Constant pressure constraint, defined

Replacement of Pext by P, however, does not require complete equilibrium mechanical equilibrium with the surroundings is sufficient. In many slow processes, the system pressure closely tracks the external pressure and can be substituted for it in Eq. (5). The most commonly encountered of these is the constant-pressure process. Because we define our constraints in the surroundings, a constant-pressure process has constant Pext. If the system has a moveable boundary and the system is initially in mechanical equilibrium with the surroundings (P, = Pcxtl), then P will remain equal to Pcxl for the following two processes ... 

Constant pressure means constant f ext) because we define our constraints with respect to the surroundings, where we control the process. In the last step, we assume mechanical equilibrium between the system and the surroundings. This is an excellent assumption for slow processes with systems with movable boundaries, for which we will generally take f ext — P- It is not a good assumption for processes in which there is a sudden change of constraints, such as an explosion or the removal of a stop which secures a piston. In such cases, the system may not even have a uniform pressure. 

Well constraints for gas flows. Consider a locus of points L defining a general well path that may be vertical, horizontal, or deviated out-of-plane and containing multiple drainholes. Let 2 denote summations performed along L. Along well paths only, in anticipation of constant pressures in the borehole, we simplify Equation 15-22 by factoring out pyjj so that... 

Equation (3) defines the equilibrium condition under the constraint that temperature and pressure are constant. A related consequence of the Second Law is that if AG < 0 the reaction of the reactant to product is thermodynamically spontaneous. Thermodynamic spontaneity means that... 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

EXAMPLE 7.4 How does the equality of pressures maxiniize the multiplicity of states Consider a gas contained on two sides of a piston as shown in Figure 7.6. The number Na of particles on the left and the number Nb on the right are each fixed. The total volume is defined by M lattice sites, and the movable piston partitions the volume into Ma sites on the left and Mg sites on the right, with the constraint M = Ma + Mb = constant. 

The Gibbs free energy is one of the most important fundamental functions. Constant temperature and pressure arc the easiest constraints to impose in the laboratory, because the atmosphere provides them. T, p, and N are the natural variables for the Gibbs free energy G = G(T, p,N), which has a minimum at equilibrium. To find the fundamental equation, start with the enthalpy, H = H(S, p,N). Now we want to replace the dS term with a dT term in the equation dH = TdS -t Vdp + Zjii Define a function G ... 

Wang and Lee [1] define the so-called Langevin and Rayleigh radiation pressures, respectively, as the mean excess pressures that either depend upon the sound wave only (i. e., with C = 0), or on the sound wave together with a constraint which determines the constant C that contributes to the pressure. The concept of the radiation pressure enables the calculation of forces acting upon material surfaces, such as an interface between two fluids or the surface of a particle or a drop in a sound field. Strictly speaking, one should use the acoustic radiation stress tensor n to calculate such forces. However, in many situations, such as when the surface is rigid or when the velocity at a surface is normal to that surface, it is convenient to use the radiation pressure rather than the full stress. 

The modeling of the space function of growth can develop in a generic way by considering the reactivity of growth, specific to a particular reaction under well-defined conditions, such as a parameter likely to vary with the intensive constraints (partial pressures, concentrations, temperature) and independent of time insofar as these constraints are maintained constant. 

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