4.0 Chapter Contents 4. THERMODYNAMIC EQUILIBRIUM

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1 24 4. THERMODYNAMIC EQUILIBRIUM Chapters 2 and 3 developed the basic armamentarium of the theory of equilibrium thermodynamics. We are now ready to address its central problem: the conditions of thermodynamic equilibrium (w 1.11). In particular, we consider the conditions under which an isolated composite system returns to a state of equilibrium after the lifting of an internal constraint (the removal of a barrier). 4.0 Chapter Contents 4.1 Representation of Equilibrium in Gibbs Space 4.2 Extremum Principles 4.3 The Extremum Principle for the Entropy 4.4 The Extremum Principle for the Internal Energy 4.5 Equivalence of the Extremum Principles for the Energy and the Entropy 4.6 Equilibrium Conditions in Terms of the Intensive Parameters: The Diathermal Case- Energy Representation 4.7 Equilibrium Conditions in Terms of the Intensive Parameters: The Diathermal Case- Entropy Representation 4.8 Equilibrium Conditions in Terms ofthe Intensive Parameters: The Diathermal Case- Partial Barrier Removal 4.9 Equilibrium Conditions in Terms of the Intensive Parameters: The Adiabatic Case 4.10 Direction of Change in the Attainment of Equilibrium 4.11 Mulfibody Thermal Equilibrium m The Zeroth Law of Thermodynamics 4.1 Representation of Equilibrium in Gibbs Space An equilibrium state (w 1.12) is represented by a point in thermodynamic configuration space. The equilibrium states accessible to a given system lie on the fimdamental surface. 4.2 Extremum Principles An extremum principle minimizes or maximizes the fundamental equation subject to certain constraints. The principle of maximum entropy and its equivalent, the principle of minimum internal energy, are the fundamental principles of equilibrium thermodynamics. Alternative extremum principles will be introduced in Chapter The Maximum Principle for the Entropy In accordance with Postulate IV, upon the removal of an internal barrier in an isolated composite system the extensive parameters of the system assume those values which maximize the entropy over the manifold of equilibrium states consistent with the remaining constraints. The extremum principle for the entropy states: "At equifibrium the value of any unconstrained parameter of an isolated thermodynamic system is such that the entropy is maximized at constant internal energy".

2 4. THERMODYNAMIC EQUILIBRIUM 25 The entropy maximum principle thus characterizes the eq~bfium state as one of maximum entropy for a given total internal energy. Figure4.3 flhstrates this for the physical simple system Fig. 4.3 The equilibrium state A as a point of maximum S for constant U i.e.~ by Mathematically the principle is expressed by the usual conditions for a maximum, (ds)v = 0 and (d2s)v < 0. (4.3) The first of these is the condition (or criterion) of thermodynamic equilibrium. The second is the condition (or criterion) of thermodynamic stabifity which ~ form the subject of Chapter The Extremum Principle for the Internal Energy An equivalent extremum principle can also be establi~ed for the internal energy as illustrated below in: Figure 4.4, again for the physical simple system. The energy minimum principle characterizes the equ~brium state as one of minimum energy for a ~en total entropy. It reads: "At equilibrium the value of any unconstrained parameter of an isolated thermodynamic system is such that the internal energy ~ minimized at constant entropy".

3 26 I. EQUILIBRIUM THERMODYNAMICS Fig. 4.4 The equilibrium state A as a point of minimum U for constant S Mathematically this is expressed by the conditions for a minimmn, i.e., by (du)s = 0 and (d 2 U)s > 0 (4.4) where the first is again the cotdition (or criterion) of thermodynamic equilibrium, and the second is the cotmition (or criterion) of thermodyt~amic stability (see Chapter 18). 4.5 Equivalence of the Extremum Principles for the Energy and the Entropy The extremum principles for the internal energy and for the entropy express the condition of equilibrium of the isolated thermodynamic system in the entropy representation and in the energy representation, respectively. They are thus equivalent and may be used interchangeably. To prove this assertion, assume that, upon the establishment of a new equilibrium, the internal energy is not minimum while the entropy is maximum It would then be possible to withdraw work from the system at constant entropy, and reinject it in the form of heat. This would restore the system to its original energy. However, the resultant increase in entropy would be inconsistent with the requirement that the equilibrium state be one of maximum entropy. Consequently, the two extremum principles imply each other.

4 4. THERMODYNAMIC EQUILIBRIUM Eq~brium Con~tions in Terms of the Intensive Parameters: The Diathermal Case- Energy Representation The conditions of thermodynamic equil~rium were stated in w and 4.4 in terms of the extensive parameters, U and S. Equilibrium conditions can, however, also be estabfished in terms of the intensive parameters. Here, and in w to 4.9 we discuss these conditions as they apply to a smgle-component simple system. Genera~ation to multicomponent simple systems and to non-single systems is straight-forward. We distinguish two cases: the diathermal and the adiabatic case. We first investigate the conditions of equilibrium in the diathermal case in the energy representation. Removal of a barrier, inside an isolated composite system, each a single-component smlple system, both composed of the same kind of matter, lifts the constraints of adiabaticity, rigidity, and impermeability between the two subsystems. Fig. 4.6 Isolated composite system consisting of two subsystems, A and El Since the composite system as a whole is isolated, the following conservation (isolatioto constraints apply: SA + SB = constant dsa = - dsb (4.6)i.~ VA + VB = constant dva = -dvb (4.6)1.2 NA + NB = constant dna = -dnb (4.6)1.3 Equations (4.6)1 gate that the total entropy, volume, and number of moles of the composite system remain constant in a virtual change of the internal energy at equilibrium. The internal energy being additive, it follows by Eq.(3.10)l and the condition of equilibrium, du = 0, that du = (TA - TB) dsa - (PA - PB) dya + (~A - ~B) dna = 0 (4.6)2 Because dsa, dva, and dna represent infinkesimal changes in independent variables, we must have

5 m 28 I. EQUILIBRIUM THERMODYNAMICS TA = TB (4.6)3.1 PA-- PB (4.6)3.2 and /-ZA =,tzb (4.6)3.3 when equilibrium is reestablished after removal of the barrier. Equations (4.6)3 are the equilibrium conditions in terms of the intensive variables in the energy representation. They express, respectively, the criteria of thermal, mechanical, and diffusional equilibrium. The corresponding stability criteria will be discussed in w 18.8 to Equilibrium Conditions in Terms of the Intensive Parameters: The Diathermal Case- Entropy Representation We now examine the conditions of equilibrium in the diathermal case in the entropy representation. The first of the conservation constraints becomes UA + UB = constant dua = - dub (4.7)1 and the other two remain unchanged. Thus now the total energy, volume, and number of moles of the composite system are constant in a virtual change of the entropy at equilibrium. It follows by the condition of equilibrium, ds = O, that ds= TA TB TA TB TA TB (4.7).2 and this furnishes the equilibrium conditions 1 1 TA TB (4.7)3.1 PA TA PB TB (4.7)3.2 ~A #B TA TB (4.7)3.3 in the entropy representation. Equations (4.6)3 and (4.7)3 are clearly equivalent.

6 4. THERMODYNAMIC EQUILIBRIUM Equffibrium Conditions in Terms of the Intensive Parameters: The Diathermal Case- Partial Barrier Removal If the barrier remains rigid and only the constraints of adiabaticity and impermeability are lifted, dva --dvb --O, and only the first and third of the equilibri~ conditions are obtained in either the energy or the entropy representation. Similarly, when the barrier remains impermeable so that only the constraints of adiabaticity and rigidity are lifted, dna = dnb = O, and only the first and second of the eqtfilibrium conditions ensue. Finally, when the barrier remains rigid and impermeable and only the constraint of adiabaticity is lifted, only the first of the equih'brium conditions results. In all four cases in which the barrier becomes diathermal, the equilibrium conditions are sufficient to characterize the equilibrium state. Although we may have no equil~rium condition in terms of the pressures or chemical potentials, the system remains fully determined because the corresponding extensive parameters, the volumes or mole numbers, are known since they can be measured before the barrier is lifted and they remain constant thereafter. 4.9 Equilibrium Conditions in Terms of the Intensive Parameters: The Adiabatic Case The special nature of heat as a form of energy exchange (w 2.20) renders the adiabatic case indeterminate. Let us again remove a barrier between two subsystems of an isolated composite system, each a single-component simple system, and both co~osed of the same kind of matter. If the barrier remains adiabatic but becomes movable and permeable, there is no heat flux and the energy transfer between the two subsystems consists only of work and/or mass action. Hence, by the energy minimum principle, du = - (PA - PB) dva + (,A -,B) dna = o. (4.9) Thus we recover the second and third of the equilibrium conditions we had found for the diathermal case, but no condition can be found for the temperatures. Clearly, if the barrier is adiabatic and impermeable, we obtain only PA = PB, while, if it is adiabatic and rigid, we can find only #A =/~B. Application of the entropy maximum principle furnishes the same indeterminate result. Therefore, if the barrier remains adiabatic, nothing can be said about the ten~eratures in the two subsystems and their entropies are not known because they cannot be measured directly. Indeed, it would be possible to withdraw a certain amount of heat from one subsystem and inject another amount into the other so that dsa - - dsb. The system could again be brought to equilibrium but the entropies of the subsystems would have changed. Thus, in the adiabatic case, the system is not completely determined Direction of Change in the Attainment of Equilibrium Lifting only the adiabatic constraint, Eq.(4.7)2 may be rex~a~en as / 1 1 \ TA TB \ /

7 30 I. EQUILIBRIUM THERMODYNAMICS where we have assumed, for the sake of simplicity, that TA --~ TB, and that the changes are finite. If TA > TB, then, since AS is necessarily positive, A UA < 0, i.e., the internal energy in subsystem A decreased, hence heat flowed from subsystem A to subsystem 13. Thus: "Heat flows from the hotter to the colder body", in accordance with common experience. Removing only the constraint of rigidity and letting TA = TB = T, Eq.(4.7}2 becomes PA--~ ds = dva. (4.10h T If PA < PB, then dva is necessarily negative, i.e., the volume of subsystem A has decreased, i.e., "All increase m pressure decreases the volume ". This again is in accordance with normal expectation. Finally, let us assume that the impermeable wall has been made permeable but stays rigid while the temperatures are the same. We then have ds = #A -- #B dna (4.10)3 T If #A ~> /-ZB, then dna must be negative. This leads to the conclusion that: "Matter flows front regions of high to regions of low chemical potential." The statement may be interpreted as saying that matter flows from regions of high concentrations to those of lower ones, once again in accordance with normal experience Multibody Thermal Equilibrium- The Zeroth Law of Thermodynamics Repeated application of the procedure outlined in w 4.6, removing adiabatic barriers only, leads to the following realization: "Two bodies that are in thermal equifibrium with a third body will be in thermal equifibrium with each other". This statement, known as the Zeroth Law of Thermodynamics, is the basis of thermometry.

CH341 Test 2 Physical Chemistry Test 2 Fall 2011, Prof. Shattuck

CH341 Test 2 Physical Chemistry Test 2 Fall 2011, Prof. Shattuck Physical Chemistry Test 2 Fall 2011, Prof. Shattuck Name 1 Constants: R = 8.3145 J K -1 mol -1 = 0.083145 L bar K -1 mol -1 R = 0.08206 L atm K -1 mol -1 1 F = 96485 C mol -1 1 atm = 1.01325 bar 1 bar