973).

(A11)

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21. A. D. Stepuhovi c and V. A. Ulickii, Lectures in Statisti cal Physics (Vysshaya Shkola, Moscow, 1978) [in Rus sian]. 22. A. M. Vasiljev, Introduction to Statistical Physics (Vysshaya Shkola, Moscow, 1980) [in Russian]. 23. L. D. Landau, and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics, Part I, 3rd ed. (Per gamon Press, Oxford, 1980). 24. Ju. L. Klimontovic, Statistical Physics (Nauka, Mos cow, 1982) [in Russian]. 25. Lj. KolarAni c, The Fundamentals of Statistical Ther modynamics, 3rd ed. (Faculty of Physical Chemistry, Univ. of Belgrade, Belgrade, 2009) [in Serbocroat]. 26. D. A. McQuarrie, Statistical Mechanics (Univ. Science Books, Sausalito, 2000). 27. B. N. Roy, Fundamentals of Classical and Statistical Thermodynamics (Wiley, West Sussex, 2002). 28. http://en.wikipedia.org/wiki/Statistical_mechanics 29. http://farside.ph.utexas.edu/teaching/sm1/lectures/lec tures.html 30. http://www.chem.arizona.edu/~salzmanr/480b/statt01/ statt01.html ˆ

11. D. A. McQuarrie, Statistical Thermodynamics (Harper and Row, New York, 1973). 12. E. S. R. Gopal, Statistical Mechanics and Properties of Matter (Wiley, New York, 1974). 13. R. Kubo, Statistical Mechanics, 4th ed. (NorthHol land, Amsterdam, 1974). 14. F. C. Andrews, Equilibrium Statistical Mechanics, 2nd ed. (Wiley, New York, 1975). 15. M. H. Everdell, Statistical Mechanics and its Chemical Applications (Academic Press, London, 1975). 16. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975). 17. D. Mušicki, Introduction to Theorical Physics, II: Statis tical Physics (ICS, Belgrade 1975) [in Serbocroat]. 18. J. E. Mayer, and M. G. Mayer, Statistical Mechanics (Wiley, New York, 1940; 2nd ed., Wiley, New York, 1977). 19. F. Reif, Statistical Physics (Nauka, Moscow, 1977) [in Russian]. 20. R. P. Feynman, Statistical Mechanics (Benjamin, MA, 1972; Mir, Moscow, 1978).

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