Residual function equilibrium calculations

As mentioned before, Approach A (also called supercritical compounds can be handled easily and that besides the phase equilibrium behavior various other properties such as densities, enthalpies including enthalpies of vaporization, heat capacities and a large number of other important thermodynamic properties can be calculated via residual functions for the pure compounds and their mbctures. For the calculation besides the critical data and the acentric factor for the equation of state and reliable mixing rules, only the ideal gas heat capacities of the pure compounds as a function of temperature are additionally required. A perfect equation of state with perfect mixing rules would provide perfect results. This is the reason why after the development of the van der Waals equation of state in 1873 an enormous number of different equations of state have been suggested. 

Figure 6. Adsorption isotherm for 2,4,5-T Capacity for adsorption of 2,4,5-T on 273-micron Columbia LC carbon at 25°C. is plotted as a function of residual concentration, C q, at equilibrium. The left ordinate gives capacity in units of fimoles of solute per gram of carbon and the right ordinate in units of milligrams of solute per gram of carbon. The points represent experimental data the line drawn through the data is the calculated Langmuir isotherm.

The integration of the equation (9.2) is easy, but in each point the equilibrium constants Kj has to be calculated by a suitable thermodynamic model, as function of pressure, temperature and composition. The trajectory obtained in this way starting with an arbitrary initial concentration describes a residue curve. The assembly of trajectories of liquid composition forms a Residue Curves Map (RCM). 

By simply knowing the phase equilibrium behavior and the composition within the beaker at the start of the experiment (x ), one can easily construct a residue curve by integrating Equation 2.8. Such integration is usually performed with the use of a numerical integration method (see later, Section 2.5.3), such as Runge Kutta type methods, remembering that at each function evaluation, a bubble point calculation must be performed in order to determine y(x). 

The Raman imaging provides information on the residual stress fields around indentations. Assuming purely elastic loading, Lucazeau and Abello [116] calculated the stress components around Vickers indentation as functions of the radius to indentation center R. The calculated hydrostatic pressure cr(R) = —l/3[ori(R) -I- CT2(R) -I- 03(/ )] was then compared with the experimentally measured shift of the main Raman band of Si-I from its equilibrium position [ A w (R) ]. The experimental pressures were found to be significantly higher than the theoretical ones at intermediate distances from the indentation center, which implied... 

Fig. 3-8 gives an estimate of the number of theoretical stages required for absorption and stripping as a function of the absorption factor and the residual gas load. Since data in Fig. 3-8 are calculated by means of Eqs. (3-17) and (3-18), which are based on linear equilibrium curve and balance line, an appropriate concentration scale for the solvent and gas has to be used. 

Thus, the equilibrium constant in the solid phase can be calculated on the basis of the molecular partition functions of vibration - i.e. the vibration frequencies of the molecules. Those values also enable us to calculate the residual energy of these molecules, and hence, in the case of perfect solutions, to determine the exponential term in relation [A2.70]. 

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