The Condition of Equilibrium

A chemical reaction is at equilibrium when the sum of the chemical potentials of the reactants is equal to that of the products. Thus for the reaction 

Lewis defined chemical activity in terms of chemical potential by the equation 

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As in Section III-2A, it is convenient to suppose the two bulk phases, a and /3, to be uniform up to an arbitrary dividing plane S, as illustrated in Fig. Ill-10. We restrict ourselves to plane surfaces so that C and C2 are zero, and the condition of equilibrium does not impose any particular location for S. As before, one computes the various extensive quantities on this basis and compares them with the values for the system as a whole. Any excess or deficiency is then attributed to the surface region. 

To understand the conditions which control sublimation, it is necessary to study the solid - liquid - vapour equilibria. In Fig. 1,19, 1 (compare Fig. 1,10, 1) the curve T IF is the vapour pressure curve of the liquid (i.e., it represents the conditions of equilibrium, temperature and pressure, for a system of liquid and vapour), and TS is the vapour pressure curve of the solid (i.e., the conditions under which the vapour and solid are in equili-hrium). The two curves intersect at T at this point, known as the triple point, solid, liquid and vapour coexist. The curve TV represents the... 

Various constraints may be put on this expression to produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, it follows immediately that... 

Guldberg and Waage (1867) clearly stated the Law of Mass Action (sometimes termed the Law of Chemical Equilibrium) in the form The velocity of a chemical reaction is proportional to the product of the active masses of the reacting substances . Active mass was interpreted as concentration and expressed in moles per litre. By applying the law to homogeneous systems, that is to systems in which all the reactants are present in one phase, for example in solution, we can arrive at a mathematical expression for the condition of equilibrium in a reversible reaction. 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

Criterion (1) is seen to be identical with Horstmann s principle it has been largely employed in the treatment of equilibria by Planck. It is, however, not always convenient in application because the systems which actually occur in practice are not isolated we shall therefore modify the relation so as to make it suitable for non-isolated systems. In this investigation we shall recover the first general method for determining the conditions of equilibrium—the principle of dissipation of energy. 

The suffix x indicates that besides T, all the variables xu a, . . . during the change of which external work is done, are maintained constant (adynamic condition). Thus, if the only external force is a normal and uniform pressure p, then x — v, the volume of the system, and (11) is the condition of equilibrium at constant temperature and volume. 

Since in the majority of cases we have to deal with constant pressure, it is most useful to express the conditions of equilibrium in terms of the potential . 

If the reaction occurs at constant temperature and volume we have as the condition of equilibrium ... 

The determination of the chemical potentials of the components of a solution, in terms ofp, T and the masses (or concentrations) is, from what precedes, equivalent to finding the conditions of equilibrium.. 

J.S. Anderson, The Conditions of Equilibrium of Nonstoichiometric Chemical Compounds, Proc. Roy. Soc. (London), A185, 69-89 (1946). 

The temperature at which this condition is satisfied may be referred to as the melting point Tm, which will depend, of course, on the composition of the liquid phase. If a diluent is present in the liquid phase, Tm may be regarded alternatively as the temperature at which the specified composition is that of a saturated solution. If the liquid polymer is pure, /Xn —mS where mS represents the chemical potential in the standard state, which, in accordance with custom in the treatment of solutions, we take to be the pure liquid at the same temperature and pressure. At the melting point T of the pure polymer, therefore, /x2 = /xt- To the extent that the polymer contains impurities (e.g., solvents, or copolymerized units), ixu will be less than juJ. Hence fXu after the addition of a diluent to the polymer at the temperature T will be less than and in order to re-establish the condition of equilibrium = a lower temperature Tm is required. 

The derivation of the quantitative relationship between this equilibrium temperature and the composition of the liquid phase may be carried out according to the well-known thermodynamic procedures for treating the depression of the melting point and for deriving solubility-temperature relations. The condition of equilibrium between crystalline polymer and the polymer unit in the solution may be restated as follows ... 

Now we focus our attention on the conditions of equilibrium for a fluid spheroid rotating about a constant axis. In this case the mutual position of fluid particles does not change and all of them move with the same angular velocity, a>. As is well known, there is a certain relationship between the density, angular velocity, and eccentricity of an oblate spheroid in equilibrium. In studying this question we will proceed from the equation of equilibrium of a fluid, described in the first section. 

Next we describe La Coste s invention and demonstrate that it allows one to make a spring with much smaller stiffness. From the condition of equilibrium it follows that... 

Taking into account the condition of equilibrium. Equation (3.161), the last equality is simplified... 

The condition of equilibrium of the charged particles at the interface between two condnctors can be formulated as the state where their electrochemical potentials are the same in the two phases ... 

The double arrow represents the condition of equilibrium that exists between the non-ionized and the ionized species of the electrolyte. Since ionization of strong electrolytes is practically complete there will not be much gain in studying this type of reaction from the point of view of equilibrium. Equilibria involving weak electrolytes, where there is only partial ionization, are of considerable importance. 

The adsorption capacity of a surface with respect to molecules of a given species is characterized by the total number N of molecules of the particular species retained by unit surface area under the conditions of equilibrium with the gas phase under the given external conditions (i.e., at a given pressure P and temperature T). An expression for N as a function of rf, rr, and 7j+ will be derived in Section II. 

A relation between the coefficients ar, ai, ctz, and a4 as well as between ai+, cc2+, a3+, and a4+ can be obtained from the conditions of equilibrium prior to illumination, which have the following form (the principle of detailed equilibrium) ... 

Van der Waals, whose theory has been further developed by Hulshoff and by Bakker, went one step further than Gibbs by assuming that there exists a perfectly continuous transition from one medium to the other at the boundary. This assumption limits him to the consideration of one particular case that of a liquid in contact with its own saturated vapour, and mathematical treatment becomes possible by the further assumption that the Van der Waals equation (see Chapter II.) holds good throughout the system. The conditions of equilibrium thus become dynamical, as opposed to the statical equilibrium of Laplace s theory. Van der Waals arrives at the following principal results (i) that a surface tension exists at the boundary liquid-saturated vapour and that it is of the same order of magnitude as that found by Laplace s theory (2) that the surface tension... 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

Brinkley, S. R. (1946). Note on the conditions of equilibrium for systems of many consituents. J. Chem. Phys., 14, 563-64. 

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