Entropy nonequilibrium conditions

The value of A is maintained at a nonzero value, and hence at nonequilibrium conditions. The entropy production per unit volume is... 

The rate of storage of chemical potential in the product B per unit of volume is J n, where the flux J = -d[A]/di = d[B]/dt. If A and B are in equilibrium, the rates of the forward and back reactions are equal and the net flux to the product J — jf — 7b = 0. Under nonequilibrium conditions where the forward reaction takes place with 7 > 0, there is a net overall entropy increase and K is the... 

In Chapter 14 (equation 14.1.16) we obtained the same equation for perturbations from the equilibrium state. Equation (18.3.6) shows that the time derivative of 5 5 has the same form even under nonequilibrium conditions. The difference is that near equilibrium SFkSJk = J2k > 0 this is not necessarily so far from equilibrium. We shall refer to this quantity as excess entropy production, but strictly speaking, it is the increase in entropy production only near the equilibrium state for a perturbation from a nonequilibiium state, the increase in entropy production is equal to 5P = 5pP -I- 5jP. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Note that Eq. (6) includes thermodynamic equilibrium (v° = 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions. 

We will assume a system initially in thermal equilibrium that is transiently brought to a nonequilibrium state. We are going to show that, under such conditions, the entropy production in Eq. (22) is equal to the heat delivered by the system to the sources. We rewrite Eq. (22) by introducing the potential energy function Gx C),... 

This reaction is actually slightly endothermic (A// = 88 kJ/mol), but the large net increase in entropy and the nonequilibrium nature of most CVD processes lead to significant tungsten deposition. As with the Ge example, the deposition mechanism involves adsorption steps and surface reactions. At low pressures and under conditions of excess hydrogen gas, the deposition rate follows the general form ... 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

In Chapter 2, we pay a renewed visit to thermodynamics. We review its essentials and the common structure of its applications. In Chapter 3, we focus on so-called energy consumption and identify the concepts of work available and work lost. The last concept can be related to entropy production, which is the subject of Chapter 4. This chapter shows how some of the findings of nonequilibrium thermodynamics are invaluable for process analysis. Chapter 5 is devoted to finite-time finite-size thermodynamics, the application of which allows us to establish optimal conditions for operating a process with minimum losses in available work. 

From the "physico-economic" standpoint convergence of the chosen method can be explained by the fact that it naturally represents the tendency of an open system with fixed conditions of interaction with the environment to equilibrium, which corresponds to minimum production of both physical and economic entropy. Optimization for the obtained "technico-economic mechanism" determines flow distribution corresponding to the minimum energy consumption, i.e., a physical mechanism. Thus, in this case the model of equilibrium thermodynamics—MEIS solves the problem of self-organization, ordering of the "physico-economic" system that is referred as a rule to the area of applications of nonequilibrium thermodynamics or synergetics. 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

In this open reaction system, the chemical potentials of reactant R and product B are maintained at a fixed value by an inflow of reactant R and an outflow of product B. The concentration of intermediate X is maintained at a nonequilibrium value, while the temperature is kept constant by the reaction exchanging heat with the environment. Determine the condition for minimum entropy production. 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

This equation describes the pressure difference because of the mass fraction difference when there is no temperature difference. This is called the osmotic pressure. This effect is reversible because AT - 0,, /2 = 0. and at stationary state J = 0. Therefore, Eq. (7.244) yields Jq = 0, and the rate of entropy production is zero. The stationary state under these conditions represents an equilibrium state. Equation (7.263) does not contain heats of transport, which is a characteristic quantity for describing nonequilibrium phenomena. 

The equality of this equation represents a system at equilibrium where JT = A = 0. The work done by the controlling system dissipates as heat. This is in line with the first law of thermodynamics. The inequality in Eq. (11.4) represents the second law of thermodynamics. The cyclic chemical reaction in nonequilibrium steady-state conditions balances the work and heat in compliance with the first law and at the same time transforms useful energy into entropy in the surroundings in compliance with the second law. The dissipated heat related to affinity A under these conditions is different from the enthalpy difference AH° = (d(Aii°/T)/d(l/Tj). The enthalpy difference can be positive if the reaction is exothermic or negative if the reaction is endothermic. On the other hand, the A contains the additional energy dissipation associated with removing a P molecule from a solution with concentration cP and adding an S molecule into a solution with concentration cs. 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

However, nonequilibrium steady states may be unstable even if the system is stable with respect to diffusion. For a nonequilibrium state, the stability condition for a chemical reaction in terms of excess entropy production is... 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Equations (A.23) and (A.25) pertain to equilibrium conditions of homogeneous systems. Such systems have constant properties over space and time and there is no entropy production. We shall now be interested in systems, away from equilibrium where properties vary as functions of location as well as time. Tb apply the results of thermodynamics to nonequilibrium systems., the principle of local (microscopic) equilibrium is invoked. For that reason it is useful to work with the thermodynamic variables on a unit volume basis. Equation (A.25) then becomes... 

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