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 d_eS into the system from the exterior i.e., the surroundings, and (ii) the internal entropy production d_iS inside the system, such that dS = d_eS + d_iS. As d_eS is nothing but the negative of the entropy change of the surroundings [i.e., d_eS = – dS_surr, 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 (dS_surr) can occur solely due to a transfer of entropy (–d_eS) from the system towards it], it is obvious that d_iS (= dS – d_eS = dS + dS_surr) is nothing but the entropy change of the universe (dS_univ). So (i.e., as d_iS = dS_univ) 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 = d_iS as d_eS 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 = dQ_1i + dQ_1e and dQ₂ = dQ_2i + dQ_2e, where dQ_1i is the heat absorbed by sub-system 1 from sub-system 2 (i.e., from the interior of the system) and dQ_1e is the heat absorbed by sub-system 1 from the surroundings (exterior), and similarly for dQ_2i & dQ_2e (obviously, dQ_1i = –dQ_2i). Thus we get that the entropy change for the whole system dS is:  dS = dQ₁/T₁ + dQ₂/T₂ = dQ_1e /T₁ + dQ_2e /T₂ + dQ_1i (1/T₁ – 1/T₂). Remembering dS = d_eS + d_iS, it becomes obvious that the part [dQ_1e / T₁ + dQ_2e / T₂] equals the entropy flow d_eS reversibly transferred from the surroundings to the system, so that the remaining part [dQ_1i (1/T₁ – 1/T₂)] is the internal entropy production d_iS arising from the irreversible heat flow inside the system; i.e., d_iS = dQ_1i (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 dQ_1i > 0 if T₁ < T₂ , dQ_1i < 0 if T₁ > T₂ and dQ_1i = 0 if T₁ = T₂. Thus dQ_1i 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 d_iS/dt, entropy production per unit time, is frequently used in irreversible thermodynamics, and is called the rate of entropy production. Here, d_iS/dt = (dQ_1i/dt)(1/T₁–1/T₂). It is obvious that this rate is never negative as d_iS is also never negative. This equation also signifies an important concept: the rate of entropy production is a product of the rate dQ_1i /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 = S_i n_i A_i (e.g., 0 = – 2CO – O₂ + 2CO₂, where the stoichiometric coefficients n_i 's are negative for the reactants A_i but positive for the products A_i) we have: TdS = dU + PdV – S_i m_i dn_i (starting from dG = VdP – SdT + S_i m_i dn_i  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) S_i m_i dn_i  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 dn_i = n_i dz where z (zeta) is the advancement of the reaction defined as n_i = n_i⁰ + zn_i, in which n_i 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), n_i changes from its initial value n_i⁰ (at z = 0) to its final value n_i⁰ + n_i ].  So we get dS = dQ/T – (1/T) S_i m_i n_i dz. This is expressed as dS = dQ/T + (A / T) dz where A is called the affinity of the chemical reaction defined as A = –S_i m_i n_i. 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 d_eS = dQ/ T from the surroundings, and the rest part A dz / T obviously being the entropy production d_iS. So d_iS = 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 d_iS/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 d_iS = (1/T) S_c A_c dz_c > 0, where A_c is the affinity of the c-th chemical reaction related to the chemical potentials per mole as A_c = – S_i n_ic m_ic. At equilibrium, all these affinities A₁, A₂, .. A_c are zero. The entropy production per unit time is here d_iS/dt = (1/T) S_c A_c v_c, 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: d_iS/dt = S_k J_k X_k where J_k are the generalized fluxes and X_k are the generalized affinities. In this standard notation, for a chemical reaction d_iS/dt = J_ch X_ch with J_ch = v and X_ch = A/T (conventional chemical-reaction affinity A divided by the temperature T). The fluxes J_k, however, depend on the affinities X_k --- 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 J_k and X_k are zero, so for systems not significantly far from equilibrium the fluxes may be safely assumed to linearly depend on the affinities: i.e., J_k = S_i L_kiX_i (with the coefficients L_ki 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 L_ki there are called phenomenological coefficients. The ‘pure’ coefficients L_ii 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 L_ki 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 d_iS/dt = J₁X₁ + J₂X₂. This gives d_iS/dt = L₁₁X₁² + (L₁₂ + L₂₁) X₁X₂ + L₂₂X₂² > 0, which implies that the 'pure' coefficients L_ii (L₁₁ & L₂₂) are all positive [e.g., for X₂ = 0, d_iS/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 d_iS/dt = S_k J_k X_k , the cross coefficients L_ik & L_ki in the relation J_k = S_i L_kiX_i must satisfy the relation L_ik = L_ki. 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) = F_x is invariant for replacement of t with (–t).

            The principle of minimum entropy production by Prigogine states that the rate of internal entropy production (d_iS/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 d_iS/dt zero, and the second derivative positive --- i.e., with d_iS/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 J_h corresponding to the heat-transport affinity X_h (equal to the difference of inverse temperature) maintained at a constant value D(1/T), whereas there also appears an affinity X_m =  – 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 J_m equals zero, though X_m is non-zero. As the system was assumed to be not far from equilibrium, the phenomenological linear relations are valid which give:

J_h = L_hhX_h + L_hmX_m ,  J_m = L_mhX_h + L_mmX_m  where  d_iS/dt = J_hX_h + J_mX_m > 0

From Onsager’s reciprocal relation, L_hm = L_mh , so that d_iS/dt = L_hhX_h² + 2L_mhX_mX_h + L_mmX_m²

Differentiating (d_iS/dt) with respect to X_m, we get the expression (2L_mhX_h + 2L_mmX_m) but as L_mhX_h + L_mmX_m i.e., J_m is zero here, so the first derivative of d_iS/dt with respect to X_m is also zero. As here X_h is kept constant, d_iS/dt depends only on (or a function only of) X_m, the lone variable specifying the process. Thus, as that first derivative is zero, here d_iS/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 (d_iS/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

L₁₁ß² + (L₁₂ + L₂₁) ß + L₂₂ > 0 where ß = X₁/X₂, i.e., ß is clearly a real quantity. However,

this inequality also means that L₁₁ß² + (L₁₂ + L₂₁) ß + L₂₂ ╪ 0 which further means that the quadratic equation L₁₁ß² + (L₁₂ + L₂₁) ß + L₂₂ = 0 can have no real roots (values) for ß, meaning that the two roots [i.e., the roots ß = {–(L₁₂ + L₂₁) + ((L₁₂ + L₂₁)² – 4L₁₁L₂₂)^½}/(2L₁₁) and ß = {–(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 J_m = dn/dt, from the high-m compartment to the low-m one), then the rate of entropy production would be d_iS/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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