Number of Moles

Conversion of a reactant is the number of moles converted per initial or feed moles of a reactant. Thus for component A 

A limiting reactant is a reactant whose concentration at the start of the reaction is the least of all reactants relative to the required stoichiometric amount needed for complete conversion. For instance, for the single reaction (7-12), A is the limiting reactant if the initial molar ratio of concentrations of A and B is less than the ratio of their stoichiometric coefficients  

Once the limiting reactant is depleted, the respective reaction stops even though other (nonlimiting) reactants may still be abundant. 

For eacn reaction or each step in a network of reactions, a unique extent of reaction t that relates the composition of components that participate in the reaction to each other can be defined. For instance, for the single reaction (7-1)  

The equilibrium constant (based on volumetric concentrations) is defined as the ratio of the forward and reverse rate constants and is related to the composition at equilibrium as follows  

When A is the limiting reactant as in Eq. (7-21), the maximum extent of reaction (with A fully converted) is 

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The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

This reaction brings about an increase in the number of moles per... 

If inert material is to be added, then ease of separation is an important consideration. For example, steam is added as an inert to hydrocarbon cracking reactions and is an attractive material in this respect because it is easily separated from the hydrocarbon components by condensation. If the reaction does not involve any change in the number of moles, inert material has no effect on equilibrium conversion. 

If the secondary reaction is reversible and involves a decrease in the number of moles, such as... 

If the byproduct reaction is reversible and inerts are present, then changing the concentration of inerts if there is a change in the number of moles should be considered, as discussed above. Whether or not there is a change in the number of moles, recycling byproducts can suppress their formation if the bj iroduct-forming reaction is reversible. An example is in the production of ethylbenzene from benzene and ethylene ... 

The selection of reactor pressure for vapor-phase reversible reactions depends on whether there is a decrease or increase in the number of moles and whether there is a system of single or multiple reactions. 

This is an endothermic reaction accompanied by an increase in the number of moles. High conversion is favored by high temperature and low pressure. The reduction in pressure is achieved in practice by the use of superheated steam as a diluent and by operating the reactor below atmospheric pressure. The steam in this case fulfills a dual purpose by also providing heat for the reaction. 

Inerts concentration. The reaction might be carried out in the presence of an inert material. This could be a solvent in a liquid-phase reaction or an inert gas in a gas-phase reaction. Figure 2.96 shows that if the reaction involves an increase in the number of moles, then adding inert material will increase equilibrium conversion. On the other hand, if the reaction involves a decrease in the number of moles, then inert concentration should be decreased (see Fig. 2.96). If there is no change in the number of moles during reaction, then inert material has no effect on equilibrium conversion. 

The relationship required is the gas expansion factor (E), and is defined for a given quantity (mass or number of moles) of gas as... 

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus... 

It must also be realized that this thin surface region is in a very turbulent state. Since the liquid is in equilibrium with its vapor, then, clearly, there is a two-way and balanced traffic of molecules hitting and condensing on the surface from the vapor phase and of molecules evaporating from the surface into the vapor phase. From the gas kinetic theory, the number of moles striking 1 cm of surface per second is... 

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... 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

If tlie arbitrary constant C is set equal to nRy where n is the number of moles in the system and R is the gas constant per mole, then the themiodynamic temperature T = 9j where 9j is the temperature measured by the ideal-gas themiometer depending on the equation of state... 

Consider two ideal-gas subsystems a and (3 coupled by a movable diatliemiic wall (piston) as shown in figure A2.1.5. The wall is held in place at a fixed position / by a stop (pin) that can be removed then the wall is free to move to a new position / . The total system (a -t P) is adiabatically enclosed, indeed isolated q = w = 0), so the total energy, volume and number of moles are fixed. 

The paradox involved here ean be made more understandable by introdueing the eoneept of entropy ereation. Unlike the energy, the volume or the number of moles, the entropy is not eonserved. The entropy of a system (in the example, subsystems a or P) may ehange in two ways first, by the transport of entropy aeross the boundary (in this ease, from a to P or vice versa) when energy is transferred in the fomi of heat, and seeond. 

We now turn to a new kind of boundary for a system, a wall penneable to matter. Molecules that pass tlirough a wall carry energy with them, so equation (A2.1.15) must be generalized to include the change of the energy with a change in the number of moles dn ... 

Two subsystems a. and p, in each of which the potentials T,p, and all the p-s are unifonn, are pennitted to interact and come to equilibrium. At equilibrium all infinitesimal processes are reversible, so for the overall system (a + P), which may be regarded as isolated, the quantities conserved include not only energy, volume and numbers of moles, but also entropy, i.e. there is no entropy creation in a system at equilibrium. One now... 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

The expression hU/i n J.s>, n signifies, by common convention, the partial derivative of U with respect to the number of moles n- of a particular species, holdmg. S, V and the number of moles n.j of all other species (/ )... 

In passing one should note that the metliod of expressing the chemical potential is arbitrary. The amount of matter of species in this article, as in most tliemiodynamics books, is expressed by the number of moles nit can, however, be expressed equally well by the number of molecules N. (convenient in statistical mechanics) or by the mass m- (Gibbs original treatment). 

If a themiodynamic system includes species that may undergo chemical reactions, one must allow for the fact that, even in a closed system, the number of moles of a particular species can change. If a chemical reaction (e.g. N2 + 3H2 INHg) is represented by the symbolic equation... 

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