Chapter03 Introduction to Thermodynamic Systems

February 13, 2019

This chapter introduces the concept of thermo- Remark 1.1. In general, we will consider three kinds dynamic systems and digresses on the issue: How of exchange between S and Se through a (possibly ab- complex systems are related to Statistical Thermo- stract) membrane: dynamics. To do so, let us review the concepts of (1) Diathermal (Contrary: Adiabatic) membrane Equilibrium Thermodynamics from the conventional that allows exchange of thermal energy. viewpoint. Let U be the universe. Let a collection of objects (2) Deformable (Contrary: Rigid) membrane that S ⊆ U be defined as the system under certain restric- allows exchange of work done by volume forces tive conditions. The complement of S in U is called or surface forces. the exterior of S and is denoted as Se. (3) Permeable (Contrary: Impermeable) membrane that allows exchange of matter (e.g., in the form 1 Basic Definitions of molecules of chemical species). Definition 1.1. A system S is an idealized body that can be isolated from Se in the sense that changes in Se Definition 1.4. A thermodynamic state variable is a do not have any effects on S. macroscopic entity that is a characteristic of a system S. A thermodynamic state variable can be a scalar, Definition 1.2. A system S is called a thermody- a finite-dimensional vector, or a tensor. Examples: namic system if any possible exchange between S and temperature, or a strain tensor. Se is restricted to one or more of the following:

(1) Any form of energy (e.g., thermal, electric, and magnetic) Deﬁnition 1.5. A thermodynamic state variable is called extensive if, within a homogeneous system S, (2) Mechanical work done by volume forces or sur- it is proportional to the mass of the system. A ther- face forces modynamic state variable is called intensive if it is (3) Any form of matter (e.g., molecules of chemical independent of the mass of the system. species)

Remark 1.2. The thermodynamic state variables Deﬁnition 1.3. A thermodynamic system S is called under consideration are either extensive or intensive. closed or isolated if there is no exchange of any form Each extensive state variable has a unique intensive with Se. state variable.

1 Remark 1.3. The set of thermodynamic state vari- Deﬁnition 1.10. A thermodynamic system S is ables for a system is determined not only by the phys- called a simple system under the following conditions: ical nature of the system S and its transformations, but also by the measurement scheme and its preci- (1) Macroscopic homogeneous, isotropic, (electri- sion. cally and magnetically) uncharged, and chem- ically inert;

(2) Sufficiently large volume having negligible sur- Definition 1.6. The set of values of the thermody- face effects; namic state variables that characterize a system S at a certain time t ∈ [t0, tf ] constitute a thermodynamic (3) Not being acted upon by exogenous electric, state Ψ(S; t) of the system. magnetic, or gravitational forces.

Deﬁnition 1.7. A system S is said to be in ther- Deﬁnition 1.11. A composite system is a combi- modynamic equilibrium if the thermodynamic state nation of two or more simple systems. A composite Ψ(S; t) does not evolve with time t. system is called closed if it is surrounded by a (possibly abstract) wall that is restrictive with respect to Remark 1.4. In general, thermodynamic systems do the total energy, total volume, and total mole num- evolve with time under the action of external stimuli. bers of each component of the composite system.

Remark 1.5. Thermostatics is the science that com- Remark 1.6. The individual simple systems within pares systems in thermodynamic equilibrium. For ex- a closed composite system need not be closed. ample, thermostatics describes the transition from a state of thermodynamic equilibrium Ψ(S) to another e S state of thermodynamic equilibrium Ψ( ). Thermo- Definition 1.12. Internal constraints of a composite dynamics, in its main sense, deals with phenomena system are the constraints that may prevent the flow outside a state of equilibrium but not too far from of energy, volume, or matter among simple systems the equilibrium. If these phenomena occur close to that comprise the composite system. the equilibrium states, then they belong to equilibrium thermodynamics; otherwise, they belong to non-equilibrium thermodynamics. Remark 1.7. If a composite system is in equilibrium with respect to certain internal constraints and Definition 1.8. The transition from one thermody- if some of these constraints are removed, then the namic state to another is called a thermodynamic composite system comes to a new equilibrium state. process. That is, certain processes that were previously disal- lowed may now become allowed; in the terminology of Classical Mechanics, they become virtual processes. Definition 1.9. A thermodynamic process is said In this context, the basic problem of thermodynam- to be reversible if the inverse evolution implies the ics is determination of the equilibrium state after re- reversal of the action of external stimuli; otherwise moval of internal constraint(s) of a closed composite the thermodynamic process is called irreversible. system.

2 2 Postulates of Equilibrium The formulation of thermodynamic formalism sug- Thermodynamics gests interchangeability of two important extensive parameters, entropy S and energy U. The notion of thermodynamic equilibrium introduced Entropy Maximization Principle: The equi- in Definition 1.7 is qualitative and suggestive rather librium value of any unconstrained internal param- than formal and precise. Consequently, it is necessary eter is to maximize the entropy of the system for to rephrase the description of thermodynamic equi- the given value of total energy. The principle of en- librium in a manner that will provide a basis for fur- tropy maximization implies that, under the equilib- ther theoretical development. Now, formal criteria of rium condition, the following two conditions hold: (i) simplicity allow the possibility of description in terms dS = 0 and (ii) d2S < 0. Furthermore, Postulate of a small number of macroscopic variables. The fol- IV (Nernst Postulate) states that S = 0 at T = 0. lowing postulates, originally proposed by Professor Two equivalent representations of the basic thermo- Laszlo Tisza of Massachusetts Institute of Technol- dynamic formalism are formulated below. ogy, make it possible to experimentally validate the Energy Minimum Principle The equilibrium theory of equilibrium thermodynamics. value of any unconstrained internal parameter is to Postulate I : There exist particular states, called minimize the energy of the system for the given value equilibrium states, of simple systems that, macro- of total entropy. scopically, are characterized by the total internal Whereas entropy maximization characterizes the energy U, the volume V , and the mole numbers equilibrium state for a given total energy, energy min- ··· N1,N1, ,Nr of the chemical components. imization characterizes the equilibrium state for a Postulate II : There exist a function, called the given total entropy. entropy S of the extensive parameters of any composite system, defined for all equilibrium states and Theorem 2.1. The principle of entropy maximiza- having the following property: tion is equivalent to that of energy minimization. The values assumed by the extensive pa- Proof. To prove that Postulate II (entropy maximiza- rameters in the absence of an internal con- tion) ⇒ energy minimization, we adopt a contraposi- straint are those that maximize the entropy tive approach: Energy Not Minimum ⇒ Entropy Not over the manifold of constrained equilib- Maximum. rium states. Let the total energy of a thermodynamic system do Postulate III : The entropy of a composite sys- not have the smallest possible value consistent with tem is additive over the constituent subsystems. The with the given entropy S, i.e., U = Umin+∆U, where entropy is continuously differentiable and is a mono- ∆U > 0. By withdrawing an amount ∆U of energy tonically increasing function of energy. from the system in the form of (reversible) work, the Postulate IV (Nernst Postulate or Third Law of entropy is maintained constant at S. Let the energy Thermodynamics): The entropy S of any system van- ∆U be returned to the system in the form of heat. ishes at absolute( ) zero temperature, i.e., in the state The entropy of the system increases by the relation- ∂U ship δQ = T dS. Thus, the system is returned to the for which ∂S = 0. V,N1,N2,··· ,Nr original energy state with increased entropy. This is a Remark 2.1. Thermodynamics is macroscopic, contradiction to the hypothesis of maximum entropy. whereas Statistical Mechanics and Quantum Mechan- To prove the other way, it will be shown that: En- ics are nominally microscopic. As the theory unfolds tropy Not Maximum ⇒ Energy Not Minimum. Let in a continuous sequence from quantum mechanical the present entropy S < Smax. Let an amount δQ to macroscopic considerations, the transition occurs of heat be added to the system to increase its en- gradually. tropy to Smax. Consequently energy of the system is

3 increased. So, the system cannot be originally be a ∑ e− n e state of minimum energy unless its entropy is maxi- value of S j=0 FjXj; and Ω0 is a normalizing con- mum. stant. In view of replacing Postulate II by Postulate IIa, Remark 2.2. Regardless of how the equilibrium con- the following observations are made: dition has been attained (e.g., by entropy maximiza- • tion with given energy, or by energy minimization The properties attributed to Postulates III and with given entropy), the final equilibrium condition IV are now interpreted to be applicable to the satisfies both extremal conditions. instantaneous entropy. • If the extensive parameters xf are assigned their The four postulates, presented earlier, correspond k respective average values X , the instantaneous to equilibrium thermodynamics, which provide a con- k entropy assumes the equilibrium value of the en- venient method for demarkation between equilibrium tropy. That is, the equilibrium entropy is de- thermodynamics and statistical mechanics. In this ( ) fined as: S X ,X , ··· ,X . way, the theory of fluctuations can be removed from 0 1 n statistical mechanics and added to thermodynamics. Specifically, Postulate II (on entropy maximiza- 3 Statistical Significance of En- tion principle) is generalized as Postulate IIa and the remaining Postulates I, II, and IV are interpreted tropy slightly differently. Let a thermodynamic system with fluctuating Qualitative interpretation of entropy at the micro f f f level provides ample insight to macroscopic thermo- extensive parameters X0, X1, ··· , Xn interact with a reservoir whose average values of the respective dynamic theory. parameters are X0,X1, ··· ,Xn. The parameters • f A macroscopic system possesses an enormous Xk, k ≤ n fluctuate by virtue of transfers to and ∼ 25 f (e.g., 10 ) number of micro coordinates. from the reservoir. The( probability) that( Xk, k ≤ n is) f f • found in the range Xk − dXk/2 to Xk + dXk/2 Transitions among different microstates occur f f f extremely rapidly, where macroscopic observa- is πdX0dX1 ··· Xn, where π is the joint probability density function. The following postulate specifies a tions are many orders of magnitude low. particular functional form of π. • Macroscopic observations respond to statistical Postulate IIa: There exists a function Se of fluc- f f f averages over micro coordinates. tuating extensive parameters X0, X1, ··· , Xn) called e f f f • Observed macroscopic measurements are aver- the instantaneous entropy, S(X0, X1, ··· , Xn), having the following property. aged effects of microscopic phenomena and are f f f obtained as extensive parameters such as energy, The probability πdX0dX1 ··· Xn that the fluctuating extensive parameters of the system are in the volume, and mole numbers. range dXf , dXf , ··· , Xf is given by the following ex- 0 1 n S S pression: Definition 3.1. Let be a system and N( ) be the number of microstates in the system. The entropy ( n ) ( ∑ ) of the system S is defined as: S(S) = kB log (N(S)), e − e − e π = Ω0 exp S FjXj Smax /kB where kB is the Boltzman constant, implying that j=0 the entropy of a macrostate is proportional to the e logarithm of the number of microstates. where kB is Boltzman constant, S is the instanta- Fundamental Law of Quantum Statistics: ∂Se neous entropy of the system; Fj ≡ is an inten- ∂Xj Every microstate, consistent with a given macrostate, sive parameter of the reservoir; Smax is the maximum is equally probable in a closed system.