16 Free energy and convex analysis. Summary Helmholtz free energy A tells us about reversible work under isothermal condition:

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1 16 Free energy and convex analysis 161 Summary Helmholtz free energy A tells us about reversible work under isothermal condition: A = W. Negative free energies are convex functions obtained from internal energy by the Legendre transformation. Convex analysis guarantees that Legendre transformations are involutions, so internal energy can be recovered from free energies. Key words Helmholtz free energy, Legendre transformation, Gibbs free energy, enthalpy, freeenergy minimum principle, convex function The reader should be able to: Understand A W. Understand the (geometrical) meaning of the Legendre transformation. In reality, the variables S, V, X, for the ordinary Gibbs relation (13.16) are often hard to control or at least awkward. For example, to keep volume constant may be more difficult than to keep pressure constant. To keep temperature constant may be easier than an adiabatic condition Isothermal system Under T constant (an isothermal condition) we must allow free exchange of heat between the system and its ambient world to maintain the system temperature. Therefore, we wish to pay attention to the right-hand side of de d Q = d W = P dv + xdx. (16.1) Under a quasistatic condition, d Q = T ds, so (16.1) reads isothermally de T ds = d(e T S) = P dv + xdx. (16.2) This implies that the introduction of the quantity A = E T S, (16.3)

2 162 Free energy and convex analysis called the Helmholtz free energy, 1 is convenient. Notice that for an isothermal process da = d W. (16.4) Thus, A is the work the system obtains by a reversible process under constant temperature (i.e., a reversible isothermal process) A by an irreversible process Work W is always measurable with the aid of mechanics. What happens if the work exchange is not reversible under isothermal conditions? 2 If we do work W to the system irreversibly (= that allows some dissipation of work), the system must discard heat to the heat reservoir to prevent the final equilibrium state from becoming hotter (recall Planck s principle in Section 13). This implies that, even if we do the actual work W, effectively the system receives less energy from the supplied work. We must conclude A W (16.5) under isothermal conditions. That is, to change the system free energy, we must provide the system with at least A of work (called the minimum work principle). Pay attention to the sign convention: coming in is +. Suppose the system does work of amount W (W < 0) to the outside. This implies that the system is supplied with work of W = W, so according to (16.5), A W must hold. Since A < 0, and A is the amount of decrease of the system free energy, when the system does work to the outside, (16.5) implies A W. (16.6) That is, the work produced by the system cannot exceed the amount of the free energy lost by the system. The maximum work we can gain from the system is A (the maximum work principle) Clausius inequality and work principle The reader might have felt that the preceding argument sounds like a hand-waving argument, so let us derive (16.5) from Clausius inequality Let I be the system and II the heat reservoir as in Fig. 14.1: S I Q/T. (16.7) Here, we assume T e = T. Q is the heat given to system I, so the heat bath loses Q or gains Q. Let us supply W to system I (however, there is no guarantee that 1 Old literatures use F instead of A. 2 What does isothermal mean? Temperature is not definable if a system is not in equilibrium, so the reader may well question what an isothermal irreversible process means. In this book any process whose initial and the final states have the same temperature is called an isothermal process. Anything can happen in between. In particular, the temperature even if well defined need not be constant. The system need not be immersed in a constant temperature heat bath during the process.

3 163 Free energy and convex analysis this work is completely received by system I as work due to its dissipation caused by irreversibility). The first law applied to heat bath II reads E II = Q. (16.8) The definition of the Helmholtz free energy and an isothermal condition imply E I = A I + T S I. (16.9) Since the total energy has been increased by W due to the supplied work, W = E = E I + E II = A I + T S I Q. (16.10) Clausius inequality implies T S I Q 0, so this implies This is what we wished to demonstrate. A I W = Q T S I 0. (16.11) 16.4 Free energy minimum principle If there is no exchange of work, irreversibility under isothermal conditions implies δa < 0. (16.12) This implies that, if there is no spontaneous change (i.e., the equilibrium state is stable), then any virtual change δa must satisfy δa 0. (16.13) That is, in the stable equilibrium state under constant T the Helmholtz free energy must be the global minimum (because, as we will learn shortly, A is convex function. This is the free energy minimum principle Gibbs relation for A The Gibbs relation now reads da = SdT P dv + xdx, (16.14) so we see, as designed, the natural set of independent thermodynamic variables is (T, V, X) instead of (S, V, X). If the reader wishes to understand the real logic behind this change of independent variables, read 16.8: Legendre transformation below Gibbs free energy It is often convenient to study systems not only under constant temperature but also under constant pressure. Then, the work due to volume change (volume work P dv ) must be freely exchanged between the system and the external world to keep the system pressure constant, so we should rewrite the Gibbs relation as de T ds + P dv = xdx +, (16.15)

4 164 Free energy and convex analysis but since T and P are constant, it is convenient to introduce the following Gibbs free energy G G = E T S + P V. (16.16) Quite an analogous argument as the case of the Helmholtz free energy tells us that under constant T and P, if no work other than due to volume changes exists, then δg < 0, spontaneous changes can occur, (16.17) δg > 0, the equilibrium is stable. (16.18) Again, this is the principle of minimum free energy (now under constant T and P ). A process with very large G > 0 looks miraculous if it happens spontaneously. In biological systems such reactions are abundant. The secret of biochemistry is to couple reactions with G > 0 to reactions sufficiently spontaneous with G < 0 to make miraculous biological processes possible Enthalpy The Gibbs free energy may be written as where is called the enthalpy. G = H T S, (16.19) H = E + P V (16.20) If there are only volume works, then d W = P dv, so under constant pressure the first law reads dh = de + P dv = d Q. (16.21) That is, the increase of enthalpy is the heat absorbed by the system under constant pressure. For example, if a chemical reaction occurs in a system, then the change of enthalpy is the reaction heat under constant pressure ( H < 0 implies an exothermic reaction; see Section 27). If a phase transition, say melting, occurs and the latent heat is Λ, it is the enthalpy change at the phase transition point (solid liquid): H = Λ. We have learned that E = E(S, V, X), the fundamental relation, is convex Its convexity allows us to recover E = E(S, V, X) from various free energies. Thus, for example, A = A(T, V, X) is also called the fundamental relation. To understand this extremely important structure of thermodynamics, we need rudiments of convex analysis. We must understand its crucial tool, the Legendre transformation, geometrically (and intuitively).

5 165 Free energy and convex analysis Fig Legendre transformation It is said that E A = E T S or E G = E T S + P V allows us to change the independent variables from (S, V, X) to (T, V, X) or (T, P, X). This is called (in most introductory textbooks) the Legendre transformation. In the transformation E(S, V, X) A(T, V, X) = E T S we use ( E/ S) V,X = T to fix T (henceforth we suppress V and X). Therefore, to obtain A for a given T we look for a point on the curve E = E(S) where its tangent has a slope T. This is equivalent to looking for a point where the difference (signed distance) between the curve E = E(S) and the line E = ST is minimized (see Fig. 16.1). Thus, we E E = E(S) E = TS If we fix X s, E is a monotone increasing convex function of S (see (14.8)). At the value of S corresponding to the vertical dotted line the difference between E(S) and T S is the smallest, where the slope of the tangent to E = E(S) is equal to T. see A = min S {E T S}. Mathematically (and aesthetically), it is far better to interpret A = min S {E T S} as A = max S {T S E}. 3 The merits are: (i) E is a convex function 4 of S, so A is a convex function of T. (ii) The Legendre transformation is an involutive transformation; 5 Legendre transformation applied twice recovers the original object: E = max T {ST ( A)} = max T {T S + A}. Thus, we can recover E from A. This is explained in the following entries. A branch of analysis studying convex functions is called convex analysis. 6 A tangent with slope T 16.9 Convex function Let f : S R, where S is a subset of R n. Let us write y = f(r) (r R n ). f is a convex function, if the line segment connecting (r 1, f(r 1 )) and (r 2, f(r 2 )) for any r 1, r 2 S is above the graph of y = f(r) (a one dimensional case is illustrated in S 3 In convex analysis max and min are always replaced by sup (= supremum) and inf (= infimum), respectively, but throughout this book we will not be meticulous. 4 convex always implies convex downward in mathematics. 5 Involution A map f whose inverse is itself (i.e., f f = 1) is called an involutive map or involution. For example, the Hermitian conjugation is an example of involutions. 6 The bible of convex analysis is: Rockafellar, R. R., (1970). Convex Analysis, Princeton: Princeton University Press (since 1997 in the series Princeton Landmarks in Mathematics). For physicists, some patience and clever reading may be required.

6 166 Free energy and convex analysis Fig. 16.2). 7 y y 1 y 2 f(r) line segment connecting any two points on the graph is above the graph r r r 1 2 Fig f is a convex function if the line segment connecting any two points on its graph (here (r 1, f(r 1 )) and (r 2, f(r 2 ))) is not below it. A convex function must be continuous, but need not be differentiable as illustrated here. A convex function is a continuous function, 8 but may not be differentiable Geometrical meaning of Legendre transformation The mathematically standard definition of the Legendre transformation for a convex function f(x) to another convex function f (α) is f (α) = max{αx f(x)}. (16.22) x A demonstration that f is also a convex function is given in There, it is also shown that f(x) = max {αx f (α)}. (16.23) α That is, the Legendre transformation is involutive: f = f. This clearly implies that Legendre transformations perfectly preserve the information in the original convex function. That f preserves all the information in f may be understood intuitively from the following geometrical meaning of the Legendre transformation: a convex curve can be reconstructed from the totality of its tangent lines (see Fig Left), where a tangent line of a convex curve is a line sharing at least one point with the curve, and all the points on the curve are on one side of the line or on it (i.e., none on the other side). A line with a slope α is specified by its y-section f (α): y = αx f (α). If this line is tangent to f, f (α) is just given by the Legendre transformation of f (Fig Right). 7 The domain S of the function f must be a convex set; a set is a convex set if the segment connecting any two points in the set is again in the set. 8 Convex functions are continuous A convex function is continuous. A rough sketch of demonstration may be: Suppose f(x) is convex on [b ε, b + ε]. Let A + = [f(b + ε) f(b)]/ε and A = [f(b) f(b ε)]/ε (note that A A + ). Then, convexity implies that on (b ε, b] (resp., on [b, b + ε)) A + (x b) + f(b) f(x) A (x b) + f(b) (resp., A (x b) + f(b) f(x) A + (x b) + f(b)), so x b implies f(x) f(b).

7 167 Free energy and convex analysis Fig y = f (x) y = f (x) f (α) y= αx l α _ α y= x f*( ) Left: The totality of tangent lines can reconstruct a convex function; if we know them, their envelope curve is the original convex function. Right: l is the maximum gap between the dotted line y = αx and the convex curve y = f(x) (we pay attention to its sign; maximum of αx f(x)). Therefore, if we choose f (α) = max x{αx f(x)}, then y = αx f (α) is the tangent line in the figure. This gives a geometrical meaning of the Legendre transformation f f f is also convex and f(x) = f (x) The function f is also convex. f can be recovered from it as f(x) = max α {αx f (α)}. This can be illustrated by Fig This graphic demonstration uses the fact that any convex function is a primitive function of an increasing function g: α (a) f( x) O x O x Fig Illustration of the relation between f and f in 1D. g α (b) f *( α) f( x) f(x) = x g(x )dx. In (a) of Fig the pale gray area is f(x). Legendre transformation maximizes the signed area αx f(x), the dark gray area, by changing x. That is, the (signed) area bounded by the α-axis, the horizontal line through α, the vertical line through x, and the graph of g(x) is maximized with respect to x. When α = g(x), this dark gray area becomes maximum; if x goes beyond this point, a negative area would be added. (b) of Fig illustrates the case just this condition is satisfied. For this particular pair (x, α) f (α) + f(x) = αx is realized; this equality is called Fenchel s equality. From these illustrations it is obvious that the relation between f and f is perfectly symmetric, so f is convex, and f(x) = max α {αx f (α)}, or f = f; is involutive. The above illustration works only for a single variable case, but the Legendre transformation and its salient features (such as f = f ) are not confined to one g

8 168 Free energy and convex analysis dimension. Read a textbook of convex analysis. Fig Legendre transformation applied to E: summary Since E is a convex function of entropy S and all the work coordinates (see 14.5), in particular it is a convex function of S. Therefore, its Legendre transform with respect to S: max S {T S E} = E = A is a convex function of T (i.e., A is concave or convex upward as a function of T ). 9 Since the Legendre transformation is an involution, we recover E = ( A) = max T {T S ( A)} = E. The usual formula we see in thermodynamics textbooks such as A = E T S is just an example of Fenchel s equality: E + ( A) = T S. Q16.1 [Basic problems] (1) In a hydrogen fuel cell H 2 + (1/2)O 2 H 2O occurs. The reaction heat is H = 286 kj/mole of hydrogen. The entropies of the substances per mole 10 are: 70 J/K for H 2O, 131 J/K for H 2, and 205 J/K for O 2. Under constant T and P (actually in the so-called standard state: 1 atm and 300 K), what is the electric energy that may be converted to extractible work? [ G = H T S = = 237 kj/mole.] (2) 1 mole of an ideal gas at 300 K is quasistatically and isothermally compressed from 5 to 25 atm. Find E, S, A and G. [ E = 0, S = log 5 = 13.4 J/K, A = E T S = T S = 4.02 kj, G = A + (P V ) = A.] Q16.2 [Joule-Thomson effect] In the Joule-Thomson experiment performed in (Fig. 16.5) a gas of pressure P 1 porous plug P P 1 2 thermometer The throttle experiment by Joule and Thomson; A gas in the left side with pressure P 1 is gently squeezed into the right with pressure P 2 (< P 1 ), and the temperature change was measured. The whole system is thermally isolated. is pushed out through a porous plug (e.g., cotton plug) to the one of pressure P 2 (< P 1) gently under thermal insulation (no exchange of heat). The process is called the Joule- Thomson process (or throttling process), and the temperature change due to this process is called the Joule-Thomson effect. (1) Show that enthalpy H is preserved in this process. (2) [This exercise is technically better done after Section 24, so come back later] Show that the Joule-Thomson coefficient ( T/ P ) H can be rewritten as ( ) T = 1 ( ( ) ) V T V. (16.24) P H C P T P 9 Warning: A is convex with respect to V and X (when T is fixed) because E is, and A is convex with respect to T (when X and V are fixed), but A is not convex as a multivariate function of T, V, X. 10 They are absolute entropies we will encounter in Section 23, but the reader may simply regard them as the entropy relative to some basic states.

9 169 Free energy and convex analysis (3) Let the equation of state of a gas be P V = RT + BP, (16.25) where B is called the second virial coefficient, and is generally a function of T. If B < 0, the interatomic interaction is attractive. Compute the Joule-Thomson coefficient for this gas using (16.24). Soln. (1) Suppose the volume V 1 on the left-hand side is extruded to the volume V 2 on the right-hand side. The first law tells us E = W = ( P 2 V 2 ) ( P 1 V 1 ) = ( P V ), so H = 0. (2) [It is better, technically, to do this type of questions after Section 24] ( ) T (T, H) = P H (P, H) = (P, T ) (T, H) (P, H) (P, T ) = 1 ( ( ) ) S T V. (16.26) C P P T Notice that Thus, (3) Immediately we get ( ) S = (S, T ) P T (P, T ) = (S, T ) ( ) (P, V ) V (P, V ) (P, T ) =. (16.27) T P ( ) T = 1 ( ( ) ) V T V. (16.28) P H C P T P ( ) T = 1 ( T db ) P H C P dt B. (16.29) Thus, generally, for lower temperatures this is positive. That is, the gas cools through the Joule-Thomson process. For many gases around the room temperature, this is the case, indicating that the interparticle interaction is generally attractive, contrary to Newton s springy molecules (see Section 2). Q16.3 [Gas under weights] A vertical cylinder of cross section A containing an ideal gas is equipped with a piston (with a negligibly small mass) and is placed in a room at temperature T. Initially, on the piston is a weight of mass M (ignore the ambient pressure, or we do this experiment in the vacuum as illustrated in Fig (I)). Now we put another identical weight on the piston. The cylinder is rigid but does not isolate the content thermally. What is the percentage of the potential energy of the weights lost as heat, etc., to the environment? (I) gas initial equilibrium state vacuum gas just after adding another weight isothermal gas final equilibrium state (II) F gas vacuum isothermal gas Fig (I) Left: the initial state; Center: just before the irreversible sinking occurs; Right: the final state. The system is kept at T. (II) By adjusting the force F, we wish to lower the weights quasistatically.

10 170 Free energy and convex analysis Soln. Let V be the initial volume and assume the gas is n moles. The initial pressure is Mg/A = P. The piston moves by V/2A, because the volume is halved with an added weight, so W = 2Mg(V/2A) = nrt is the potential energy lost from the weights between the initial and the final states. The increase of the free energy of the gas is (notice E = 0 for isothermal process for an ideal gas) A = E T S = T S = T nr log V/2 = nrt log 2 (16.30) V Hence, (W A)/W 100 % = = 31%. We can directly obtain A as well since da = SdT P dv. T is constant, so V /2 nrt V A = dv = nrt log = nrt log 2. (16.31) V V V/2 Let us do this process gently by applying an appropriate force F (Fig (II)). Then, the work W rev (reversible work) done to the gas is a reversible work, so A = W rev. W rev is the potential energy difference the work we do through F, so clearly W > W rev = A. Without our assistance, it is clear that the potential energy of the weights is lost as heat (and perhaps sound), and the loss should be W A.

The Second Law of Thermodynamics (Chapter 4)

The Second Law of Thermodynamics (Chapter 4) First Law: Energy of universe is constant: ΔE system = - ΔE surroundings Second Law: New variable, S, entropy. Changes in S, ΔS, tell us which processes made