NON-EQUILIBRIUM THERMODYNAMICS:
AN ULTRA-CONDENSED INTRODUCTION
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(Abridged lecture notes on the topic for M.Sc. Chemistry students at Cotton College)

Abstract: Non-equilibrium thermodynamics or irreversible thermodynamics, the branch of science mostly founded by Ilya Prigogine (born 1917, Russia; Nobel prize in Chemistry, 1977) discusses about irreversible (non-equilibrium) processes. In any irreversible process the entropy of the universe increases: this gives the concept of entropy production here. Any irreversible process always occurs because of some generalized forces or affinities (e.g., difference in chemical potential, difference in temperature, difference in electrical potential etc.) which result in some generalized fluxes or flows (chemical reaction and diffusion, heat flow, current flow etc. respectively; such fluxes indicate the speed of the irreversible process). Non-equilibrium thermodynamics puts forward some generalized linear relations among such forces and fluxes (called phenomenological relations; those relations are generalizations from observed irreversible phenomena), and looks for the possible interrelation among the coefficients involved there. While looking for such interrelations, it notes that the irreversible phenomena at a microscopic level are governed by reversible laws, i.e., the mechanical equations of motion of individual particles are symmetric with respect to time; this concept is called that of microscopic reversibility.

 

                For a system within which an irreversible process occurs, the infinitesimal entropy change dS because of that process can be expressed as a sum of two parts: (i) the entropy flow deS into the system from the exterior i.e., the surroundings, and (ii) the internal entropy production diS inside the system, such that dS = deS + diS. As deS is nothing but the negative of the entropy change of the surroundings [i.e., deS = – dSsurr, this is because the irreversible process under consideration is assumed to occur within the system only, thus implying that the entropy change in the surroundings (dSsurr) can occur solely due to a transfer of entropy (–deS) from the system towards it], it is obvious that diS (= dS – deS = dS + dSsurr) is nothing but the entropy change of the universe (dSuniv). So (i.e., as diS = dSuniv) the internal entropy production is never negative (as per the 2nd law of thermodynamics stated in terms of entropy); it is positive for irreversible processes and zero for reversible (equilibrium) processes. For an isolated system dS = diS as deS obviously equals 0.

 

                As an example of entropy production in an irreversible process, let us consider a system consisting of two sub-systems 1 and 2, maintained respectively at two uniform temperatures T1 and T2 (see figure).

                                                

Entropy being an extensive property, it is obvious that dS = dS1 + dS2. Let the heat absorbed by each sub-system be divided into two parts as: dQ1 = dQ1i + dQ1e and dQ2 = dQ2i + dQ2e, where dQ1i is the heat absorbed by sub-system 1 from sub-system 2 (i.e., from the interior of the system) and dQ1e is the heat absorbed by sub-system 1 from the surroundings (exterior), and similarly for dQ2i & dQ2e (obviously, dQ1i = –dQ2i). Thus we get that the entropy change for the whole system dS is:  dS = dQ1/T1 + dQ2/T2 = dQ1e /T1 + dQ2e /T2 + dQ1i (1/T1 – 1/T2). Remembering dS = deS + diS, it becomes obvious that the part [dQ1e / T1 + dQ2e / T2] equals the entropy flow deS reversibly transferred from the surroundings to the system, so that the remaining part [dQ1i (1/T1 – 1/T2)] is the internal entropy production diS arising from the irreversible heat flow inside the system; i.e., diS = dQ1i (1/T1 – 1/T2).

           It is obvious that the internal entropy production is always positive here: from our familiar zeroth law of thermodynamics, we find that dQ1i > 0 if T1 < T2 , dQ1i < 0 if T1 > T2 and dQ1i = 0 if T1 = T2. Thus dQ1i and (1/T1 – 1/T2) will have the same sign or, at most, both will be zero, excluding the possibility for any negative value for the internal entropy production. It can be zero when thermal equilibrium is established, i.e. when T1 = T2. The quantity diS/dt, entropy production per unit time, is frequently used in irreversible thermodynamics, and is called the rate of entropy production. Here, diS/dt = (dQ1i/dt)(1/T1–1/T2). It is obvious that this rate is never negative as diS is also never negative. This equation also signifies an important concept: the rate of entropy production is a product of the rate dQ1i /dt of the irreversible process of heat flow (a generalized flow or flux), and of a generalized driving force or affinity {(1/T1 – 1/T2), the difference of inverse of temperatures} which is a function of state of the system. The direction of the irreversible process is determined by the sign of this function, and this driving force may be considered the macroscopic cause of the irreversible process. [It is common experience that the magnitude of the generalized flow is more when its aforesaid cause is more in magnitude; in the later part of this discussion we'd even talk about (experimentally observable) proportionality laws relating the flux and the causative generalized force.]

                As another example, let us consider the entropy production due to (irreversible) chemical reactions. For a chemical reaction expressed as 0 = Si ni Ai (e.g., 0 = – 2CO – O2 + 2CO2, where the stoichiometric coefficients ni 's are negative for the reactants Ai but positive for the products Ai) we have: TdS = dU + PdV – Si mi dni (starting from dG = VdP – SdT + Si mi dni  and G = U + PV – TS implying dG = dU + PdV + VdP – TdS – SdT, then equating these two expressions for dG) i.e., dS = dQ/T – (1/T) Si mi dni  as dU + PdV equals the heat absorbed dQ (using the first-law relation dQ + dw = dU --- it is being assumed here that in the reaction no non-mechanical work is done, so that dw = –PdV). 

Now, for the reaction, we have dni = ni dz where z (zeta) is the advancement of the reaction defined as ni = ni0 + zni, in which ni is the number of moles of i-th substance at time t [by the end of the chemical reaction (i.e., when z becomes 1), ni changes from its initial value ni0 (at z = 0) to its final value ni0 + ni ].  So we get dS = dQ/T – (1/T) Si mi ni dz. This is expressed as dS = dQ/T + (A / T) dz where A is called the affinity of the chemical reaction defined as A = –Si mi ni. Considering an example of a reaction, such as the carbon monoxide combustion reaction mentioned above, it would obvious to the attentive reader that this affinity A is nothing but the negative of DG, the Gibbs free energy change for the chemical reaction. Similar to the difference of temperature-inverse values in the last example, affinity A can also be thought of as the macroscopic cause of the reaction: if the total chemical potential of the products are lesser than that of the reactants, the affinity A will be positive (with DG being negative) and the reaction will then proceed in the forward direction, and vice-versa.

            Here also, the entropy change consists of two parts: the entropy flow deS = dQ/ T from the surroundings, and the rest part A dz / T obviously being the entropy production diS. So diS = A dz / T, which is always positive for spontaneous processes as dz and A always have the same sign (as, clearly, dz > 0 implies forward reaction and dz < 0 implies backward reaction). If A = 0, the system is at equilibrium and so dz = 0. The rate of entropy production per unit time is diS/dt = Av/T where v = dz/dt is the speed of the chemical reaction (the rate of the chemical reaction is r = (1/V)(dz/dt) = v/V, V being the volume. Here also we see that the rate of entropy production is a product of A / T (a function of state, the generalized affinity) and the rate v (the generalized flow) of the irreversible process. (So A/T should have been called the affinity, but just for historical reasons A is being called affinity for a chemical reaction.)

            This discussion can be extended to the case of several simultaneous reactions. We then get diS = (1/T) Sc Ac dzc > 0, where Ac is the affinity of the c-th chemical reaction related to the chemical potentials per mole as Ac = – Si nic mic. At equilibrium, all these affinities A1, A2, .. Ac are zero. The entropy production per unit time is here diS/dt = (1/T) Sc Ac vc, this being greater than zero except at equilibrium. Here, however, it may happen that a system undergoes two simultaneous reactions such that, say, A1v1 < 0 (i.e., A1 & v1 unusually being of opposite signs) and A2v2 > 0, provided that the sum A1v1 + A2v2 > 0. Thus in such cases one of the reactions (in this example the first one) might progress in a direction opposite of its own affinity, provided the other one doesn't do so. The pair of reactions is then called 'coupled' reactions. Coupled reactions are of great importance in biological processes: many biosynthetic processes (such as synthesis of the complex enzyme macromolecules from the amino acids) are non-spontaneous processes made possible by virtue of being coupled with other spontaneous processes (such as oxidation of glucose).

              Let us now consider the relations that arise among the generalized fluxes on one hand, and the generalized affinities on the other. For a generalized set of several irreversible processes occurring simultaneously within a system, the entropy production can be written as a sum of the products of generalized affinities and the corresponding fluxes: diS/dt = Sk Jk Xk where Jk are the generalized fluxes and Xk are the generalized affinities. In this standard notation, for a chemical reaction diS/dt = Jch Xch with Jch = v and Xch = A/T (conventional chemical-reaction affinity A divided by the temperature T). The fluxes Jk, however, depend on the affinities Xk --- we have already come across the idea that a flux is but the consequence of the corresponding affinity, and have the same sign as the affinity. Now, as at equilibrium all Jk and Xk are zero, so for systems not significantly far from equilibrium the fluxes may be safely assumed to linearly depend on the affinities: i.e., Jk = Si LkiXi (with the coefficients Lki s being constants). Examples of such linear relations include Fourier's law for heat flow, Fick's law for diffusion, Ohm's law for electric current etc., each relating only one type of flux and the corresponding affinity. However, such linear relations also include Soret effect, Dufour effect etc. --- each dealing with two or more simultaneously-occurring sets of fluxes and affinities. As such laws are generalizations from observed irreversible phenomena, they are called phenomenological* relations, and the coefficients Lki there are called phenomenological coefficients. The ‘pure’ coefficients Lii stand for the relation of a flux with the same type of affinity, e.g., the coefficients for heat conductivity, electrical conductivity, chemical drag etc., while the cross coefficients Lki with k ╪ i describe the interference of two irreversible processes i & k, such as thermodiffusion. Considering only two arbitrary irreversible processes 1 & 2 happening together, we get J1 = L11X1 + L12X2 and J2 = L21X1 + L22 X2, while diS/dt = J1X1 + J2X2. This gives diS/dt = L11X12 + (L12 + L21) X1X2 + L22X22 > 0, which implies that the 'pure' coefficients Lii (L11 & L22) are all positive [e.g., for X2 = 0, diS/dt = L11X12 > 0, giving L11 > 0], while the 'cross' coefficients L12 & L21 may be positive or negative, as they need only satisfy (L12 +L21)2 < 4L11L22 , because** of the above inequality.

              Lars Onsager (born 1903, Norway) has shown that for fluxes and affinities defined obeying the standard relation diS/dt = Sk Jk Xk , the cross coefficients Lik & Lki in the relation Jk = Si LkiXi must satisfy the relation Lik = Lki. This relation is known as the Onsager reciprocal relation. The derivation of this relation involves the concept of microscopic reversibility. Microscopic reversibility means the symmetry of all mechanical equations of motions of individual particles with respect to time, i.e., is invariant for replacement of t with (–t). For example, the Newton's second law m(d2x/dt2) = Fx is invariant for replacement of t with (–t).

            The principle of minimum entropy production by Prigogine states that the rate of internal entropy production (diS/dt) is a minimum for a non-equilibrium stationary state not far from the equilibrium. For small deviations or fluctuations around a stationary state, the rate of entropy production can only decrease with time (i.e., its time derivative is negative), so that the stationary state with the minimum rate of entropy production is re-attained. On the other hand, while in the stationary state, the time derivative of the rate of entropy production is zero. Thus, such stationary states are stable states with of the first derivative of diS/dt zero, and the second derivative positive --- i.e., with diS/dt minimum.

            To understand this principle, let us consider a non-equilibrium stationary state. Familiar examples include (i) a one-component (pure) gaseous system with two compartments separated by a small hole or capillary, while the two compartments are being maintained (using a heating device at one chamber and cooling device at the other chamber, see fig.) at two different constant temperatures T1 and T2           

                                                        

AND (ii) a continuously charged chemical-reaction chamber (as is found in any continuous-process chemical manufacturing unit) where two substances such as SO3 and H2O are being continuously charged into, and the product such as H2SO4 are being continuously withdrawn. It is obvious that in both cases familiar stationary states appear and remain, verifying the concept of stable stationary states.

 

Considering the first example, we see that there is a heat flow Jh corresponding to the heat-transport affinity Xh (equal to the difference of inverse temperature) maintained at a constant value D(1/T), whereas there also appears an affinity Xm =  – D(m / T) corresponding to*** matter transport, that reflects itself in the so-called thermo-molecular pressure difference between the two phases. Because of the existence of stationary state, there's no net matter flow, and so Jm equals zero, though Xm is non-zero. As the system was assumed to be not far from equilibrium, the phenomenological linear relations are valid which give:

Jh = LhhXh + LhmXm ,  Jm = LmhXh + LmmXm  where  diS/dt = JhXh + JmXm > 0

From Onsager’s reciprocal relation, Lhm = Lmh , so that diS/dt = LhhXh2 + 2LmhXmXh + LmmXm2

Differentiating (diS/dt) with respect to Xm, we get the expression (2LmhXh + 2LmmXm) but as LmhXh + LmmXm i.e., Jm is zero here, so the first derivative of diS/dt with respect to Xm is also zero. As here Xh is kept constant, diS/dt depends only on (or a function only of) Xm, the lone variable specifying the process. Thus, as that first derivative is zero, here diS/dt is either a maximum or a minimum. As it couldn't have been a maximum as it is always a finite positive quantity, so rate of internal entropy production (diS/dt) is a minimum in the stationary state.

 

 

Notes:

* The word phenomenological means 'of experimental origin'.


** To derive the relation (L12 +L21)2 < 4L11L22 let us consider the (very general) case of 

(X1 ╪ 0, X2 ╪ 0) for which L11X12 + (L12 + L21) X1 X2 + L22X22 > 0, obviously meaning

L11ί2 + (L12 + L21) ί + L22 > 0 where ί = X1/X2, i.e., ί is clearly a real quantity. However,

this inequality also means that L11ί2 + (L12 + L21) ί + L22 ╪ 0 which further means that the quadratic equation L11ί2 + (L12 + L21) ί + L22 = 0 can have no real roots (values) for ί, meaning that the two roots [i.e., the roots ί = {–(L12 + L21) + ((L12 + L21)2 – 4L11L22)½}/(2L11) and ί = {–(L12 + L21) – ( (L12 + L21)2 – 4L11L22)½}/(2L11)] of that quadratic equation can only be complex. This, however, would mean that {(L12 + L21)2 – 4L11L22}½ must be a purely imaginary number, i.e., that (L12 + L21)2 – 4L11L22 < 0, which means (L12 +L21)2 < 4L11L22.     

Using the Onsager reciprocal relation L12 = L21, this immediately gives
|L12| < (L11L22)½, meaning that the magnitude of the 'cross' coefficient L12 has to be less than the geometric mean (L11L22)½ of the two 'pure' coefficients L11 & L22.

 

*** It may be shown that if, because of solely a difference Dm in the chemical potential m, diffusion of matter occurs (at the rate Jm = dn/dt, from the high-m compartment to the low-m one), then the rate of entropy production would be diS/dt = – (Dm / T). dn/dt, thus meaning that the generalized affinity here would be nothing other than –(Dm / T) i.e., –D(m / T). [The negative sign here signifies that matter diffuses from a region of high m to one of low m].

 

 

© 2005-2009. Rituraj Kalita, Guwahati (India).

 

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