Summer Lecture Notes Thermodynamics: Fundamental Relation, Parameters, and Maxwell Relations

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1 Summer Lecture Notes Thermodynamics: Fundamental Relation, Parameters, and Maxwell Relations Andrew Forrester August 4, The Fundamental (Difference or Differential) Relation of Thermodynamics Conservation of Energy A physical system is simply a set of physical objects, real or imagined. I define the complement of a system to be the environment of that system. For a system of total energy E (with respect to an inertial reference frame) and its environment of energy E, E = E, so energy does not appear or disappear from the universe. (Thus, for an isolated system E = 0.) Note that this is not a continuity relation; presumably energy may disappear from one system if the same amount of energy immediately appears in another system, no matter how distant that other system is. If we add continuity of energy, or local conservation of energy, that will state that energy is transferred via local physical processes only. Note that energy is not something that can flow from place to place or literally be transferred. It always refers to physical states of systems or motions and configurations of particles and waves, or wavicles 2, and is only meaningful when relating those states. Only differences in energy, or transfer of energy, is physically significant. 1.2 Heat and the Relation The heat absorbed by a system (from its environment) is defined as Q E + W, where W is the work performed by the system on its environment. (Using conservation of energy, this gives us E = W Q as well.) This definition is actually trickier than it looks, because authors such as Callen [1] and Reif [2] generalize the notion of work (to include electrochemical work, for example) and restrict the meaning of heat, while it is also perfectly reasonable to restrict the meaning of work to its definition in Newtonian mechanics and allow heat to encompass any energy exchanged by nonmechanical means. (Perhaps all generalized work really does derive from mechanical work...) Most unrestrictedly, heat could encompass the disappearing/reappearing energy mentioned earlier. However, if we provide a physical theory of heat as energy exchanged in specific ways and describe its mechanisms, then such teleportation of energy would presumably not be included, and this 1 This relation should not be confused with other thermodynamic fundamental relations, to be discussed in the next section. 2 I call things that are easily modeled with quantum mechanical wavefunctions wavicles due to their simultaneous wave and particle nature. 1

2 equation would not be a definition, but rather, a postulated equation stating that energy may only be exchanged by means of work and heat. Thus, the so-called fundamental relation of thermodynamics, has three interpretations: tautology, definition of heat (restated), or E = Q W, conservation of energy (if a theory of heat is expounded rather than Q being defined mathematically). So, it seems that the name Fundamental Thermodynamic Relation would only make sense in the third case (which is not how Reif presents it). It should also be said that this difference relation is often used in its differential form de = d Q d W, since this is the relevant form for quasi-static processes, which are a major focus in thermodynamics. 2 Thermodynamic Parameters 2.1 Equilibria, Parameter Types, Ensembles Thermodynamics started with an interest in relating heat, in its intuitive colloquial sense, and work (ergo thermo and dynamics ). Hence, whether anyone knew it or not, the interest was in the energy associated with macroscopic physical systems. To say anything significant, we must of course relate the energy to the parameters or variables that describe the system, using equations or inequalities. (The energy must be a function of the parameters.) But we simplify the investigation by examining systems in thermodynamic equilibrium (simultaneous thermal, mechanical, chemical, electromagnetic equilibria) or near equilibrium (in quasi-static processes), where the parameters are constant and well-defined 3 (or smoothly varying during quasi-static processes). Furthermore, the parameters that describe equilibrium states are not allowed to depend on past states, so we are free of hysteresis and other memory-retaining effects. (See Callen, Section 1-5.) Macroscopic physical systems are sets of many (meaning on the order of Avagadro s number or larger) particles or wavicles, which are microscopic. Using this assumption, we immediately distinguish between macroscopic states or macrostates, which describe the system as a whole without referring to the microscopic particles, and microscopic states or microstates of the individual particles. Of course, we should know what states means as well. By states of systems, I mean particular configurations and motions of the system and the constituents of the system that can be characterized by measurable parameters. (When we get into quantum states, however, things become more abstract and we may (perhaps?) actually know more about the energy of states than the actual configurations and motions of the constituent wavicles.) Accessible states are those that do not defy the constraints placed upon the system in question. So a system in equilibrium is by definition in one macrostate, but its constituents may be in many different microstates (and the arrangements of microstates could conceivably be changing or evolving without detection (true?)). State parameters are parameters that describe the state of a system. Parameters that merely relate states (such as work W and heat Q) are not state parameters. If a state parameter is considered 3 By well-defined, one might mean that before one can talk about the volume or the pressure of a system, there must be very little fluctuations over space in the instantaneous values of those parameters, or at least that those definitions are meaningful in some sense. 2

3 to be a dependent variable of other independent parameters that form a complete set, then it is called a state function. (Complete sets are discussed in a moment.) An external parameter, or equivalently, a macroscopic parameter, characterizes a system or one of its subsystems or components, without reference to the individual particles that make up the system. An internal parameter, or microscopic parameter, is a hidden parameter that characterizes the microstates. Macroscopic/External Parameters Energy (See Callen, page 13), Entropy (why?), Temperature, Volume, Pressure, Number of particles (See Callen, page 13), Chemical potential, applied Electric and Magnetic Fields, etc. Microscopic/Internal Parameters Quantum numbers, Particle energy, and, as far as I can tell, particle Position and Momentum (The word external leads one to question why entropy or energy would be considered external and to be suspicious about the external-ness of number-of-particles.) It seems that intensive and extensive parameters can only be rigorously defined after we ve defined what a complete set of parameters is. In a loose sense, intesive parameters are independent of the size of the system while extensive parameters scale linearly with the size of the system. For example, if a system doubles in volume and in its number of particles simultaneously, any extensive parameter of that system will double as well while the intensive parameters will remain at their previous values. Note that parameters are not necessarily either intensive or extensive, but to first order or further, it seems that all thermodynamic parameters are one or the other. 2.2 Complete Sets Parameters (Internal and External...) The energy of a system in equilibrium must be a function of external parameters, completely describable by macroscopic characteristics. (Using the fundamental postulate of thermodynamics, ensembles, and partition functions, it is simultaneously a function of internal and external parameters, and you can use the internal parameters to derive the functions that only include external parameters.) In general, the energy must be a function of internal parameters as well, but we limit our scope to equilibrium states. Once you ve found experimentally a set of parameters that fully specify the energy, you ve found a complete set. NOTE: There is no known way to algorithmically determine what makes a complete set for a given system and its contraints.... E = E(X 0, X 1, X 2,..., X c ) Incidentally, the energy of a system A with constraints C always depends on what that system is and what the constraints are and what state we ve chosen as our reference state R (since only change in energy is physically meaningful): E = E(A, C, R; X 1, X 2,..., X c ) However we ll take A, C, and R as defined and understood whenever we write the letter E. It also turns out that, to first order, energy is additive amongst subsystems,... (?) so energy should be a homogeneous function of order one, and all of the parameters in the complete set must be extensive. (What does this imply for the other thermodynamic potentials, which are functions of intensive and extensive parameters?) λe = E(λX 0, λx 1, λx 2,..., λx c) We also use our microscopic theory of matter to create internal parameters and relate them to themselves and the external parameters. We come up with a set of functions (where each of them is apparently traditionally signified by ρ, ω, or n) called the ((parameter-space-adjective) 3

4 number-) density of states 4 and refers to the number of microstates in an infinitesimal region of a particular parameter-space, where the parameter(s) in question may be internal or external (such as the particle energy or momentum or the system energy(?)). (When X is the energy, we usually use the letter ω 5, unless we re already using that letter for frequency.) The density of states can be written as ρ(x a, X b,..., X z X j = X j0, j a, b,..., z), or more simplified as ρ(x a, X b,..., X z ) given that the remaining parameters are understood to be held constant. Using the fundamental postulate of (modern micro-statistical) thermodynamics 6, we get the probability density w(x a, X b,..., X z X j = X j0, j a, b,..., z), usually simplified notationally to w(x a, X b,..., X z ): w(x a, X b,..., X z ) P (X a < X a < X a + dx a ) = ρ(x a, X b,..., X z ) C ρ(x a, X b,..., X z) dx a dx b dx z E = w(x a, X b,..., X z )E r (X 1, X 2,..., X c?; x 1, x 2,...x q )) C ω(e) E f ω = ω ( {x 1 }, {x 2 },..., {x p } ; E, X 1, X 2,..., X c ) S k ln(ω) 2.3 Generalized Forces and Displacements; Conjugate Pairs de = 1 dx dx c dx c P k k de = P k dx k k=0 (P k must be intensive) Callen (page 284) defines the P k s without the negative sign and calls them the intensive parameters. β 1 kt ln ω(e) β 1 (E 1 ) = β 2 (E 2 ), T 1 (E 1 ) = T 2 (E 2 ) Proven below: ln ω(e) k = βp k 4 One could also call it the (parameter-space-adjective) number-of-states density. 5 Reif and most texts tend to mainly speak of Ω(E) rather than ω(e), where Ω(E) = ω(e)δe for some unspecified, very small amount of energy δe. 6 I would phrase the postulate as The constituents of an isolated system in equilibrium are equally likely to be in any of their accessible microstates. 4

5 2.4 Important Conjugate Pairs P dv We assume a non-viscouse fluid with pressure P is confined in a closed two-dimensional surface in three-dimensional space. Any arbitrary infinitesimal deformation of the surface can be seen as equivalent to many tiny cylindrical bulges or divets at the surface (tangential motion or shearing does not matter since we assume the fluid within the boundary is non-viscous). Each of the little cylinders will have associated with it its own work d 2W = df ds = P da ds, and integrated over the whole surface, you get d 2W S = P S da ds = P dv : d W = P dv In a case of uniform expansion, you d have something like V = A s + A s, which yields dv = A ds. T ds (This section was slapped together using Reif, section 3.8 and other sections.) Given a system made of particles, each particle has an energy that assumes a particular value from a set of accessible state energies E s, where s indicates a particular accessible microscopic state. We assume that the number of accessible states is countable and that there there is a ground state with a minimum energy E min for this set. Define Φ(E) to be the number of accessible microstates with E s E. (Φ is a stepped function that is discontinuous at E = E s for each s.) Let the energetic number-of-microstates density 7 or density of states 8 be defined by ω(e) Φ(E + de) Φ(E) Φ(E + δe) Φ(E) lim = δ Φ(E) de δe de δe δe, where δe is chosen to be as small as possible such that that ω is continuous over E. We may call δ/δe a thermodynamic derivative with respect to E, or, more mathematically, a discontinuous derivative with respect to E. Since each E s is a function of the parameter(s) X, Φ and ω depend on X: Φ(E + δe; X) Φ(E; X) δ Φ(E; X) ω(e) = ω(e; X) δe δe Note that E does not depend on X; only the individual E s energies do. Let Φ(E, Y ; X) be the number of accessible microstates that have E s E and s Y. ω(e; Y ; X) δ Φ(E, Y ; X) δe Y I m going to leave that density unnamed. Note that ω(e; X) = Ymax = Y min ω(e, Y ; X) dy = δ Φ(E, Y ; X) Y δe ω(e, Y ; X) dy We define Y, the mean value of Y over all accessible states (where each state is considered equally likely) as Ymax Y Y (E; X) = min ω(e, Y ; X) Y dy Ymax Y min ω(e, Y ; X) dy = 1 ω(e; X) = = P 7 I made this name up for clarity. 8 There are more kinds of density of states than this one. ω(e, Y ; X) Y dy = s 5

6 If we examine ω with a small change in X, dx, small enough that we can neglect changes in Y, we see ω(e, Y ; X + dx) = ω(e Y dx, Y ; X) ω(e, Y ; X) ω(e, Y ; X + dx) ω(e, Y ; X) = lim dx 0 dx = lim dx 0 ω(e de, Y ; X) ω(e, Y ; X) de ω(e, Y ; X) = lim dx 0 = ω(e, Y ; X) Y Y dx dx de dx de Y dx So, using the definition of Y, ω(e; X) = [ ω(e, Y ; X) dy ] ω(e, Y ; X) = dy ( ) ω(e, Y ; X) = Y dy = ω(e, Y ; X) Y dy = ( ) ω(e; X) Y = ω(e; X) Y ω(e; X) Y *The second and fourth equalities should be justified when I have time and space. This implies that = 1 ω(e; X) ω(e; X) 1 = ω(e; X) ω(e; X) = Y Y = β(e; X) Y Y = βp + P Y Y 6

7 So, in general, d ln ω(e; X k ) = de + dx k k ( = β de + βp k + P ) k dx k = β de + β P k dx k + dx k = β de + βd W + = β(de + d W ) + = β d Q + thus, using kt 1/β and S = k ln ω(e; X k ), we have ( d Q = 1 β d ln ω(e; X k ) dx k ( ) = T d k ln ω(e; X k ) kt = T ds kt dx k dx k dx k dx k ) dx k However, if we use ω(e; X) E f, where the number of degrees of freedom f is of the order of Avogadro s number, and Y / Y /E (justify this later), then = ln Ef Y Y E Y Y f 1 E Y Y E (f + 1) Y E f Y E ln Ef Y = βp Y and we get rid of the second term in the equation above: d Q = T ds 7

8 This relation is of course true in the special case where there is no work done and de = d Q. We can also use a Taylor expansion of ln ω to show that for a heat reservoir (See Reif page 107). µ dn 2.5 Other Conjugate Pairs dw Q = T S d W c = µ dn In general, we seem to avoid conservative work and the associated potentials. F g dy Presumably, if E is total energy rather than internal energy, then it could include gravitational potential and there could be a conservative gravitational work or potential term, although I m not sure if it would be important: Electric, Magnetic terms 3 Maxwell Relations No time for this, dudes. References dw g = F g dy [1] Herbert B. Callen: Thermodynamics and an Introduction to Thermostatistics, Second Edition, John Wiley & Sons (1985) [2] F. Reif: Fundamentals of Statistical and Thermal Physics, McGraw-Hill (1965) 8