Mole number

In Equation (11), V is the total volume containing n moles of component i, n moles of component j, etc. The differentiation is carried out such that, in addition to temperature and pressure, all mole numbers (except n ) are held constant. 

Because mole numbers are additive, it follows that the product A. A V will be an additive quantity provided that a for each species remains constant during the course of the reaction. This last condition implies, essentially, that the effective dipole moments and hence the orientation of each species remain constant, which is most likely to be the case at constant film pressure. Then... 

Here, denotes the total number of moles associated with the adsorbed layer, and N and are the respective mole fractions in that layer and in solution at equilibrium. As before, it is assumed, for convenience, that mole numbers refer to that amount of system associated with one gram of adsorbent. Equation XI-24 may be written... 

Consider a closed, nonreacting PTT system containing n moles of a homogeneous fluid mixture. The mole numbers of the individual chemical species sum to... 

The subscript indicates that all mole numbers are held constant. Comparison with equation 46 shows that... 

All terms in this equation have the units of moles moreover, in contrast to equation 60, the enthalpy rather than the entropy appears on the right-hand side. Equation 167 is a general relation expressing G/RT as a function of all of its coordinates, T, P, and mole numbers. Because of its generaUty, equation 167 may be written for the special case of an ideal gas ... 

If the T and P of a multiphase system are constant, then the quantities capable of change are the iadividual mole numbers of the various chemical species / ia the various phases p. In the absence of chemical reactions, which is assumed here, the may change only by iaterphase mass transfer, and not (because the system is closed) by the transfer of matter across the boundaries of the system. Hence, for phase equUibrium ia a TT-phase system, equation 212 is subject to a set of material balance constraints ... 

Equation 215 asserts that only N of the 2N mole numbers are independently variable. If the independent mole numbers are chosen as it follows from equation 212 that a set of Aiconditions on at the equiUbrium state can be written as follows, where i = 1,2, - , N ... 

The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reactionj ... 

Now the total change in mole number is just the sum of the changes resulting from the various reactions. Thus, by equation 233,... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

The mole-number derivative of nB is given by Eq. (4-195) the corresponding derivative of nC, similarly found from Eq. (4-184), is... 

Methanol synthesis will be used many times as an example to explain some concepts, largely because the stoichiometry of methanol synthesis is simple. The physical properties of all compounds are well known, details of many competing technologies have been published and methanol is an important industrial chemical. In addition to its relative simplicity, methanol synthesis offers an opportunity to show how to handle reversible reactions, the change in mole numbers, removal of reaction heat, and other engineering problems. 

If the reaetion mixture is large enough that the mole numbers eorresponding to the stoiehiometrie numbers reaet, then the eomposi-tions remain unehanged. If these mole numbers reaet at equilibrium, then the overall ehange in Gibbs funetion is... 

Number of Grams Number of Moles Number of Molecules Number of O Atoms... 

Since there is a decrease of 0.03035 gram/mole number of nucleons in the nucleus. This calcula-of helium formed in this reaction, an equivalent tion provides us with the binding energy per... 

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... 

Equation (5.5) is an extended form of the fourth Gibbs equation that applies when the mole numbers become variables. Under the condition of constant T and p, it becomes... 

The same result can be obtained from an application of Euler s theorem, explained in more detail in Appendix 1. The thermodynamic quantities, Z, are homogeneous functions of degree one with respect to mole numbers.c At constant T and p, one can use Euler s theorem to write an expression for Z in terms of the mole numbers and the derivatives of Z with respect to the mole numbers. The result isd... 

Mass-mole-number calculations often involve atoms yvithin a compound as well as the compound itself. The chemical formula provides the link between moles of a compound and the number of moles of the compound s individual elements ... 

Example 3-12 illustrates mass-mole-number conversions involving both elements and compounds. 

These diagrams are established by drawing the boundary lines representing the geometrical locus where the mole numbers of two adjacent species are equal, with n being defined as ... 

There are three possibilities of letting the neutral species appear in the ionic partition diagram. Indeed, as the partition coefficient of the neutral species is neither potential- nor pH-dependent, the variable can be either the neutral species in the aqueous phase or that in the organic phase. However, to account for the experimental reality, it is more convenient to introduce the total mole number of neutral species [297], defined as ... 

LSER linear solvation energy relationship MLP molecular lipophilicity potential n mole number N neutral species... 

This equation is valid for all species Ah a fact that is a consequence of the law of definite proportions. The molar extent of reaction is a time-dependent extensive variable that is measured in moles. It is a useful measure of the progress of the reaction because it is not tied to any particular species A. Changes in the mole numbers of two species j and k can be related to one another by eliminating between two expressions that may be derived from equation 1.1.4. 

The variable / depends on the particular species chosen as a reference substance. In general, the initial mole numbers of the reactants do not constitute simple stoichiometric ratios, and the number of moles of product that may be formed is limited by the amount of one of the reactants present in the system. If the extent of reaction is not limited by thermodynamic equilibrium constraints, this limiting reagent is the one that determines the maximum possible value of the extent of reaction ( max). We should refer our fractional conversions to this stoichiometrically limiting reactant if / is to lie between zero and unity. Consequently, the treatment used in subsequent chapters will define fractional conversions in terms of the limiting reactant. 

This equation is extremely useful for calculating the equilibrium composition of the reaction mixture. The mole numbers of the various species at equilibrium may be related to their values at time zero using the extent of reaction. When these values are substituted into equation 2.6.9, one has a single equation in a single unknown, the equilibrium extent of reaction. This technique is utilized in Illustration 2.1. If more than one independent reaction is occurring in a given system, one needs as many equations of the form of equation 2.6.9 as there are independent reactions. These equations are then written in terms of the various extents of reaction to obtain a set of independent equations equal to the number of unknowns. Such a system is considered in Illustration 2.2. 

From the stoichiometry of the reaction it fs possible to determine the mole numbers of the various species in terms of the extent of reaction and their initial mole numbers. 

The first step in the analysis is to determine if the chemical equations A to C are independent by applying the test described above. When one does this one finds that only two of the reactions are independent. We will choose the first two for use in subsequent calculations. Let the variables a and B represent the equilibrium degrees of advancement of reactions A and B, respectively. A mole table indicating the mole numbers of the various species present at equilibrium may be prepared using the following form of equation 1.1.6. 

Species Initial mole number Equilibrium mole number... 

The fact that none of the mole numbers can ever be negative places maximum values of 1/2 on a and 1 on B. 

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