8. INTRODUCTION TO STATISTICAL THERMODYNAMICS

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1 n * D n d Fluid z z z FIGURE 8-1. A SYSTEM IS IN EQUILIBRIUM EVEN IF THERE ARE VARIATIONS IN THE NUMBER OF MOLECULES IN A SMALL VOLUME, SO LONG AS THE PROPERTIES ARE UNIFORM ON A MACROSCOPIC SCALE 8. INTRODUCTION TO STATISTICAL THERMODYNAMICS The objective o statistical thermodynamics is to give a molecular basis or thermodynamics. Thus, it is necessary to deine the concepts o thermodynamics at the molecular level. Thermodynamics is built on the concept o equilibrium. (see Section 1. on page 1). Suppose an isolated, simple luid system has been let undisturbed until it has arrived in a state where no macroscopic, spontaneous changes are taking place, and all o the intensive properties are spatially uniorm on a macroscopic scale within the volume. For the present, we reer to such a system as being in a state o stable equilibrium (see Section 1.2 on page 2). The luid is hypothesized to consists o molecules that are in a state o motion. Thus, there are microscopic changes taking place at all times. For example, i n is the num- System n n * ber o molecules in volume ~d 3, and n* is the number o molecules in D 3, where d is on the order o a ew molecular diameters and D is the on the order o 10 or more molecular diameters, there would be variations as a unction o time in the number o molecules within d 3, but in an equilibrium system there would be negligible variations in the number FEBRUARY 15,

2 o molecules within D. 3 Thus, there can be luctuations about the equilibrium state, but that system is none-the-less in equilibrium in the sense that there are no macroscopic changes. 8.1 IS MOLECULAR DESCRIPTION NEEDED? I the system is in a thermodynamically metastable state, luctuations can give rise to a phase change in the system. For example, consider a large volume o a liquid phase. On a macroscopic scale suppose the liquid phase is homogeneous. Thus, luctuations can exist in the system on the a microscopic scale, but the system may none-the-less be viewed as being in a state o thermodynamic equilibrium. Vapor 2R Liquid Isolated System with Homogeneous Liquid Phase Liquid Three Phase Isolated System FIGURE 8-2. Suppose a spherical bubble o radius R orms and once it orms it is held at that size until the heterogeneous system is in equilibrium. For the homogeneous liquid phase S S 0 ( U, V, N) (8-1) and according to the Euler relation FEBRUARY 15,

3 S U PL V µ T + T T --- N (8-2) I a bubble orms in an isolated system, the constraint requires that N N L + N V + N LV U U L + U V + U LV V V L + V V (8-3) Hence the entropy o the homogeneous system may be written 1 S ( U T L + U V + U LV P ) --- ( V T L + V V µ + ) --- ( N T L + N V + N LV ) (8-4) The entropy o the isolated system when the bubble o radius expressed SR ( ) S L + S V + S LV or in terms o the individual phases R is present may be SR ( ) T L U L P + L U LV T LV V L µ -----N L L + T L T L γ LV µ V T LV A LV i N LV T LV U V T V + P V V V µ V T V N V T V (8-5) Now suppose that the isolated system is suiciently large so that there is no change in any o the intensive properties o the liquid phase as a result o the bubble ormation. Then P L T L P T µ L µ (8-6) and the change in the entropy o the system could be written FEBRUARY 15,

4 SR ( ) S U V U LV T V T L P V T V P L T L T LV T L V V γ LV A LV µ V T V µ L T L T LV T LV µ L T L N V µ LV N LV (8-7) I the bubble is held at radius R until equilibrium is restored then µ V µ L µ LV (8-8) T V T L T LV T (8-9) then v L P V P exp ( P L P RT ) (8-10) The radius at which the bubble is being held is not necessarily equal to the value o, but R c P V P L 2γ LV R c (8-11) or R c 2γ P exp v L RT L ( P L P ) P L (8-12) R c R c ( P L, T L ) (8-13) FEBRUARY 15,

5 Thus, since the pressure in the liquid does not change as a result o the nucleation o the R c bubble, is constant. It should be noted that the expression or is the same as that or R C R eq given in Eq. (6-30), and as indicated in Figure 6-7, its value depends on the thermodynamic state o the system and its value can be either negative or positive. I the pressure in the liquid phase is greater than the saturation vapour pressure, R C is negative, and the pressure in the liquid phase is less than the saturation vapour pressure, its value is positive. Ater substituting Eq. (8-11) into Eq. (8-7) and making use o Eqs. (8-8) and (8-9) one inds the change in entropy may be written S S 0 4πγ LV T R 2 2R R c (8-14) Two possibilities now arise: SR ( ) S 0 1.I the value o R C were negative, R c then SR ( ) S 0 has the shape indi- 0 FIGURE 8-3. FOR R C < 0 R cated in Figure 8-3. The shape o the entropy curve shown there indicates that a luctuation could not give rise to a change o phase in this case. I a bubble o radius R ormed as a result o a luctuation, the system would move in the direction o increasing entropy; thus the system move toward the macroscopically homogeneous state. Any luctuation would dissipate. FEBRUARY 15,

6 SR ( ) S 0 0 FIGURE 8-4. FOR R c > 0 R c R 2.However, i the value o R C were positive the shape o the entropy would be that indicated in Figure 8-4. In this case the system is metastable. The interpretation is very dierent in this case. I the, the thermody- luctuations produced a bubble with a radius that was greater than namic tendency would be or the bubble to grow to a new phase. However, i the entropy postulate is taken to mean that in an isolated system the entropy can never decrease, then one would conclude that a bubble could not orm spontaneously in an isolated system. A detailed experimental investigation has been perormed to determine i bubble nucleation does take place homogeneously as a result o luctuations in density. The liquid studied was ethyl ether. The results strongly support the concept o luctuations, and the concept that they are responsible or homogeneous nucleation o a new phase. 1, 2 Thus, interpreting the second postulate as orbidding luctuations does not seem to be tenable position. R C THE SECOND POSTULATE OF THERMODYNAMICS In thermodynamics, the property entropy is postulated to exists and or a composite system satisying certain constraints it is postulated that the system will partition the 1. T. W. Forest and C. A. Ward J. Chem. Physics 66, , T. W. Forest and C. A. Ward J. Chem. Physics 69, , FEBRUARY 15,

7 total energy, molecules and the volume so as to maximize this unction (Section on page 3). Thus, thermodynamics does not deine the entropy unction. It only claims that the unction entropy exists and that it has certain properties. When this postulate is used to consider a constant volume, constant mole number system surrounded by a reservoir with which the composite system exchanges thermal energy, it is ound that the entropy postulate requires the Helmholtz unction: F U TS to be a minimum (Section 4.1 on page 74) when the system is in a stable equilibrium state. are I the expression or this unction were known in terms o its independent variables T, V, N 1 N r, then all o the macroscopic inormation about a simple system would be known. For example, suppose we wanted to know an equation o state or a substance: P P( T, V, N 1 N r ) (8-15) I the Helmholtz unction were known, then the equation o state could be simply obtained by dierentiating the Helmholtz unction: P F V T, N 1 N r (8-16) and it would be o the orm indicated in Eq. (8-15). The other equations o state: S F T V, N 1 N r (8-17) F µ i N i T, V, N j (8-18) FEBRUARY 15,

8 Thus, in order to have a complete description o the thermodynamic system, the expression or the Helmholtz unction may be constructed. However, note that this unction would not allow us to deine the luctuation about the equilibrium state. To obtain a description that is suiciently general to allow us to predict the luctuations about an equilibrium state, we shall hypothesize that thermodynamic describes the average properties o a macroscopic system in some as yet undeined sense, and compare the predictions that ollow rom that hypothesis with measurements. The problem or statistical thermodynamics then is deine how this average is to be deined. At the microscopic level, no such property as temperature or entropy is deined. That is, when a system is deined with either classical mechanics and quantum mechanics, there no property entropy or temperature deined. So, these properties are not averages since they are not deined in the microscopic description o the system. A task or statistical thermodynamics is thus to deine these properties in terms o microscopic variables. 9. MICROSCOPIC DESCRIPTION OF THERMODYNAMIC SYSTEM 9.1 THE CLASSICAL MECHANICS DESCRIPTION 1, 2 One possibility to be considered is classical mechanics. I this approach is adopted, one o the irst questions to be considered is that o the level at which the system is to be described, or in other words how ine the description is to be. The system could be described at the molecular level, at the atomic level, at the electronic level, or at the nuclear level. At the chosen level, the particles will be approximated as mass points and 1.R. C. Tolman The Principles o Statistical Mechanics Oxord, London, D. N. Zubarev Nonequilibrium Statistical Thermodynamics CB, NY, FEBRUARY 15,

9 the independent variables required to describe the system at that level will be assumed to the independent variables o the classical mechanics description. Consider a macroscopic system that consists N-particles that each has a mass m and suppose the particles are conined within a volume V. We neglect any interaction between the system and the surrounding. Note that the value o energy o the system is not known, but it is constrained to be constant. From a thermodynamic point o view, the description is not complete, we only know N and V. The system coniguration at any time can be deined in terms o the coordinates o each mass point at that time. I the minimum number o coordinates required or this purpose is, then the coniguration o the system at any instant can be described in terms o the generalized coordinates, q 1, q 2, q, where the coordinates can depend on time q i q i () t (9-1) To deine the motion o the system we could use Newton s laws or we could use the Lagrangian 1, L where L K Φ (9-2) where K is the kinetic energy and the potential energy is Φ. For the system o particles the kinetic energy may be expressed K i q 2m i 2 (9-3) 1.C. Lanczos The Variational Principles o Mechanics U. Toronto Press, Toronto, FEBRUARY 15,

10 We shall assume the potential energy may be expressed in terms o the generalized coordinates: Φ Φ( ) q (9-4) where q denotes ( q 1, q 2 q ) The independent variables o the Lagrangian are L L(, q ) q and the equation o motion may be written d L dt q L 0 i 1 q i i (9-5) It is more convenient to work in terms o the Hamiltonian or the system because, as will be seen, this unction is constant during the system motion. The generalized momenta are deined as p i L q i (9-6) Then rom Eqs. (9-2) and (9-3) p i q i mq i j 1 m ---q 2 j 2 Φ( ) q (9-7) Then as the expression or the kinetic energy, we ind the amiliar FEBRUARY 15,

11 K 1 2m p 2 i (9-8) To orm the expression or the classical Hamiltonian,, a Legendre transorm is perormed on the Lagrangian with the objective o introducing as the independent variables,. In view o Eq. (9-6), the classical Hamiltonian H cl is given by q p H cl H cl pi i 1 L + q i (9-9) and ater dierentiating dh cl i 1 L q i dqi i 1 L dqi q i + + i 1 q idpi + i 1 p i dq i (9-10) and ater making use o the deinition o the generalized momenta. dh cl i 1 L q i dqi + p i dq i + i 1 i 1 q idpi + i 1 p i dq i (9-11) or dh cl i 1 L dqi + q q i idpi i 1 (9-12) Thus, H cl H cl (, ) q p (9-13) H cl q i L q i (9-14) FEBRUARY 15,

12 and rom the Lagrangian equation o motion or the system, Eq. (9-5), and the deinition o the generalized momenta d L ---- p dt i i 1 q i (9-15) Hence d H ---- p cl dt i q i (9-16) Also rom Eq.(9-12), we have q i H cl p i (9-17) The Hamiltonian equation o motion or the system then are Eqs (9-16) and (9-17). Example When a gas atom adsorbs on a solid surace, it is sometimes modeled as a harmonic oscillator. The adsorption mechanism is thought to involve the atom being trapped by the surace potential. The gas atom is then modeled as an a harmonic oscillator. As a simple limit, suppose a system consists o a single atom o mass m that only has one degree o reedom. Use the Hamiltonian equations to ormulate the equations o motion or the system. Suppose the atom is initially displaced a distance q 0 expression or the position o the mass as a unction o time. Solution 1. The classical Hamiltonian may be expressed rom its equilibrium position. Give the FEBRUARY 15,

13 H cl K + Φ p Φ 2m (9-18) To determine the expression or the potential, we require that Φ F ( κq) (9-19) where κ is the spring constant. Then Gas Φ q κq (9-20) Solid Model and ater integrating and requiring that potential to vanish when the atom is at the origin. FIGURE 9-1. Φ κq (9-21) It should noted that the expression or the potential is obtained rom an assumption. There is no method available or deriving this unction. Thus, rather than making the assumptions o thermodynamics to describe the system, the mechanics description o the system is based on assumptions that allow one to derive an expression or the potential. When the atom is displaced to a position q 0, it is being given an initial energy o κq H (9-22) Hence the Hamiltonian is given by FEBRUARY 15,

14 p H 2 κq cl m 2 (9-23) The Hamiltonian equations o motion are dp dt dq dt H cl κq q H cl p p --- m (9-24) and ater dierentiating the latter d dt d q dt 1 () --- ( κq) m (9-25) Note that Eq. (9-25) is Newton s second law and it has been obtained rom the Hamiltonian equations o motion. Ater making use o the initial condition q( 0) q 0 (9-26) or κ q q 0 cos ---t m (9-27) dp dt κ κ q 0 cos ---t m (9-28) The momentum o the atom is then K pt () Kq 0 sin --- t m --- K ( mk)q m K 0 sin ---t m (9-29) FEBRUARY 15,

15 and the classical Hamiltonian is H cl () t p 2 Kq K q 2 K 2m sin ---t 2 K --- K q m 2 0 cos ---t 2 + m K ---q (9-30) H cl K ---q (9-31) or p 2 2m Kq K ---q (9-32) Thus, the oscillator can be assigned any energy (i.e. value o q 0 ), and the system does not change its energy. To illustrate the motion o the system, consider a Euclidean space with coordinates pq,. Such a space is sometimes a phase or since the system consist o only one particle a µ -space p q q 0 Model FIGURE 9-2. CLASSICAL DESCRIPTION OF THE MOTION OF A HARMONIC OSCILLATOR. It only changes its orm rom potential and then back again. Also as may be seen rom Eq. (9-27), the maximum displacement o the mass is. We will compare these properties o a classical oscillator with the properties o a quantum mechanical oscillator ater quantum mechanics is ormulated. q 0 FEBRUARY 15,

16 9.1.1 MOTION OF A SYSTEM WITH F DEGREES OF FREEDOM Consider a thermodynamic system that has N particles and suppose that each particle has α degrees o reedom. Then or the system there are a total o Nα degrees o reedom: Nα (9-33) The Hamiltonian Eqs. o motion are d H ---- cl p dt i i 1 q i (9-34) and q i H cl i1 p i (9-35) The Hamiltonian has as its independent variables H cl H cl (, ) q p Then i this unction is dierentiated dh cl dt i 1 H cl H cl dq i dp i q i dt p i dt (9-36) and ater combining the Hamiltonian Eqs. o motion, one inds dh cl ṗ dt ( i q i q i ṗ ) i 0 i 1 (9-37) Thus, the Hamiltonian is a constant o the motion o the system. FEBRUARY 15,

17 To see what this means, consider a Euclidean space that has 2 axis or each degree o reedom. Hal o the coordinates corresponds to the particle positions and the other hal correspond to the generalized momenta. Such a space is also called a phase space or Γ - space. What inormation would be required to place a point in this space? At one time one would have to know all the positions o all o the particles q and the momenta o all o the particles. I this inormation were known, then since the potential energy has been p assumed to only be a unction o particle position Φ Φ( q 1 q ) (9-38) the potential energy o the system o particles would be ixed at the time t, although the value o the potential energy would not be known. Similarly, the kinetic energy o the system would be determined and its value could actually be calculated i the particle mass were known: K i p2 2m i (9-39) q t t+ p FIGURE 9-3. GAMMA SPACE FOR A SYSTEM OF N PARTICLES FEBRUARY 15,

18 Then since H cl is given by H cl K + Φ (9-40) Thus, i, q p were known at time t. the value o the Hamiltonian would be ixed. What Eq. (9-37) shows is that the value o the Hamiltonian would not change as the system moves in phase space. Since Hamiltonian is the total mechanical energy o the system, this is the result we would expect. When a quantity has this property, it is said to be a constant o motion o the system. The energy o the system is not known, but once its value has been assigned, according to classical mechanics, its value does not change provided the Hamiltonian can be written as indicated in Eq. (9-13).Thus i p2 2m i + Φ( q 1 q ) E (9-41) where E is a constant, the value o the which is unknown. But Eq. (9-41) shows that there is a relation between the p and the. Thus the motion o the system in phase space must q be on a surace in phase space (in a space o -1 dimensions). Also, the phase point can not move to an arbitrary position in phase space. The values o q are limited to being in the volume V and Eq. (9-41) must be satisied. Thus, the surace on which the phase point moves is limited. FEBRUARY 15,