Infinitesimal processes

Two subsystems a. and p, in each of which the potentials T,p, and all the p-s are unifonn, are pennitted to interact and come to equilibrium. At equilibrium all infinitesimal processes are reversible, so for the overall system (a + P), which may be regarded as isolated, the quantities conserved include not only energy, volume and numbers of moles, but also entropy, i.e. there is no entropy creation in a system at equilibrium. One now... 

For a virtual infinitesimal process consisting of heat flow across the wall, or displacement of the wall, at equilibrium there is no change in entropy, dS = 0, and hence the conditions for equilibrium are... 

The computation of chemical equilibria can be posed as an optimization problem with linear side conditions. For any infinitesimal process in which the amounts of species present may be changed by either the transfer of species to or from a phase or by chemical reaction, the change in the Gibbs free energy is... 

The surface tension, y, and the mechanical equilibrium at interfaces have been described in the literature in detail (Adamson and Gast, 1997 Chattoraj and Birdi, 1984 Birdi, 1989, 2002, 2008). The surface has been considered as a hypothetical stretched membrane, which is termed as the surface tension. In a real system undergoing an infinitesimal process, it can be written that... 

A thermodynamic process is said to have taken place if a change is observed to have taken place in any macroscopic property of the system. An infinitesimal process is a process in which there is only an infinitesimal change in any macroscopic property of the system. A natural process is an infinitesimal process that occurs spontaneously in real systems an unnatural process is one that cannot occur spontaneously in real systems. Reversible processes are either natural or unnatural processes which can occur in either direction between two states of equilibrium... 

For a single-component, single-phase system or a system at material equilibrium, the change of internal energy is completely determined by the change in two state variables. Thus, Eq. (20) is valid for any process that goes between the initial and final states of the infinitesimal process its application is not limited to reversible processes. It would, for example, apply to a Joule-Thomson expansion, a distinctly nonreversible process. 

Because of the simplicity of these relationships, we sometimes say that the natural variables of U are S and V, of // arc S and P, of A are V and T, of G are P and T, and of S are U and V. It is noteworthy that the natural variables of U are both extensive variables and those of G are both intensive variables the natural variables of H and A are mixed—one extensive and one intensive variable for each. Because Eqs. (20)-(24) only hold for systems at material equilibrium, they will become our criteria for material equilibrium under each set of conditions. For example, at constant T and P, for a system to be at material equilibrium for a process, dG must equal zero for the (infinitesimal) process. 

The first and second laws of thermodynamics and the Helmholtz and Gibbs free energies are rearranged to obtain the relationships between the state functions (i.e., E, H, A, and G) and temperature, pressure, and volume. For an infinitesimal process the first law is given by ... 

In the classical formulation, the second law of thermodynamics states that there exist an absolute scale for the temperature T and an extensive function 5(p, V, called the entropy, such that for an infinitesimal process in a closed system... 

The heat of reaction (sometimes called the differential or partial heat of reaction to distinguish it from that defined previously) may then be defined as the heat evolved in this infinitesimal process per unit change of e, whence the relation dQ = dH and equation (40) show that the heat of reaction may be written as... 

One should note how the temperatures of the system and surroundings occur in the above expression. Eq. (1.12.6a) can then be trivially rearranged to solve for the heat transfer for any given infinitesimal process under irreversible conditions ... 

The first law of thermodynamics is conservation of energy, which states that there exists an extensive function U = U p, V, N,), called the internal energy, having the property that for a closed system (one that does not exchange material with its surroundings) the heat added to the system in an infinitesimal process is... 

Consider a thermal machine consisting of an insulated piston-cylinder assembly attached to a container as shown in Kg. 1.2. Initially, the matter in the cylinder is separated from that of the container by a partition. The partition is ruptured and, following an infinitesimal process, the mass A mf within the cylinder is slowly pushed by the piston into the container. Assume the container to be a control volume. During this process, the heat received and the shaft work done by the control volume, respectively, are Agcv and AWcv> subscript cv denoting the control volume. We wish to find the rate of the first law of thermodynamics for this control volume. 

There are two important inexact differentials in thermodynamics. If a system undergoes an infinitesimal process (one in which the independent variables specifying the state of the system change infinitesimally), dq denotes the amount of heat transferred to the system and dw denotes the amount of work done on the system. Both of these quantities are inexact differentials. For a fluid system undergoing a reversible process,... 

It follows at once from this equation that if an infinitesimal process occurs at constant volume, and only PV work is involved,... 

The relationships valid for constant-pressure processes may readily be deduced from equation (4.17). For an infinitesimal process at constant pressure the heat absorbed dq/> is given by... 

Heat 8Q added to a system in an infinitesimal process is used to increase the internal energy by dE and to perform an amount of work 5IV ... 

Consider a rectangular rubber sample to be stretched in the x-direction by an external tension /. The law of thermodynamics for the infinitesimal process of stretching the sample from the length L to L + is... 

A mathematical expression of the first law of thermodynamics for an infinitesimal process is as follows... 

In equations 4.9, we stated that a system was at equilibrium if AG = 0 or, equivalently for an infinitesimal process, dG = 0. For chemical equilibrium, we require that the derivative in equation 5.3, defined as the Gibbs energy of reaction A G, be zero ... 

Before doing so, it is instructive to consider a special case for the above. Let a system undergo an infinitesimal process from state A to state B at constant volume and composition, and in the absence of other work. Equation (1.10.2a) then reduces to... 

Next, we investigate a similar setup in the configuration of Figure 5.4.1(b), in which the applied pressure, Pb, is uniformly and reversibly exerted on the piston in channel B. Now there is a change in total volume, namely —dV,. The work exerted in this infinitesimal process is... 

The criteria for finite processes are completely analogous to those for infinitesimal processes. For example, for a simple system at constant pressure and temperature, a finite spontaneous process must obey... 

The work done by the surroundings on the system is for the infinitesimal process ... 

Under load (irreversible conditions) the corresponding efficiencies are smaller because of polarization effects on the electrodes and ohmic drops. A quantitative derivation was given for an infinitesimal process on the basis of irreversible thermodynamics by van Rysselberghe [3,4]. 

If the two subsystems are not in equilibriiun, we say that they are out of equihbrimn or in a state of disequilibrium. When the composite system is out of equilibrium the second law of thermodynamics requires the entropy change dS associated with an infinitesimal process to be positive. After the composite system, originally in a state of disequilibrimn, has exhausted all allowed spontaneous transfer processes, the entropy will reach a maximum and the conditions of equilibrimn consistent with the imposed restraints will be realized. Ar r further changes in the composite system will be reversible and the equality dS = 0 will hold. 

---