Saturation vapor pressure, hypothetical

Combining Equations 4, 5, 6, and 7 an expression for the hypothetical saturation vapor pressure of the pure monomer is obtained ... 

The most convenient mathematical method of describing pervaporation is to divide the overall separation processes into two steps, as shown in Figure 40. The first is evaporation of the feed Hquid to form a (hypothetical) saturated vapor phase on the feed side of the membrane. The second is permeation of this vapor through the membrane to the low pressure permeate side of the membrane. Although no evaporation actually takes place on the feed side of the membrane during pervaporation, this approach is mathematically simple and is thermodynamically completely equivalent to the physical process. The evaporation step from the feed hquid to the saturated vapor phase produces a separation, which can be defined (eq. 13) as the ratio of... 

Hydrothermal synthesis of a-alumina has been well studied. Since the hydro-thermal reaction of aluminum compound yields boehmite at relatively low temperatures (approximately 200°C), transformation of boehmite was examined and it was reported that more than 10 hours is required for complete conversion into a-alumina, even with a reaction at 445°C in a 0.1 N NaOH solution and in the presence of seed crystals. On the other hand, under glycothermal conditions, a-alumina is formed at 285°C for 4 h. The equilibrium point between diaspore (another polymorph of AlOOH) and a-alumina under the saturated vapor pressure of water was determined to be 360°C. However, near the equilibrium point, the transformation rate is very sluggish, and only a small conversion of diaspore is observed. Therefore complete conversion of diaspore into a-alumina requires a much higher temperature. Since boehmite is slightly less stable than diaspore, the hypothetical equilibrium point between boehmite and a-alumina would be lower than that for diaspore-alumina. However, a-alumina would not be formed by a hydrothermal reaction at such a low temperature as has been achieved in the glycothermal reaction. 

Here is the vapor pressure of pure liquid solute at the same temperature and total pressure as the solution. If the pressure is too low for pure B to exist as a liquid at this temperature, we can with little error replace with the saturation vapor pressure of liquid B at the same temperature, because the effect of total pressure on the vapor pressure of a liquid is usually negligible (Sec. 12.8.1). If the temperature is above the critical temperature of pure B, we can estimate a hypothetical vapor pressure by extrapolating the liquid-vapor coexistence curve beyond the critical point. 

Figure 12.19 Effect of temperature on fugacity of a pure saturated liquid. Vapor-phase nonidealities (cpf) lower from the pure vapor-pressure curve, but the variation of /j-"with 1/T remains roughly linear. At supercritical temperatures, jnue vapor pressures do not exist nevertheless, for (0.9 < r /T < 1), we may choose the hypothetical pure liquid for the standard state and obtain a value of f° by extrapolation. These values were comjnited for pure water using data from steam tables [14].

Solution The problem really asks for the calculation of the difference in the free energy of formation between two standard states. The gas standard state is in the hypothetical ideal-gas state at T°, P°, and the liquid is in the pure liquid state at the same temperature and pressure. We construd the following path from the hypothetical ideal-gas state at P°, T° (g), to the saturated vapor at T, to the saturated liquid at T°, to the pure liquid at T, P°. To obtain G°(Z) we add corresponding changes to G°(g) ... 

Despite widespread use of the ideal K-value concept in industrial calculations, particularly during years prior to digital computers, a sound thermodynamic basis does not exist for calculation of the fugacity coefficients for pure species as required by (4-85). Mehra, Brown, and Thodos discuss the fact that, for vapor-liquid equilibrium at given system temperature and pressure, at least one component of the mixture cannot exist as a pure vapor and at least one other component cannot exist as a pure liquid. For example, in Fig. 4.3, at a reduced pressure of 0.5 and a reduced temperature of 0.9, methane can exist only as a vapor and toluene can exist only as a liquid. It is possible to compute vl or f v for each species but not both, unless vl = vy, which corresponds to saturation conditions. An even more serious problem is posed by species whose critical temperatures are below the system temperature. Attempts to overcome these difficulties via development of pure species fugacity correlations for hypothetical states by extrapolation procedures are discussed by Prausnitz. ... 

---