Equations of equilibrium

For vapor-liquid equilibria, the equations of equilibrium which must be satisfied are of the form... 

For a vapor phase (superscript V) and a liquid phase (superscript L), at the same temperature, the equation of equilibrium... 

For multicomponent vapor-liquid equilibria, the equation of equilibrium for every condensable component i is... 

In multicomponent liquid-liquid equilibria,the equation of equilibrium, for every component i, is... 

For a lower bound on the apparent Young s modulus, E, load the basic uniaxial test specimen with normal stress on the ends. The internal stress field that satisfies this loading and the stress equations of equilibrium is... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

Antisymmetric angle-ply laminates were described in Section 4.3.3 and found to have extensional stiffnesses A i,A.,2, A22, and Agg bending-extension coupling stiffnesses B. g and 625 and bending stiffnesses D.. , D. 2, D22. and Dgg. Thus, this laminate exhibits a different type of bending-extension coupling than does the antisymmetric cross-ply laminate. The coupled governing differential equations of equilibrium are... 

It is often necessary to compute the forces in structures made up of connected rigid bodies. A free-body diagram of the entire structure is used to develop an equation or equations of equilibrium based on the body weight of the structure and the external forces. Then the structure is decomposed into its elements and equilibrium equations are written for each element, taking advantage of the fact that by Newton s third law the forces between two members at a common frictionless joint are equal and opposite. 

P = F(r,T). (10), and since this fixes the external conditions (viz., the one that the pressure on the system must have a given value) in order that the system may be in equilibrium, with chosen values of the independent variables specific volume and temperature, it may be called the equation of equilibrium of the fluid. 

The thermodynamic theory of equilibrium was first stated, in a general way, by Horstmann in 1873 (cf. 50), who also obtained explicit equations of equilibrium in the case where it is established in a gas, and showed that these were in agreement with the data available at that time, and with his own experiments. 

In the applications of the thermodynamic equations of equilibrium to gaseous systems we shall take in ... 

An application of Eq. (11) is shown in Fig. 2, which gives the solubility of solid carbon dioxide in compressed air at a low temperature. The solubility is calculated from the equation of equilibrium... 

In Fig. 2, we showed the solubility of a solid component in a compressed gas as calculated from Eq. (11). A similar calculation can be made for the solubility of a liquid component in a compressed gas which is only slightly soluble in the liquid. For the liquid component, when xt 1, the equation of equilibrium is... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

The fundamental idea of this procedure is as follows For a system of two fluid phases containing N components, we are concerned with N — 1 independent mole fractions in each phase, as well as with two other intensive variables, temperature T and total pressure P. Let us suppose that the two phases (vapor and liquid) are at equilibrium, and that we are given the total pressure P and the mole fractions of the liquid phase x, x2,. .., xN. We wish to find the equilibrium temperature T and the mole fractions of the vapor phase yu y2,. .., yN-i- The total number of unknowns is N + 2 there are N — 1 unknown mole fractions, one unknown temperature, and two unknown densities corresponding to the two limits of integration in Eq. (6), one for the liquid phase and the other for the vapor phase. To solve for these N +2 unknowns, we require N + 2 equations of equilibrium. For each component i we have an equation of the form... 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

To simplify matters, let us assume that at the system temperature, solvents 1 and 3 have negligible volatility the gas phase, then, contains only component 2. The solubility of component 2 in the liquid is governed by the equation of equilibrium... 

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. 

It may be useful to assume that we know the equation of equilibrium for a fluid and demonstrate how a model of two channels allows us to evaluate the flattening of earth. First, consider the channel B, where we have... 

Substitution of the latter into Equation (3.173) gives the equation of equilibrium in terms of the second derivatives of the potential at the origin of coordinates ... 

In order to simplify this equation of equilibrium we take into account the following. First of all, the lever is usually made from a thin and light tube, with masses at its ends, and they can be treated as elementary masses. This means that the distance along axis is much larger than the dimensions of masses >t]. This allows us to... 

Hence the difference between the tensions at the two solid-fluid interfaces which is the quantity always involved in equations of equilibrium can be expressed in terms of the fluid-fluid tension and an angle, called the angle of contact which is plainly susceptible of direct measurement. 

The functional relationship that expresses how U depends on other extensities Xj is called the fundamental equation of equilibrium thermodynamics ... 

For an element in equilibrium with no body forces, the equations of equilibrium were obtained by Lame and Clapeyron (1831). Consider the stresses in a cubic element in equilibrium as shown in Fig. 2.3. Denote 7y as a component of the stress tensor T acting on a plane whose normal is in the direction of e and the resulting force is in the direction of ej. In the Cartesian coordinates in Fig. 2.3, the total force on the pair of element surfaces whose normal vectors are in the direction of ex can be given by... 

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