Second law inequality

In Chapter 9 we will see that the second-law inequality of Equation (4.13) will form a cornerstone of the constraint-based approach to analyzing biochemical networks. 

Evans and Searles reviewed the transient fluctuation relations, how steady state relations can be derived from them, their implications and applications, and experimental tests available in 2002. A shorter review paper highlighted the main results, and a derivation of the second law inequality. Recently, Sevick et al reviewed the ES FR, the Crooks FR and the Jarzynski equality, highlighting the similarities and differences between the two FRs and also discussing experimental work that has been carried out to test these results. 

The ES FR can be used to provide a simple proof of the Second Law Inequality ... 

This is, in fact, the most significant result that can be obtained for incompressible Newtonian fluids from the second-law inequality. If we do not restrict ourselves to incompressible or isothermal conditions, inequality (2-82) can be satisfied for arbitrary motions and arbitrary temperature fields only if... 

For an arbitrarily large ensemble of experiments from some iiutial time r = 0, consequence of the fluctuation theorem is that an ensemble average of the entropy production cannot be negative for any value of the averaging time f. (a ) > 0. This inequality is called the second law inequality. It can be proved for systems with time-dependent fields of arbitrary magnitude and time dependence. However, It does not imply that the ensemble averaged entropy production is nonnegative at all times. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

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. 

Still another statement of the second law is the Clausius inequality which states that... 

Jarzynski s identity, (5.8), immediately leads to the second law in the form of (5.6) because of Jensen s inequality, (e x) > e. Moreover, in the limit of an infinitely fast transformation, r —> 0, we recover free energy perturbation theory. In that limit, the configurations will not relax during the transformation. The average in... 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

Note that the true free energy difference is A A = AAlDO, and the other limit gives the average work, W) = AA. The second law of thermodynamics indicates that AAoq = A A < (W) = AA. In fact, due to the monotonic behavior of the Boltzmann factor in (6.84), one has the general inequality... 

Equation (6.87) is a condensed mathematical statement of the second law the inequality applies to any real process, which is necessarily irreversible, and the equality applies to the limiting case of the reversible process. 

Generalized Second Law for Steady-State Transitions. From the inequality in Eqs. (18) and (94), we obtain... 

The simple inequality (4.10) captures the essence of the second law. Its general consistency with universal inductive experience will be established in Section 4.4, and its further consequences (culminating in the final form of the second law as expressed by Clausius) will be developed in Sections 4.5-4.7. Thus, Carnot s remarkable principle provides virtually complete answers to the questions posed at the beginning of this chapter, although the relationship of (4.10) to these broader issues will certainly not become obvious until the following section. 

Clausius proceeded to demonstrate the power of entropy to express the deep consequences of the second law. We begin by introducing the inequality of Clausius, which complements Carnot s theorem (4.25) for the irreversible case. 

The general inequality (4.48) leads to the famous Clausius formulation of the second law ... 

Table 5.1 summarizes the various constraint conditions and the associated thermodynamic potentials and second-law statements for direction of spontaneous change or condition of equilibrium. All of these statements are equivalent to Carnot s theorem ( dq/T < 0) or to Clausius inequality ([Pg.164]

For any chemical system the second law of thermodynamics furnishes useful equalities and inequalities which characterize the state of that system. At a state of equilibrium in a closed system, the total internal... 

Equations 27 and 28 permit a simple comparison to be made between the actual composition of a chemical system in a given state (degree of advancement) and the composition at the equilibrium state. If Q K, the affinity has a positive or negative value, indicating a thermodynamic tendency for spontaneous chemical reaction. Identifying conditions for spontaneous reaction and direction of a chemical reaction under given conditions is, of course, quite commonly applied to chemical thermodynamic principle (the inequality of the second law) in analytical chemistry, natural water chemistry, and chemical industry. Equality of Q and K indicates that the reaction is at chemical equilibrium. For each of several chemical reactions in a closed system there is a corresponding equilibrium constant, K, and reaction quotient, Q. The status of each of the independent reactions is subject to definition by Equations 26-28. 

This is the Helmholtz inequality, a variant of the Second Law and thus an acknowledgment of the consequences of irreversible processes. 

From the first law (energy conservation) of thermodynamics we have dQi = dW + dQ2, and the second law (entropy creation) of thermodynamics gives us dQx Tx) + dQ2IT2) a 0, where equality is for a reversible heat engine and inequality for an irreversible one. We then have the efficiency dW/dQx) for the reversible heat engine and the efficiency... 

We shall now state the second law in a mathematical form which is very commonly used. We let S denote the entropy. Our previous statement is then dS dQ/T, or T dS dQ, the equality sign holding for the reversible, the inequality for irreversible, processes. But now we use the first law, Eq. (2.1), to express dQ in terms of dll and dW. The inequality becomes at once... 

When an enzyme-catalyzed biochemical reaction operating in an isothermal system is in a non-equilibrium steady state, energy is continuously dissipated in the form of heat. The quantity J AG is the rate of heat dissipation per unit time. The inequality of Equation (4.13) means that the enzyme can extract energy from the system and dissipate heat and that an enzyme cannot convert heat into chemical energy. This fact is a statement of the second law of thermodynamics, articulated by William Thompson (who was later given the honorific title Lord Kelvin), which states that with only a single temperature bath T, one may convert chemical work to heat, but not vice versa. 

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. 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

Since die second law of themiodynaniics requires that So > 0, it follows tliat Wlost 0-When a process is completely reversible, the equality holds, and die lost work is zero. For irreversible processes the inequality holds, and the lost work, i.e., die energy drat becomes unavailable for work, is positive. 

The second law of thermodynamics formalizes the observation that heat is spontaneously transferred only from higher temperatures to lower temperatures. From this observation, one can deduce the existence of a state function of a system the entropy S. The second law of thermodynamics states that the entropy change dS of a closed, constant-volume system obeys the following inequality... 

Now consider a closed system that can alter its volume V. In this case, the work performed by the system is 5FK = pdY- Combining the first and the second laws of thermodynamics for a closed system (i.e. inserting the inequality in Eq. (1.2) into Eq. (1.1)), we obtain... 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

In fluid d mamics there is no specific use of the transport equation for entropy other than being a physical condition indicating whether a constitutive relation proposed has a sound physical basis or not (nevertheless, this may be a constraint of great importance in many situations). In this connexion we usually think of the second law of thermodynamics as providing an inequality, expressing the observation that irreversible phenomena lead to entropy production. 

The entropy equation can now be used to express the Clausius form of the second law of thermodynamics for open flow systems (e.g., [7] [145], p. 126). The inequality expresses that irreversible phenomena (diffusive momentum... 

The pioneering work in the direction of the second law of thermodynamics is considered to be performed in 1825 by Sadi Carnot investigating the Carnot cycle [51] [40]. Carnot s main theoretical contribution was that he realized that the production of work by a steam engine depends on a flow of heat from a higher temperature to a lower temperature. However, Clausius (1822-1888) was the first that clearly stated the basic ideas of the second law of thermodynamics in 1850 [13] and the mathematical relationship called the inequality of Clausius in 1854 [51]. The word entropy was coined by Clausius in 1854 [51]. 

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