Phases in Equilibria

The general problem is to determine at given conditions of temperature and pressure, the quantities and compositions of the two phases in equilibrium starting from an initial quantity of material of known composition and to resolve the system of the following equations ... 

The surface tension is calculated starting from the parachor and the densities of the phases in equilibrium by the Sugden method (1924) J... 

In order to specify fhe size of fhe sysfem, af leasf one of fhese variables ought to be extensive (one that is proportional to the size of the system, like n or the total volume V). In the special case of several phases in equilibrium several extensive properties, e.g. n and Vfor two phases, may be required to detennine the relative amounts of the two phases. The rest of the variables can be intensive (independent of the size of the system) like T,p, the molar volume V = V/n,or the density p. For multicomponent systems, additional variables, e.g. several ns, are needed to specify composifion. 

When, for a one-component system, one of the two phases in equilibrium is a sufficiently dilute gas, i.e. is at a pressure well below 1 atm, one can obtain a very usefiil approximate equation from equation (A2.1.52). The molar volume of the gas is at least two orders of magnitude larger than that of the liquid or solid, and is very nearly an ideal gas. Then one can write... 

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

Experimental results describing limited mutual solubility are usually presented as phase diagrams in which the compositions of the phases in equilibrium with each other at a given temperature are mapped for various temperatures. As noted above, the chemical potentials are the same in the equilibrium phases, so Eqs. (8.53) and (8.54) offer a method for calculating such... 

At low-to-moderate pressure ranges typical of most industrial apphcatious, the fundamental composition relationship between the vapor and hquid phases in equilibrium can be expressed as a function of the total... 

Enhancement of gas storage capacity through adsorption occins when the overall storage density is increased above that of the normal gas density at a given pressure. The adsorbed phase has a greater density than the gas phase in equilibrium with it. However, enhancement in a storage system of fixed volume can only happen if a greater amount of gas is adsorbed compared to the volume of gas displaced by the adsorbent volume. 

C concentration on the solid phase in equilibrium with the feed concentration. 

Systems in which the phases in equilibrium with the solid... 

Figure 3. Bromine concentration in the aqueous phase in equilibrium with complexes of different MEP MEM ratios 1 1 3 1, 9 1. Taken from Ref. [66]...

System Starting composition Phases in equilibrium Voltage (mV) vs. Li Li (mol/kg"1)... 

The points a, /3, on the horizontal lines, represent the compositions of liquid and vapour phases in equilibrium at a given temperature or pressure. 

Under certain pressure and temperature conditions, a system can contain two or more phases in equilibrium. An example is the temperature and pressure where solid and liquid are in equilibrium. We refer to this condition as (solid + liquid) equilibrium, and the temperature as the melting temperature. This temperature changes with pressure and with composition. The melting temperature when the... 

For a pure substance, having three phases in equilibrium results in a triple point that is invariant. When pure solid, liquid, and gaseous water are in equilibrium, the temperature is fixed at a value of 273.16 K, and the pressure of the gas is fixed at the vapor pressure value (0.6105 kPa). 

A useful relationship between the temperature and pressure of phases in equilibrium can be derived from the condition for equilibrium. We start with equilibrium between phases A and B written as... 

From these kinds of considerations, it is easy to see why a one component system can have no more than three phases in equilibrium. Any number of phases greater than three would yield a negative number of degrees of freedom, a situation that makes no physical sense. 

Two phases are present in the region between the two curves the compositions of the two phases in equilibrium with each other are given by the intersection of a horizontal tie-line with the vapor and liquid curves. Lines cb and fd in Figure 8.13 are two examples. One degree of freedom is present in this region. Thus, specifying the pressure fixes the compositions of the phases in equilibrium conversely, specifying the composition of one of the phases in equilibrium sets the pressure and the composition of the other phase.w... 

Since, in this two-phase region, the compositions of the phases in equilibrium are fixed by a horizontal tie-line (for example, line def), specifying the pressure, which specifies the position of the tie-line, sets the composition, as expected with one degree of freedom present. 

Figure 8.18 (Liquid + liquid) equilibria at p = 0.1 MPa for (.vin-QHu + Y2CH3OH). The intersections of the tie-lines with the equilibrium curve give the compositions of the phases in equilibrium,...

In a two-phase region, the composition oC the two phases in equilibrium are given by the end points of tie-lines. Thus, under curve ac. the phases are pure solid benzene ( v — 0) and a liquid with composition given by line ac. 

Use the phase diagram for helium in Exercise 8.13 (a) to describe the phases in equilibrium at each of helium s two triple points (b) to decide which liquid phase is more dense, helium-1 or helium-II. 

The Volta potential is defined as the difference between the electrostatic outer potentials of two condensed phases in equilibrium. The measurement of this and related quantities is performed using a system of voltaic cells. This technique, which in some applications is called the surface potential method, is one of the oldest but still frequently used experimental methods for studying phenomena at electrified solid and hquid surfaces and interfaces. The difficulty with the method, which in fact is common to most electrochemical methods, is lack of molecular specificity. However, combined with modem surface-sensitive methods such as spectroscopy, it can provide important physicochemical information. Even without such complementary molecular information, the voltaic cell method is still the source of much basic electrochemical data. 

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