Dilution, free energy

When large differences in the CEDs of the components occur, x exceeds the critical value and the two components segregate into two macroscopic phases. Values of x < 5 characterize instead thermodynamically good solvents in which polymer segments experience chain expansion and solvation. The term j — x describes the excess mixing (dilution) free energy, which attains pseudo ideal values when [48]... 

To proceed fiirther, to evaluate the standard free energy AG , we need infonnation (experimental or theoretical) about the particular reaction. One source of infonnation is the equilibrium constant for a chemical reaction involving gases. Previous sections have shown how the chemical potential for a species in a gaseous mixture or in a dilute solution (and the corresponding activities) can be defined and measured. Thus, if one can detennine (by some kind of analysis)... 

Thus, if the activities of the various species can be detennined or if one can extrapolate to infinite dilution, the measurement of the emf yields the standard free energy of the reaction. 

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

II The increment in the free energy, AF, in the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are for a gas, fugacity (approximately equal to the pressure) of 1 atm for a pure liquid or solid, the substance at a pressure of 1 atm for a substance in aqueous solution, the hyj)othetical solution of unit molahty, which has all the properties of the infinitely dilute solution except the property of concentration. 

The process described above is usually called osmosis and this usually imphes a flow of fluid in one direction or the other. If the permeating species, usually called the solvent, flows from the pure compartment to the mixture compartment then it is called osmosis pure and simple. This seems the natural process since the solvent dilutes the solution and this involves an increase in entropy and/or a decrease in free energy, so the resultant flow is spontaneous and the system tends to equihbrium. However, the starting conditions may be such that the difference of pressure... 

Solvation can be studied by thermodynamic methods, often combined with ex-trathermodynamic assumptions so as to express results for individual ions (rather than for neutral electrolytes). The solvation energy is the free energy change upon transferring a molecule or ion from the gas phase into a solvent at infinite dilution. This sometimes can be obtained from a consideration of the following processes, written for a 1 1 electrolyte ... 

This value is in excellent agreement with the calculated free energy value by the consideration of various nonbonded interactions in the epimers (69 and 70) [(2 X 1,3-diaxial Me—H interaction) — (1 x 1,3-diaxial Me—H interaction + 1 X l,2- < Me—H interaction) = (0.9 x 2) — (0.9 + 0.6) = 0.3 kcal/mole]. Hydrolysis of the enamine with dilute acetic acid gave a 3 2 mixture of cis and trans isomers of the ketone, thus confirming the assignments made to the enamine components. 

We next consider the synthesis and chemical reactions of the oxides of chlorine. Because the compounds are strongly endothermic and have large positive free energies of formation it is not possible to prepare them by direct reaction of CI2 and O2. Dichlorine monoxide, CI2O, is best obtained by treating freshly prepared yellow HgO and CI2 gas (diluted with dry air or by dissolution in CCI4) ... 

At finite temperature the chemical potentials can be calculated as follows. In the dilute solution approximation, the Gibbs free energy is given by ... 

The critical hydrogen content for the ductility loss increased with increasing hydrogen solubility in the alloy. The fracture surfaces were not characteristic of those found under conditions of SCC. In terms of hydrogen and deuterium solubility in a similar series of bcc alloys, the equilibrium constants were determined at infinite dilution as a function of temperature The free energy function was expressed in terms of the bound-proton model. 

Consider next the process depicted in Fig. 10. If an ionic crystal is in contact with a dilute solution, and we take q additional ion pairs into the solution, there will be a change in the cratic term, and at the same lime the change in the free energy AF will receive the contribution qL, that is to say, a contribution consisting of q units each equal to L. 

We may say then that in each of those processes the change AF in the free energy consists of two parts, a unitary part and a communal part. When an ionic solution is not extremely dilute, the free energy of the solution receives a contribution from the interionic forces this quantity depends on the concentration of the solute and is a communal quantity. When, however, the solution is extremely dilute, the interionic contribu-... 

The Disparity of a Solution. We may begin to use the word disparity in a technical sense, for the quantity defined above, and to speak of d as the disparity of the solution when the mole fraction of the solute is x. In dilute ionic solutions the sign of d is always negative. The effect of the interionic forces is that ions added to a dilute solution always lose more free energy than they would when added to the corresponding ideal solution hence the total communal term is less than the cratic term. 

The same remarks can be made about each part of the c.m.f., separately. The unitary part of the e.m.f., expressed in volts, is numerically equal to the unitary change in the free energy, expressed in electron-volts per ion pair. At the same time, the cratic term in the e.m.f., expressed in volts, is numerically equal to the cratic change in the free energy, expressed in electron-volts per ion pair. A similar statement can be made about the interionic part but we are usually interested in the value of the e.m.f., extrapolated to extreme dilution, where this part is negligibly small. 

The heat of solution tends at extreme dilution to the value +4207 cal/mole. Calculate the conventional free energy of solution at 25°C and the conventional entropy of solution. 

We shall be interested in pairs of cells, in which the mole fraction of the solute in one solvent is equal to its mole fraction in the other solvent. Suppose then that a series of such pairs of cells is made up, with the solute at progressively greater dilutions. When the members of these pairs of cells are placed back to back, the resultant e.m.f. s will contain progressively smaller contributions from the intcrionic forces and, on extrapolation to extreme dilution, this contribution will be negligibly small. Since the mole fraction on each side is the same, the difference between the eratic terms will be zero. In any such scries of cells, the measured e.m.f. s when extrapolated to extreme dilution thus yield the unitary part of the change in free energy. 

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