Entropy as a state property

The denominator of the right hand side of Eq. 3.1 is relevant to the total number of the microscopic energy states of the system and is called the particle partition function z  

Statistical thermodynamics has defined, in addition to the particle partition function z, the canonical ensemble partition function Zas follows  

For a system consisting of the total number of particles N and maintaining its total energy U and volume V constant, statistical thermodynamics defines the entropy, S, in terms of the logarithm of the total number of microscopic energy distribution states Q N,V,U) in the system as shown in Eq. 3.6  

The classical definition of entropy based on the second law of thermodynamics has given the total differential of entropy in the form of dQrev / / . With a reversible heat transfer into a closed system receiving a differential amount of heat dQrev, the system changes its entropy by the differential amount of dS as shown in Eq. 3.8  

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The paradox here is that if entropy is a state property of a system it cannot depend on what we happen to know about the system. Quantum mechanics has a similar-sounding, but quite different epistemological problem, which, in principle, placed limits on the precision by which certain pairs of properties are measured. Since measurement involves experimental design and choice of parameters of interest, in the quantum framework the observer is required to complete the phenomenon. In statistical thermodynamics, however, entropy is microscopic uncertainty and if we interpret entropy as lack of microscopic information about the macroscopic thermodynamic state we seem to get involved in the identity of that state alone, which would be a conflicting standpoint. Therefore, let us discuss all such viewpoints and inherent differences often arising from not fully congruous ideas, which try to enlighten the true interdisciplinary of the notion of entropy. 

Equation (2.66) indicates that the entropy for a multipart system is the sum of the entropies of its constituent parts, a result that is almost intuitively obvious. While it has been derived from a calculation involving only reversible processes, entropy is a state function, so that the property of additivity must be completely general, and it must apply to irreversible processes as well. 

For a closed system the first law of thermodynamics has defined an energy function called internal energy U, which is expressed as a function of the temperature, volume, and number of moles of the constituent substances in the system U = u(t, V, n, nc). Furthermore, the second law has defined a state property, called entropy S, of the system, which is also expressed as a function of state variables S =s(T,V,nl---nc). Thermodynamics presumes that the functions t/(r,V,n, " nj and 5(7, y, I nc) exist independent of whether the system is closed or open. The energy functions of U, H, F, and G, then, apply not only to closed systems but also to open systems. 

Equation (1.60) illustrates a very important thermodynamic property in which the quantity dqrev / T becomes zero when the cyclic process is completed regardless of the paths taken from the initial to final states. Such a property is known as a state function, as are P, V, T, E, and H. Clausius suggested defining a new thermodynamic state function, called entropy and denoted as S, where dqrev / T = dS, so that... 

Note that since entropy is a state propeny, once two properties of a one-phase system, such as temperature and pressure, are fixed, the value of the entropy is also fixed. Consequently, the entropy of steam can be found in the steam tables or the Mol Her diagram, and that of methane, nitrogen, and HFC-134a in the appropriate figures in Chapter 3. In the next section we consider entropy changes for an ideal gas. and in Chapter 6 we develop the equations to be used to compute entropy changes for nonideal fluids. 

Just as the first law led to the definition of the energy, so also the second law leads to a definition of a state property of the system, the entropy. It is characteristic of a state property that the sum of the changes of that property in a cycle is zero. For example, the sum of changes in energy of a system in a cycle is given by dU = 0. We now ask whether the second law defines some new property whose changes sum to zero in a cycle. 

For entropy to be of any practical value, we must be able to relate it to quantities that can be measured experimentally. Here is how we develop this relationship. Since entropy is a state function, we can express it as a mathematical function of two intensive properties. We choose internal energy U, and volume V, and write S = S(U, V). This unusual choice is perfectly permissible.iThe differential of entropy in terms of these independent variables is... 

It is well known from thermodynamic principles that energy transferred as work is more useful than energy transferred as heat. Work can be completely converted to heat, but only a fraction of heat can be converted to work. Furthermore, as the temperature of a system is decreased, heat transferred from the system becomes less useful and less of the heat can be converted to work. A state property that accounts for the differences between heat and work is entropy, S. When heat is transferred into a closed system at temperature T, the entropy of the system increases because entropy transfer accompanies heat transfer. By contrast, work transfer (shaft work) is not accompanied by entropy transfer. When heat is transferred at a rate Q from a surrounding heat reservoir at a constant temperature, Treservoir, into a system, the heat reservoir experiences a decrease in entropy given by... 

On account of the second law— the existence of entropy as a function of state—the Joule-Thomson coefficient can be related to other measurable properties of the gas. Thus from (2 95)... 

Entropy, as formulated here and in the previous chapter, encompasses all aspects of matter transformations changes in energy, volume and composition. Thus, every system in Nature, be it a gas, an aqueous solution or a living cell, is associated with a certain entropy. We shall obtain explicit expressions for entropies of various systems in the following chapters and study how entropy production is related to irreversible processes. At this stage, however, we shall note some general properties of entropy as a function of state. 

Worth mentioning is also the associated and intriguing means of irreversibility, or better, of the entropy increase, which are dynamical and which often lie outside the scope of standard edification. Notwithstanding, the entropy as the maximum property of equilibrium states is hardly understandable unless linked with the dynamical considerations. The equal a priori probability of states is already in the form of a symmetry principle because entropy depends symmetrically on all permissible states. TTie particular function of entropy is determined completely then by symmetry over the set of states and by the requirement of extensivity. Consequently it can be even shown that a full thermodynamic (heat) theory can be formulated with the heat, being totally absent. Nonetheless, the familiar central formulas, such as dlS = dQ/T, remains lawful although dQ does not acquire to have the significance of energy. Nevertheless, for the standard thermophysical studies the classical treatises are still of the daily use so that their basic principles and the extent of applicability are worthy of brief recapitulation. 

The second law of thermodynamics means that the processes of energy transformation can occur spontaneously only provided that energy passes from its concentrated (ordered) form to a diffused (disordered) one. Such energy redistribution in the system is characterized by a quantity which has been named as entropy, which, as a function of state of the thermodynamic system (the more energy irreversible dissipates as heat, the higher entropy is. Whence it follows that any system whose properties change in time aspires to an equilibrium state at which the entropy of the system takes its maximum value. In this coimection, the second law of thermodynamics is often called the law of increasing entropy, and entropy (as a physical quantity or as a physical notion) is considered as a measure of disorder of a physicochemical system. 

Transient, or time-resolved, techniques measure tire response of a substance after a rapid perturbation. A swift kick can be provided by any means tliat suddenly moves tire system away from equilibrium—a change in reactant concentration, for instance, or tire photodissociation of a chemical bond. Kinetic properties such as rate constants and amplitudes of chemical reactions or transfonnations of physical state taking place in a material are tlien detennined by measuring tire time course of relaxation to some, possibly new, equilibrium state. Detennining how tire kinetic rate constants vary witli temperature can further yield infonnation about tire tliennodynamic properties (activation entlialpies and entropies) of transition states, tire exceedingly ephemeral species tliat he between reactants, intennediates and products in a chemical reaction. 

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