Hypothetical mixture, changes

Solubility. Sohd—Hquid equihbrium, or the solubiHty of a chemical compound in a solvent, refers to the amount of solute that can be dissolved at constant temperature, pressure, and system composition in other words, the maximum concentration of the solute in the solvent at static conditions. In a system consisting of a solute and a solvent, specifying system temperature and pressure fixes ah. other intensive variables. In particular, the composition of each of the two phases is fixed, and solubiHty diagrams of the type shown for a hypothetical mixture of R and S in Figure 2 can be constmcted. Such a system is said to form an eutectic, ie, there is a condition at which both R and S crystallize into a soHd phase at a fixed ratio that is identical to their ratio in solution. Consequently, there is no change in the composition of residual Hquor as a result of crystallization. 

Table III. Changes in fraction composition of a hypothetical mixture due to decreasing concentration or decreasing sensitivity.

An ideal liquid solution is a hypothetical mixture of liquids in which there is no special force of attraction between the components of the solution and for which no change in internal energy occurs on mixing. Under these circumstances no change in the character of the liquids is caused by mixing, merely a dilution of one liquid by the other. 

The different units (sites or cells) of the sorbent will be localized in the crystal and, therefore, are distinguishable. If they are independent in the sense that there are no interactions between sorbate particles in different units, we may hypothetically group all units of one kind together in space without changing the sorption behavior. The sorption characteristics would be the same as those of a macroscopic mixture of different solids, each containing sorption units of only one kind (index i) and obeying a sorption isotherm d (p). 

Hypothetically, there are two ways this could happen The stereocenter could be destroyed by some process that totally changes the structure of the compound, or the compound could undergo a chemical process that causes R/S randomization at the stereocenter, leading to a racemic mixture. Let s see which is more likely. Reaction with base (say, hydroxide) removes the acidic proton from the asymmetric carbon and converts the molecule into its conjugate base (Chapter 2) ... 

To obtain a physical interpretation for the residual Gibbs energy, we start with an ideal-gas mixture confined to a closed vessel. As the process, we consider the reversible isothermal-isobaric conversion of the ideal-gas molecules into real ones. Although this process is hypothetical, it is a mathematically well-defined operation in statistical mechanics the process amounts to a "turning on" of intermolecular forces. We first want to obtain an expression for the work, but since the process involves a change in molecular identities, we must start with the general energy balance (3.6.3). For a system with no inlets and no outlets, (3.6.3) becomes... 

To obtain a physical interpretation for the excess Gibbs energy, we consider a Lewis-Randall ideal solution confined to a closed vessel, and we determine the reversible isothermal-isobaric work involved in converting the ideal solution into a real mixture. Again this is a hypothetical process all intermolecular forces are initially the same (but they are nonzero), and the process changes the forces into those of real molecules. 

When we choose a standard state, we are merely identifying a particular ideal solution on which to base an activity coefficient. The standard state may be real or hypothetical, so long as it is weU-defined and so long as a value for its fugacity can be obtained. Ultimately the choice of standard state is made for computational convenience normally this means either that reliable models for y, exist, or else that the value of Yi is close to unity over the states of interest. When neither of these conditions pertain, we should consider changing the standard state. In many situations the appropriate choice is one of the possibilities discussed in 10.2.1-10.2.3 however, when the mole fraction is not a convenient measure of composition, such as occurs for mixtures of electrolytes or of polymers, then other standard states may be preferred. 

For processes that involve phase transformation, the general approach is to use the ideal-solution equation for the enthalpy of the liquid and treat the vapor phase as an ideal-gas mixture. This reduces the problem to a calculation of the enthalpies of pure liquid and pure vapor components. If the calculation involves states near the phase boundary, hypothetical states maybe involved, whose properties must be calculated by extrapolation from known real states. As an example, consider the constant-pressure heating of a solution that contains 30% acetonitrile in nitromethane, at 1 bar. This is shown by the line LVon the Txy graph in Figure 11-1. The enthalpy change for this process is... 

The problem, therefore, is reduced to the calculation of enthalpy changes of pure components. Notice that two of the states involved in this equation are hypothetical pure acetonitrile at the bubble temperature of the solution is a vapor and pure nitromethane is liquid at the dew temperature of the mixture. Accordingly, Hli refers to the enthalpy of a hypothetical liquid at Tbubwe, P, and Hv refers to a hypothetical vapor at Tdew, P. The properties of these hypothetical states may be approximated as those of the corresponding saturated phases Hli is taken to be the enthalpy of the pure saturated liquid at Tbubwe, and Hv as the enthalpy of the pure saturated vapor at Tdew. 

An equilibrium is disturbed when the concentration of one or more of its components is changed. As a result, the concentration of all species will change, and a new equilibrium mixture will be established. Consider the hypothetical equilibrium represented by the equation... 

Figure 9.3(B) depicts the absorption, circular dichroism and optical rotatory dispersion spectra of the mirror image of the hypothetical molecule considered in Figure 9.3(A). Note that vdille the absorption spectra in both cases are identical, the circular dichroism and optical rotatory dispersion spectra have changed signs. If a racemic mixture of this hypothetical compound is subjected to the above experiment, the absorption spectrum would still be identical but the circular dichroism and optical rotatory dispersion spectra would cancel out.

Consider the special case of a mixture containing charged species, for example an aqueous solution of the electrolyte KCl. We could consider the constituents to be either the substances H2O and KCl, or else H2O and the species K+ and Cl . Any mixture we can prepare in the laboratory must remain electrically neutral, or virtually so. Thus, while we are able to independently vary the amounts of H2O and KCl, we cannot in practice independently vary the amounts of K+ and Cl in the mixture. The chemical potential of the K" " ion is defined as the rate at which the Gibbs energy changes with the amount of K+ added at constant T and p while the amount of Cl is kept constant. This is a hypothetical process in which the net charge of the mixture increases. The chemical potential of a ion is therefore a valid but purely theoretical concept. Let A stand for H2O, B for KCl, - - for K+, and — for CP. Then it is theoretically valid to write the total differential of G for the KCl solution either as... 

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