Equation component liquid molar

Table I. Component Liquid Molar Volumes (Vs) and Isothermal Compressibilities (ks) at 100 K and Vapor Pressure (Ps) Which Were Used along with Shape Factors (c or a) to Determine Equation-of-State Parameters a and b...

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. 

Critical constant increment in Lydersen equation (Equation 8.27) Special diffusion volume coefficient for component i (Table 8.5) Liquid molar volume Mass fraction (weight fraction)... 

The exponential term in Equation 7-13 is a correction factor for the effect of pressure on liquid-phase fugacity and is known as the Poynting factor. In Equation 7-13, V[ can be replaced by the partial molar volume of component i in the liquid solution for greater accuracy. Eor low to moderate pressure, V is assumed as the saturated liquid molar volume at the specified temperature. Equation 7-13 is simplified to give... 

Each of equation sets 3.32 and 3.34 encompasses NC equations, and equation system 3.33 includes N equations. Altogether there are N(1C + 1) simultaneous equations. The unknown variables are the vapor and liquid molar rates of each component in each stage (V, and Ly, 7 = 1,. .N and C) and the temperatures in... 

Variables Ly, and Vji refer to component molar flow rates in the liquid and vapor phases. The second subscript identifies the component. The component liquid and vapor enthalpies, both at saturated conditions, are related by the equation... 

It is remarkable that no empirical mixture parameters and no experimental data are required to use the equation. The only parameters in the Flory-Huggins equation are the hard core volumes V, which are a pme-component property, and the atomic or group contribution values are found in standard compilations. Since the v/s are significant in the FH equation only in terms of their ratios, pure-liquid molar volumes are often used for V in place of hard core volumes. For solutions of polymers of the same chemical formula, molecular masses are legitimate substitutes for V , for the same reason. Thus the volume fractions ( ) can be substituted by mass fractions W . Either volume fraction or mass fraction is directly related to laboratory data. To avoid mole fractions, the activity tti from Equations (4.368) and (4.369) can be used to calculate by / = aj. ... 

The equation is discussed in Section 4.3.3. Two parameters are required for each component v, is the pure i liquid molar volume at 298°K, and 5 is the solubility parameter of component i. The solubility parameter of the mixture 8 , is given by the volumetric average of the components 8 , = Zpi ,v,8,/2 pi ,v,. Parameters are available for numerous substances. Additional parameters can be readily determined from their definitions, when the need arises, except for light gases. 

The ability of the equations of state to fit VE data does depend to some extent on the component a and b values. In previous studies (5,6) a and b were determined for the LHW model by fitting a saturated liquid molar volume and a heat of vaporization for each component. Using these a and b values instead of the ones from Table II led to standard deviations for the binary VE fits which were larger than those in Table III (cf. Ref. 11). The maximum increase was from 0.024 to 0.048 cm3 mol 1... 

From the condition of mass equilibrium for a unit volume of gas-liquid mixture with volume distribution of drops n(V, t), there follow equations for the molar concentration of components in the gaseous phase ... 

In these equations N. is molar part of the component q. is its coeflBcient of viscosity in pure liquid state additional viscosity, which, in accordance with Frenkel, reflects the difference in energies of interaction of particles by the first and the second kinds between themselves and between the particles of the same kind. 

The value of a can be simply read off a VLE curve as the value of a corresponding to the interface partial pressure of the vol tile component, at the bulk-liquid molarity. Therefore, a quant tative check of the validity of Equation 51 can be performed. 

Equation (8.37) is a practical working relationship for the infinite dilution aqueous molecular component partial molar enthalpy as a function of temperature. There is, however, an underlying assumption made. The relationship derived above assumes equilibrium between liquid and vapor, or, in other words a saturated liquid-vapor situation. In a subcooled liquid the enthalpy predicted by equation (8.37) should be corrected for the enthalpy difference between the pure component at the prevailing pressure and the saturation pressure. 

In the entropic-free volume model, the activity coefficient of the solvent is given by Eqs. (44)-(48) with p = 1 [52]. The residual contribution is represented by the residual contribution of the UNIFAC model with temperature-dependent interaction parameters [53]. The liquid molar volumes needed for the calculation of the free volume of a component can be taken from experiment or calculated from the Tait equation [4] or by the group contribution method of Elbro et al. [56]. This model is relatively easy to use. 

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... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

As the mole fraction of A in the mixture iucreases toward uuity, the secoud term iu Eq. (2.17) teuds toward zero, aud the chemical poteutial of A teuds toward the standard chemical potential, Pa = G, the molar Gibbs euergy (or chemical poteutial) of A iu the realizable standard state of pure A, in the sense that pure liquid A is a known chemical substance. A similar equation holds for the component B. 

While the solubility parameter can be used to conduct solubility studies, it is more informative, in dealing with charged polymers such as SPSF, to employ the three dimensional solubility parameter (A7,A8). The solubility parameter of a liquid is related to the total cohesive energy (E) by the equation 6 = (E/V) 2, where V is the molar volume. The total cohesive energy can be broken down into three additive components E = E j + Ep + Ejj, where the three components represent the contributions to E due to dispersion or London forces, permanent dipole-dipole or polar forces, and hydrogen bonding forces, respectively. This relationship is used... 

In equation 33, the superscript I refers to the use of method I, a T) is the activity of component i in the stoichiometric liquid (si) at the temperature of interest, AHj is the molar enthalpy of fusion of the compound ij, and ACp[ij] is the difference between the molar heat capacities of the stoichiometric liquid and the compound ij. This representation requires values of the Gibbs energy of mixing and heat capacity for the stoichiometric liquid mixture as a function of temperature in a range for which the mixture is not stable and thus generally not observable. When equation 33 is combined with equations 23 and 24 in the limit of the AC binary system, it is termed the fusion equation for the liquidus (107-111). 

In the dynamic rate-based stage model, molar holdup terms have to be considered in the mass balance equations, whereas the changes in both the specific molar component holdup and the total molar holdup are taken into account. For the liquid phase, these equations are as follows ... 

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