Molar fraction, definition

A quantitative study of the cholesteric induction and of the chiral transfer from dopant to phase requires the definition of the helical twisting power 3. This quantity expresses the ability of a chiral dopant to twist a nematic phase and can be numerically expressed in Eq. (2) where p is the cholesteric pitch, c the dopant molar fraction, and r its enantiomeric excess its sign is taken to be positive or negative for right-handed (P), or left-handed (M) cholesterics, respectively. This relation holds for molar fractions <0.01-0.05 ... 

From a thermodynamic viewpoint the theoretical framework (5-6) and the experimental measurements of such equilibria are well established. Two approaches are currently in use. They only differ in the definition of the ion activity at the surface, which is either expressed as an equivalent fraction (5) or as a molar fraction (6-8). 

The diagrammatic form of figure 5.68 is that commonly adopted to display intracrystalline distributions (see also figures 5.39 and 5.40). However, this sort of plot has the disadvantage of losing definition as the compositional limits of the system are approached. A different representation of intracrystalline disorder is that seen for olivines (figures 5.10 and 5.12 section 5.2.5) the distribution constant is plotted against the molar fraction of one of the components in the mixture. 

For conversions lower than xgei the average molar mass of the polymer exhibits a continuous increase. The first two moments of the molar mass distribution are the number-average molar mass, Mn, and the mass-average molar mass, Mw, respectively. Mn is defined in terms of the number contribution of every species to the whole population. The weight factor used to define this average is the molar fraction. Mw is defined in terms of the mass contribution of every species to the whole mass, so that the mass fraction is the weight factor used in its definition. 

As already mentioned, the definition of phases in articular cartilage is not unambiguous, because the mechanical, chemical and electrical roles of proteoglycans (PG s) may dictate contradictory choices. In fact, if the phase criterion was kinematically based (that is on velocity), PG s would be classified as part of the solid phase. However, its osmotic effect is important, not so much because of its concentration or molar fraction itself, but because of its effective charge and the latter should be involved in the electroneutrality condition of the extrafibrillar phase. 

We now discuss the partial differential with respect to pressure p and temperature T of the equilibrium constant Kx(T,p) in terms of the molar fraction. From the definition in Eq. 6.3 we obtain immediately Eq. 6.18 for the partial differential of the logarithm of Kx with respect to pressure p ... 

EDP or EPW is a mixture of ethylene-diamine (ED or E), pyrocatechol (P), and water (W). This solution usually operates at 110-120 °C (about boiling point) [93], The typical compositions and etch rates can be seen in Table 5. Figure 23 shows the etch rate as a function of the water content [92]. No etching occurs in solutions without the presence of water and the maximum etch rate is observed with a water molar fraction of about 0.6. The etch rate increases with increasing pyrocatechol content to about 5 mol% above which etch rate becomes constant. There is a definite etch rate without addition of pyrocatechol,... 

Interpretation of the second and third virial coefficients, A2 and A3, in terms of Floiy-Huggins theory is apparent from Eq. (3.82). The second virial coefl[icient A2 evidently is a measure of the interaction between a solvent and a polymer. When A2 happens to be zero, Eq. (3.82) simplifies greatly and many thermodynamic measurements become much easier to interpret. Such solutions with vanishing A2 may, however, be called pseudoideal solutions, to distinguish them from ideal solutions for which activities are equal to the molar fractions. Inspection of Eq. (3.83) reveals that A2 vanishes when the interaction parameter X equal to. We should also recall that %, according to its definition given by Eq. (3.40), is inversely proportional to temperature T. Since x is positive for most polymer-solvent systems, it should acquire the value at some specific temperature. 

There are attempts to motivate the definition of ideal mixture by a simpler way, e.g. it is possible to show [149, 150] that if the chemical potential of each constituent depends (besides temperature and pressure) only on the molar fraction of that constituent then this dependence is logarithmic as in (4.437) (it is assumed also that the partial internal energy and volume of at least one constituent depends on temperature and pressure only and that the number of constituents must be 3 as a minimum). 

A useful definition is provided by Haraya et alP and Takabaand Nakao, who used a unique coefficient expressed in terms of molar fractions to measure the polarization on both the membrane sides. However, the expressions... 

The availability of the local compositions allows one to calculate a quantity called preferential solvation. There is no generally established definition of preferential solvation (see Section 3.1 in Chapter 3). For a binary mixture, the excess local molar fraction has been usually considered synonymous with preferential solvation the more positive the higher the preference of i to be solvated by j than by... 

In liquid systems with chemically non-interacting components, molar polarization, P, according to definition, is additive value expressing composition in molar fractions ... 

To quantify this we will develop a definition of relative volatility from some basic equations of state. Firstly Raoult s Law states that the partial pressure of a component i in the vapour (p,) is proportional to its molar fraction in the liquid (x,). 

In view of the definition of the molar fractions, this relation is written as ... 

Consider a solution. We label the values relative to the solvent with the subscript 0, and those relative to the solutes with the subscript s. By definition of the molar fractions, we can write the ratio ... 

This equation is the link between tables of thermodynamic data (such as Table 4.2.1), which allow the evaluation of ArG°, and the equilibrium constant Kr of the reaction (sometimes also denoted as reaction quotient Qr), which is a function of the composition of the system in terms of concentration, molar fractions, and so on. The value and definition of Kr depends on the choice of the standard state and the ideality of the system, as shown subsequently for ideal and real gases, liquids, and gas-solid systems. 

By definition, the equilibrium coefficient of the component i is the ratio Yi/xi of the molar fraction in the gaseous phase to the molar fraction in the liquid phase. 

By taking into account the definitions of molar fraction. Equations 3.35 and 3.36, the concentrations in Equations 3.39 and 3.40 can alternatively be... 

Formally, Equation 3.61 is obtained by dividing Equation 3.57 with 3.59 or by dividing Equation 3.58 with 3.60. One should, however, recall that has a dissimilar definition for continuous (Equation 3.55) and discontinuous (Equation 3.56) reactors. For continuous reactors, xoi denotes the molar fraction of component i at the reactor inlet, whereas in the case of discontinuous reactors, it is the molar fraction at the initial state. 

The definition of molar fraction gives us, starting from Equations 3.71 and 3.72, as well as from Equations 3.73 and 3.74, an expression for the molar fraction x/ ... 

For a system with multiple chemical reactions, the expression for the total concentration (Equation 3.87) is naturally still valid. By inserting expressions of the molar fractions. Equations 3.76 and 3.85, into the definition of the concentration. Equations 3.44 and 3.45, the following expressions are obtained for the concentrations ... 

In order to obtain thermodynamic data, all of these constants have to be transformed into thermodynamic constants, Ky. Because of co-operative effects in the micelles, the solution is not ideal and the activity of the solute has to be used instead of the molar fraction or concentration. This approach requires the precise definition of standard states. In the articles of Ben-Naim (21) or of Tanford (22), some different standard states are proposed for the solute, i.e. a pure solute or an infinitely dilute state. 

We can also write, on the basis of the definitions of the molar fractions and the fractions of sites ... 

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