Composition variable

In this section, definitions of a number of composition variables are listed. Useful relations among these variables are given, and the approximate values of some of these variables for systems in which one component predominates are presented. 

The most commonly used composition variables are as follows  

Total mass of an r-component system w = Wf Weight fraction = w,/w Molarity concentration Ci = 1000 i/K Partial densities pi = w,/K 

Mole ratio with respect to reference component 1 rj = njni Molalitywithrespecttoreferencecomponentl = (1000/Afi)( i/ i) A number of relations between these variables are the following  

In systems in which component 1 predominates, the following approximate relations are valid  

A homogeneous mixture is a phase containing more than one substance. This chapter discusses composition variables and partial molar quantities of mixtures in which no chemical reaction is occurring. The ideal mixture is defined. Chemical potentials, activity coefficients, and activities of individual substances in both ideal and nonideal mixtures are discussed. 

Except for the use of fugacities to determine activity coefficients in condensed phases, a discussion of phase equilibria involving mixtures will be postponed to Chap. 13. 

A composition variable is an intensive property that indicates the relative amount of a 

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When a condensable solute is present, the activity coefficient of a solvent is given by Equation (15) provided that all composition variables (x, 9, and ) are taicen on an (all) solute-free basis. Composition variables 9 and 4 are automatically on a solute-free basis by setting q = q = r = 0 for every solute. 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Theory shows that these equations must be simple power series in the concentration (or an alternative composition variable) and experimental data can always be fitted this way.)... 

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Use of equation 247 for actual calculations requires expHcit iatroduction of composition variables. As ia phase-equiUbrium calculations, this is normally done for gas phases through the fugacity coefficient and for Hquid phases through the activity coefficient. Thus, either... 

Postiilate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. Tnese molar or unit-mass properties, represented by the plain symbols U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V U S ) do depend on system size, and are extensive. For a system containing n moles of fluid, M = nM, where M is a molar property. 

The application of Eq. (4-342) requires explicit introduction of composition variables. For gas-phase reactions this is accomphshed through the fugacity coefficient ... 

From the above list of rate-based model equations, it is seen that they total 5C -t- 6 for each tray, compared to 2C -t-1 or 2C -t- 3 (depending on whether mole fractious or component flow rates are used for composition variables) for each stage in the equihbrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. 

The theoiy enables a reasonable estimate of sample quantity needed to attain specified accuracy of a composition variable. The result is an ideal quantity—not realized in practice. Actual quantities for practical estimation are larger by an appropriate multiple to account for the reality that material is incompletely mixed when stored in stockpiles or carried on conveyors. Sample quantity to accommodate incompletely mixed sohds can be specified through evaluating variance by autocorrelation of data derived with a series of stockpile samples, or from multiple sample extractions taken from a moving stream (Gy, Pitard). 

The entries in the table are arranged in order of increasing reaction coordinate or distance along the reaction path (the reaction coordinate is a composite variable spanning all of the degrees of freedom of the potential energy surface). The energy and optimized variable values are listed for each point (in this case, as Cartesian coordinates). The first and last entries correspond to the final points on each side of the reaction path. 

Wetness of a metal surface The lime of wetness of the metal surface is an exceedingly complex, composite variable. It determines the duration of the electrochemical corrosion process. Firstly it involves a consideration of all the means by which an electrolyte solution can form in contact with the metal surface. Secondly, the conditions under which this solution is stable with respect to the ambient atmosphere must be considered, and finally the rate of evaporation of the solution when atmospheric conditions change to make its existence unstable. Attempts have been made to measure directly the time of wetness , but these have tended to use metals forming non-bulky corrosion products (see Section 20.1). The literature is very sparse on the r61e of insoluble corrosion products in extending the time of wetness, but considerable differences in moisture desorption rates are found for rusted steels of slightly differing alloy content, e.g. mild steel and Cor-Ten. 

Composition Variables on Acceleration-Induced Burning-Rate Augmentation of Solid Propellants , NASA TND-6923 (1972) 13) G.W. 

NC content is a major compositional variable affecting the physical properties of tensile... 

But much of chemistry involves mixtures, solutions, and reacting systems in which the number of moles or mole number, of each species present can be variable. When this happens, the extensive properties, Z = V, S, U, H,A or G become functions of the composition variables, as well as two of the state variables as described earlier.a We can express this mathematically as... 

Because p and T are held constant, and only a single composition variable is changed in each derivative, the partial derivatives in equation (5.18) are equivalent to those in equation (5.14), and equation (5.18) becomes the same as equation (5.17). ... 

The number of independent variables for each phase are the two that we have considered earlier among p, T, V, U, H, A, and G, plus (C—1) composition variables.y Thus the total number of independent variables for P phases is (C -1+2) per phase (P phases) = (C+ 1)P. 

Clearly, we must determine F or p as a function of composition. The integration will be easier if is treated as the composition variable rather than a since this avoids expansion of the derivative as a product d Va) = Vda- -adV. The numerical methods in subsequent chapters treat such products as composite variables to avoid expansion into individual derivatives. Here in Chapter 2, the composite variable, Na = Va, has a natural interpretation as the number of moles in the batch system. To integrate Equation (2.32), F or p must be determined as a function of Na- Both liquid- and gas-phase reactors are considered in the next few examples. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

Equation (2.42) represents a set of M ODEs in M independent variables, /, //,. It, like the redundant set of ODEs in Equation (2.38), will normally require numerical solution. Once solved, the values for the s can be used to calculate the N composition variables using Equation (2.40). 

Solution The analysis could be carried out using mole fractions as the composition variable, but this would restrict applicability to the specific conditions of the experiment. Greater generality is possible by converting to concentration units. The results will then apply to somewhat different pressures. The somewhat recognizes the fact that the reaction mechanism and even the equation of state may change at extreme pressures. The results will not apply at different temperatures since k and kc will be functions of temperature. The temperature dependence of rate constants is considered in Chapter 5. 

The sum of k i and ki2 is so high relative to the rate of micromixing that A and B will not coexist. Thus the number of ordinary mass balance differential equations requiring integration can be reduced from five to four by defining the following composite variables ... 

A complete list of the reaction conditions tested for this response surface design can be found in [76], The center point reaction condition was repeated six times. This was done to measure the variability of the reaction system. Also, the space velocity is kept constant, as it was the least important factor predicted by screening design, for all the reaction conditions. The purpose of this nested response surface design was to develop an empirical model in the form of Eqn (5) to relate the five reaction condition variables and the three catalyst composition variables to the observed catalytic performance. 

In more complex cases when several reactions are occurring simultaneously in the system under observation, calculations of the composition of the system as a function of time will require the knowledge of a number of independent composition variables equal to the number of independent chemical equations used to characterize the reactions involved. 

Although there have been attempts to evaluate the mechanical properties of trimodal elastomers, this has not been done in any organized manner. The basic problem is the large number of variables involved, specifically three molecular weights and two independent composition variables (mol fractions) this makes it practically impossible to do an exhaustive series of relevant experiments. For this... 

The analytical composition of the system must be known and its degree of variability determined. This is the chemical composition variable (see Figure 2.1). 

As in Example 3-3, cB is not independent of cA, but is related to it through equation 3.4-5, to which we add the extent of reaction to emphasize that there is only one composition variable ... 

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