Equation pure-component

Xj. = Adjustable parameter in the UNIFAC equation = Pure-component constant. 

Equation (15) requires only pure-component and binary parameters. 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Pure-component parameters required in Equations (16) through (23) are... 

When pure component i constitutes the solid phase, the liquid-solid equiiibrium obeys the following equation ... 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

Equation (8.19) can be integrated using the convention that aj = 1 for the pure component, which has the vapor pressure p.° ... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubiHty parameters are evaluated from pure-component data. Results based on these equations should be treated as only quaHtative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). AppHcations of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processiag, such as N2, H2, CO2, and H2S. Values for 5 and H can be found ia many references (1—3,7). 

Enthalpy of Vaporization The enthalpy (heat) of vaporization AHv is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liqmd of a pure component i.e., at a temperature (below the critical temperature) anci corresponding vapor pressure. AHy is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation ... 

A mixing rule developed by Kendall and Monroe" is useful for determining the liquid viscosity of defined Iiydi ocai bon mixtiai es. Equation (2-119) depends only on the pure component viscosities at the given temperature and pressure and the mixture composition. 

Liquids For pure component hydrocarbon liquids at reduced temperatures between 0.25 and 0.8 and at pressures below 3.4 MPa, an equation based on the methods of Pachaiyappan et al. and RiedeP may be used ... 

Equations of state are also used for pure components. Given such an equation written in terms of the two-dimensional spreading pressure 7C, the corresponding isotherm is easily determined, as described later for mixtures [see Eq. (16-42)]. The two-dimensional equivalent of an ideal gas is an ideal surface gas, which is described by... 

Equation (16-36) with y = 1 provides the basis for the ideal adsorbed-solution theoiy [Myers and Prausnitz, AIChE J., 11, 121 (1965)]. The spreading pressure for a pure component is determined by integrating Eq. (16-35) for a pure component to obtain... 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

Note that for this discussion now, Pim, just above and in equations to follow, refers to the pure component vapor pressure of the immiscible liquid being distilled [127], When steam is added to the still [127] ... 

Figures 10-85, 10-86, and 10-86A and Equation 10-115A represent the effective reduction of the pure component (condensahle) when inert gases are present, resulting in the reduced effective heat transfer for condensing the mixture.

If we examine the first column of the matrix in equation [23] we see that each Kw, is the absorbance at each wavelength, w, due to one concentration unit of component 1. Thus, the first column of the matrix is identical to the pure component spectrum of component 1. Similarly, the second column is identical to the pure component spectrum of component 2, and so on. 

In equation [24], A is generated by multiplying the pure component spectra in the matrix K by the concentration matrix, C, just as was done in equation [20]. But, in this case, C will have a column of concentration values for each sample. Each column of C will generate a corresponding column in A containing the spectrum for that sample. Note that equation [24] can also be written as equation [22]. We can represent equation [24] graphically ... 

To produce a calibration using classical least-squares, we start with a training set consisting of a concentration matrix, C, and an absorbance matrix, A, for known calibration samples. We then solve for the matrix, K. Each column of K will each hold the spectrum of one of the pure components. Since the data in C and A contain noise, there will, in general, be no exact solution for equation [29]. So, we must find the best least-squares solution for equation [29]. In other words, we want to find K such that the sum of the squares of the errors is minimized. The errors are the difference between the measured spectra, A, and the spectra calculated by multiplying K and C ... 

We can see that the estimated spectra, while they come close to the actual spectra, have some significant problems. We can understand the source of the problems when we look at the spectrum of Component 4. Because we stated in equation [40] that we will account for all of the absorbance in the spectra, CLS was forced to distribute the absorbance contributions from Component 4 among the other components. Since there is no "correct 1 way to distribute the Component 4 absorbance, the actual distribution will depend upon the makeup of the training set Accordingly, we see that CLS distributed the Component 4 absorbance differently for each training set. We can verify this by taking the sum of the 3 estimated pure component spectra, and subtracting from it the sum of the actual spectra of the first 3 components ... 

Recognizing the difficulty satisfying the requirements for successful CLS, you may wonder why anyone would ever use CLS. There are a number of applications where CLS is particularly appropriate. One of the best examples is the case where a library of quantitative spectra is available, and the application requires the analysis of one or more components that suffer little or no interference other than that caused by the components themselves. In such cases, we do not need to use equation [33] to calculate the pure component spectra if we already have them in a library. We can simply construct a K matrix containing the required library spectra and proceed directly to equation [34] to calculate the calibration matrix K., . 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

To obtain an analytic function / in Eq. (55), Chueh uses the Redlich-Kwong equation however, since the application is intended for liquids, the two constants in that equation were not evaluated (as is usually done) from critical data alone, but rather from a fit of the pure-component saturated-liquid volumes. The constants a and b in the equation of Redlich and Kwong are calculated from the relations... 

---