Equilibrium isobars

A schematic representation of these several conditions can be seen from Fig. 3. If the continuous lines AB and CD represent true equilibrium isobars for van der Waals and chemisorption respectively, it is obvious that, on raising and lowering the temperature in the CD region the same experimental values will be obtained. If the dotted line BC represents values obtained while the temperature is being raised it is obvious that these are below the true equilibrium values because the latter cannot be reached in practicable times of observation since the velocity is too low. If the... 

FIGURE 113. Equilibrium isobars (a) and isosteres (b) showing temperature and pressure dependence of equilibrium adsorption capacity for water on 4A sieve. (Courtesy of W. R, Grace/... 

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

It should be noted that we have here considered the system at constant pressure. If we are not considering the system at isobaric conditions, the invariant equilibrium becomes univariant, and a univariant equilibrium becomes divariant, etc. A... 

At a given temperature and pressure eqs. (4.7) and (4.8) must be solved simultaneously to determine the compositions of the two phases a and P that correspond to coexistence. At isobaric conditions, a plot of the composition of the two phases in equilibrium versus temperature yields a part of the equilibrium T, x-phase diagram. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

As an example of more complex systems and descriptions, the Ni-Mg system is shown in Fig. 2.32 (adapted from Levinsky 1997). In (a) an isobaric section of the diagram is shown (a low pressure has been considered in order to have a certain extension of the gas phase which consists essentially of Mg vapour). In Fig 2.32(b) there is an isothermal section of the diagram at 700°C. Notice, for different values of pressure, the change in the sequence of phases stable at different compositions. A value of the pressure close to atmosphere is approached at the top of the figure. In Fig 2.32(c) the usual Tlx diagram is shown. This can be considered an isobaric phase diagram if pressure is relatively low but still higher than the sum of the equilibrium partial pressures of the components. 

Figure 2.31. Schematic representation of the P/T equilibria in a simple two-component system (forming continuous solid and liquid solutions). In (a) a perspective view of the P-T-X diagram is shown in (b) its projection on the P/T plane. Notice the two single-component systems represented, for instance, for the component B by the three lines SB/G (sublimation line of B representing the gas/so lid equilibrium), SB/LB (melting equilibrium of B) and the boiling line LB/G. The solid solution is indicated by a. Notice in (a) the isobaric and isothermal sections of the diagrams (compare with Fig. 2.1).

Figure 17.4. Equilibrium data for the adsorption of ammonia on charcoal1 10) (a) Adsorption isotherm (b) Adsorption isobar (c) Adsorption isostere...

The reader will find adsorption equilibrium relationships presented in any of three typical forms. The form of equilibrium most frequently presented is the isotherm, the partial pressure as a function of the loading at a given constant sorbent temperature. An isobar implies a chart of the loading as a function of the temperature while the partial pressure of the sorbate is held constant. 

Aucejo, A., Loras, S., Munoz, R., Reich, R., and Segura, H. Isobaric vapor-liquid equilibrium in the systems 2-methylpentane-r methyl 1,1-dimethylethyl ether,-r ethyl 1,1-dimethylethyl ether, and-r methyl l,l-dimethylpropylether,7 Chem. Eng. Data, 43(6) 973-977, 1998. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

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