Equilibrium problems pressures

Example 12.4 illustrates a principle that you will find very useful in solving equilibrium problems throughout this (and later) chapters. As a system approaches equilibrium, changes in partial pressures of reactants and products—like changes in molar amounts—are related to one another through the coefficients of the balanced equation. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

The question may then be raised as to whether insoluble monolayers may really be treated in terms of equilibrium thermodynamics. In general, this problem has been approached by considering (i) the equilibrium spreading pressure of the monolayer in the presence of the bulk crystalline surfactant, and (ii) the stability of the monolayer film as spread from solution. These quantities are obtained experimentally and are necessary in any consideration of film thermodynamic properties. In both cases, time is clearly a practical variable. 

Reaction between an absorbed solute and a reagent lowers the equilibrium partial pressure of the solute and thus increases the rate of mass transfer. The mass transfer coefficient likewise may be enhanced which contributes further to increased absorption rate. Three modes of contacting gas and liquid phases are possible The gas is dispersed as bubbles in the liquid, the liquid is dispersed as droplets, the two phases are contacted on a thin liquid film deposited over a packing or wall. The choice between these modes is an important practical problem. 

Note You could also solve this problem using equilibrium partial pressures of the gases ... 

Note that the question tells you there are 2.00 moles of the compound. This information is irrelevant to solving the problem since equilibrium vapor pressure is independent of the amount of compound. 

Remember, in working Le Chatelier problems, pressure effects are important only for gases that are involved in the equilibrium. 

Certainly the condition in Eq. (74) is valid since there must be no accumulation of solute at the interface. But the condition for equilibrium at the interface in Eq. (75) may not be adequate for the description of many mass transfer processes. It is not, for example, difficult to imagine that in the evaporation of a liquid, the vaporization may take place so rapidly that the concentration of vapor just above the liquid surface is considerably less than the concentration corresponding to the equilibrium vapor pressure. The problem of obtaining a quantitative theoretical description of this process has been attacked by Schrage (S4), who has suggested several molecular theories for describing gas-liquid and gas-solid systems. 

Although UHV is required for LEED measurement, there is considerable interest in applying this technique to surfaces that carry adsorbed species. In view of our discussion of adsorption equilibrium above in the chapter, there is a difficulty here since adsorbed molecules imply an equilibrium gas phase. One way around this problem is to study surfaces at a sufficiently low coverage that the equilibrium gas pressure is compatible with the LEED technique. When higher pressures are desired, the surface is first equilibrated, then the excess gas is pumped out, and the surfaces before and after adsorption are compared through LEED. Chemisorption is better suited for study by this method than physical adsorption because the adsorbed layer remains intact when the equilibrium gas is removed. 

Equilibrium vapor pressure is the vapor pressure of a system in which two or more phases or a substance coexist in equilibrium. In meteorology, the reference is to water substance, unless otherwise specified, If the system consists of moist air in equilibrium with a plane surface of pure water or ice, the more specialized term saturation vapor pressure is usually employed, in which case, the vapor pressure is a function of temperature only. In the atmosphere, the system is complicated by the presence of impurities in liquid or solid water substance (see also Raoult s Law), drops or ice crystals or both, existing as aerosols and, in general, the problem becomes one of nucleation. For example, the difference in vapor pressure over supercooled water... 

At system pressures up to several tens of MPa, the fugacity coefficients, < > and (j), and the Poynting factor, 7Zp are usually near unity. A simplified version of equation 19 can therefore be used for the majority of vapor—liquid equilibrium problems ... 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

We have applied a global optimization technique, based on interval analysis, to the high-pressure phase equilibrium problem (INTFLASH). It does not require any initial guesses and is guaranteed, both mathematically and computationally, to converge to the correct solution. The interval analysis method and its application to phase equilibria using equation-of-state... 

One further vapor/liquid equilibrium problem is the flash calculation. origin of the name is in the change that occurs when a liquid under press passes through a valve to a pressure low enough that some of the liquid vapori or flashes, producing a two-phase stream of vapor and liquid in equilibri We consider here only the P.f -flash, which refers to any calculation of quantities and compositions of the vapor and liquid phases making up a two-ph system in equilibrium at known P, T, and overall composition. 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

A typical equilibrium problem involves finding the equilibrium concentrations (or pressures) of reactants and products, given the value of the equilibrium constant and the initial concentrations (or pressures). 

In principle, knowing the molar entropy of the perfect gas (Section 1.17), and by measuring the change of equilibrium gas pressure as a function of temperature, one can determine the molar entropy of the adsorbed phase. The problem here is that the experiment has to be carried out at constant 0, a problematic task. Methods for circumventing this difficulty are shown below. Meanwhile, for completeness, we observe that at equilibrium the chemical potentials of the gas and adsorbate must match then Hg — Hs — T(Sg — Ss), so that we obtain the alternative formulation... 

In the chapters so far we have considered the phase rule somewhat intuitively for example, in solving equilibrium problems we used the obvious principle that an equilibrium problem can be solved if for n unknowns (e.g., activities or concentrations of n species) n equations are available. For example, in a closed dissolved carbonate system we need to define the system (H2CO3, HC03 , CO H, OH ) and two concentration conditions (e.g., Ct and pH, or [Aik] and H2CO ]), in addition to temperature and pressure, because the five species are interconnected by three mass laws (two acid-base equilibria and the ion product of H2O). In the example given P = 1 (aqueous solution), C = 3 le.g., HCO, H", H20(l)], andF = 4 (pressure, temperature, and two concentration conditions). 

The first step in the theoretical procedure is to iteratively solve the reaction equilibrium problem for composition of the equilibrium reaction mixture assuming that the reactant phase is initially critical or supercritical. Critical properties of the reaction mixture and fugacities are then estimated. Operating temperature and pressure are then re-evaluated such that the reaction mixture is constrained to the supercritical region. The iterative process is continued until satisfactory convergence is achieved. 

Calculate the equilibrium partial pressures of all species involved in a gas-phase chemical or gas-solid reaction from the initial pressure(s) of the reactants (Section 14.5, Problems 27-32). 

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