Mixing, entropy, gases

The apolar contribution to AS0, ASap, is better characterized than AHap. The value of Tt has been shown to be a universal temperature for all processes involving the transfer of an apolar surface into water and has a value of 112°C (Murphy et al., 1990). At this temperature the AS0 of transfer, ASf, represents the mixing entropy of the process. The universal value of Tt was determined using mole fraction concentration units, so that the liquid transfer ASf takes on a value of zero. The value of Tt remains the same using the local standard state of Ben-Naim (i.e., molar concentration units) (Ben-Naim, 1978), but the value of Ais increased by R ln(55.5), where R is the gas constant and 55.5 is the molarity of water. 

Just because this is the simple mixture, the partial entropy s may be interpreted as specific entropy of pure (ideal) gas at a density equal to those in the mixture (see (4.426) and below), and the mixing entropy may be calculated as the sum of entropy changes at the expansion of pure (ideal) gases a (with masses Wa) from starting density (before mixing) to final density (as in the mixture). 

The mixing entropy in an ideal mixture is therefore the same as in an ideal gas mixture (4.435) but it is valid more generally, e.g. in the liquid the ideal mixture is formed from liquid pure constituents. 

In this equation Gp and Gq are the Gibbs energies of the two components P and Q and Xp and Xq are the concentrations of the components in the mixture. This equation can be rewritten by expressing the mixing term in the mixing enthalpy (zero in ideal mixtures) and the mixing entropy (G = H - TS is used). For the entropy its expression in terms of the concentration x in an ideal gas mixture is substituted in the equation. For the present purpose this is an acceptable approximation. The atomic concentration Xp is called x and for a binary system the... 

Here A Ui is a residual (redundant) chemieal potential, ASi and ASi are redundant combinatorial and non-combinatorial mixing entropies eorrespondingly, AHi is a redundant mixing entropy, R is the universal gas constant, Ui and U2 are volume fraetions of components. 

For example, the expansion of a gas requires the release of a pm holding a piston in place or the opening of a stopcock, while a chemical reaction can be initiated by mixing the reactants or by adding a catalyst. One often finds statements that at equilibrium in an isolated system (constant U, V, n), the entropy is maximized . Wliat does this mean ... 

Also of importance is the effect of temperature on the gas solubility. From this information it is possible to determine the enthalpy and entropy change experienced by the gas when it changes from the ideal gas state (/z and ) to the mixed liquid state ( andT,). 

Figure 2.13 Mixing of ideal gas A with ideal gas B at constant temperature and constant total pressure. The entropy change AS is given by equation (2.78).

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

In crystals for which n0 is large, such as iodine, the lowest symmetric and the lowest antisymmetric state have practically the same energy and properties, and each corresponds to one eigenfunction only. As a result a mixture of symmetric and antisymmetric molecules at low temperatures will behave as a perfect solid solution, each molecule having just its spin quantum weight, and the entropy of the solid will be the translational entropy plus the same entropy of mixing and spin entropy as that of the gas. This has been verified for I2 by Giauque.17 Only at extremely low temperatures will these entropy quantities be lost. 

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

Cantor and SchimmeP provide a lucid description of the thermodynamics of the hydrophobic effect, and they stress the importance of considering both the unitary and cratic contributions to the partial molal entropy of solute-solvent interactions. Briefly, the partial molal entropy (5a) is the sum of the unitary contribution (5a ) which takes into account the characteristics of solute A and its interactions with water) and the cratic term (-R In Ca, where R is the universal gas constant and ( a is the mole fraction of component A) which is a statistical term resulting from the mixing of component A with solvent molecules. The unitary change in entropy 5a ... 

Here is Boltzmann s constant, or the gas constant per molecule, R/Nq, where No is Avogadro s number (or Na + Nb for one mole of solution) and va and are the volume fractions of solvent and polymer. Comparison of Eq. (2.76) with the entropy of mixing presented earlier in the chapter by Eq. (2.30) shows that they are similar in form, except that now the volume fractions of the components, va and vg, are found to be the most convenient way of expressing the entropy change for polymers, rather than the mole fraction used for most small molecules. This change arises from the differences in size between the large polymer molecules and the small solvent molecules which would normally mean mole fractions close to unity for the solvent, especially when dilute solutions are being studied. 

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