Entropies, partial molal

Heat of Precipitation. Entropy of Solution and Partial Molal Entropy. The Unitary Part of the Entropy. Equilibrium in Proton Transfers. Equilibrium in Any Process. The Unitary Part of a Free Energy Change. The Conventional Standard Free Energy Change. Proton Transfers Involving a Solvent Molecule. The Conventional Standard Free Energy of Solution. The Disparity of a Solution. The E.M.F. of Galvanic Cells. 

Entropy of Solution and Partial Molal Entropy. If the heat of solution of a solute is known, and the free energy of solution is known at some low concentration, then the entropy of solution AS at the same concentration can at once be found from the relation... 

In the case of a sparingly soluble substance, if each of the quantities in (64) is divided by Avogadro s constant, we confirm the statement made above— namely, that, if AS at per ion pair is added to the contribution made to the entropy of the crystal by each ion pair, in this way we evaluate the contribution made by one additional ion pair to the entropy of the saturated solution and it is important to grasp that this contribution depends only on the presence of the additional pair of ions in the solution and does not depend on where they have come from. They might have been introduced into the solution from a vacuum, instead of from the surface of a solid. In (64) the quantities on the right-hand side refer to the solution of a crystal, but the quantity (S2 — Si) does not it denotes merely a change in the entropy of a solution due to the presence of additional ions, which may have come from anywhere. When Si denotes the entropy of a sufficiently large amount of solution, (S2 — Si) is the partial molal entropy of the solute in this solution. 

When solid AgCl is in contact with its saturated aqueous solution, we have found that, if additional ion pairs are transferred from the surface of the crystal to the solution, the total change of entropy is equivalent to 52.8 e.u Since the entropy of the solid is 23.0 e.u., we find that the partial molal entropy of AgCl in its saturated aqueous solution at 25°C is... 

Since the saturated solutions of AgT and AgCl are both very dilute, it is of interest to examine their partial molal entropies, to see whether we can make a comparison between the values of the unitary terms. As mentioned above, the heat of precipitation of silver iodide was found by calorimetric measurement to be 1.16 electron-volts per ion pair, or 26,710 cal/mole. Dividing this by the temperature, we find for the entropy of solution of the crystal in the saturated solution the value... 

The entropy of solid Agl is a little larger than that of AgCl, namely, 27.1 e.u., as compared with 23.0 e.u. Using (64) we find for the partial molal entropy of Agl in its saturated solution the value... 

Conventional Partial Molal Ionic Entropies. Correlation between Ionic Entropy and Viscosity. Conventional Partial Molal Entropy of (H30)+ and (OH)-. The Conventional and the Unitary Entropy of Solution. Solutes in Aaueous Solution. Solutes in Methanol Solution. 

Suppose that we arbitrarily set the partial molal entropy of the K+ ion equal to zero. This means that we assign to the Cl- ion the whole of the partial molal entropy of the ion pair (K+ + Cl-) that is to say, we assign to the Cl- ion, not only the unitary term for the Cl- ion, but also the cratic term for both ions, and also the unitary term for the potassium ion. 

A procedure like this has been adopted in the literature, except that it is the value for the hydrogen ion that has been set equal to zero. This involves a slightly more difficult concept for, when a proton is added to water, it converts an H20 molecule into an (HsO)+ ion. The entropy of the original water molecule is replaced by the entropy of the (HsO)+ ion and its co-sphere. When the partial molal entropy of HC1 in aqueous solution has been determined, the whole is assigned to the Cl- ion that is to say, the value for the hydrogen ion is set equal to zero, and the values for all other species of ions are expressed relative to this zero. 

Fto. 54. Viscosity S-coofHoients from Table 24 plotted against partial molal entropies from Table 25 (on scale of 0.0 for H+). 

Conventional Partial Molal Entropy of (H30)+ and (OH)-. Let us now consider the partial molal entropy for the (1I30)+ ion and the (OH)- ion. If we wish to add an (HsO)+ ion to water, this may be done in two steps we first add an H2O molecule to the liquid, and then add a proton to this molecule. The entropy of liquid water at 25°C is 16.75 cal/deg/mole. This value may be obtained (1) from the low temperature calorimetric data of Giauque and Stout,1 combined with the zero point entropy predicted by Pauling, or (2) from the spectroscopic entropy of steam loss the entropy of vaporization. 2 Values obtained by the two methods agree within 0.01 cal/deg. 

The molar entropy of solution of a crystal—the AS0 of (168)—may be regarded as the difference between (1) the molar entropy of the crystal, and (2) the partial molal entropy of the solute in a solution of a certain Concentration. The question arises In a solution of what concentration Now we notice that (168) may be written in the form... 

Accordingly, the conventional partial molal entropies of ions in solution are often said to refer to the ions in a one-molal solution (m = 1) hot in a real one-molal solution, but in a hypothetical ideal one-molal solution, where the contribution from the interionic forces is taken to be zero, and the cratic term replaces the communal term. 

Table 30. The Unitary Partial Molal Entropy in Aqueous Solution...

Taking the values for the Ag+ ion and the Cl" ion from Table 25, and adjusting the cratic term, find the value of the partial molal entropy of silver chlorido in an aqueous solution having a molality equal to 10 8. 

As mentioned in Sec. 90 we may think of this AS0 as being the difference between the molar entropy of solid 1 j2C03 and the partial molal entropies of one (C03) ion and two Lif ions, each in its hypothetical standard aqueous solution. We propose now to compare the AS of the reaction... 

From Table 44 it will be seen that the molar entropy of solid Li2C03 amounts to 21.60 cal/degree. In Sec. 104 it was pointed out that, if the AS0 for the process (192) is added to the entropy of solid Li2C03, the sum will be equal to the partial molal entropies of the COJ" ion and two Li+ ions. For this sum then we obtain... 

Table 45. Conventional Partial Molal Entropies of Ions in Water at 25°C...

Electrostriotion, 188, 190-192 Energy levels, 34, 151-152 Entropy, of crystals, 95, 180, 211, 267 partial molal (see Partial molal entropy) of solution (see Solution)... 

Frank, H. S. Evans, M. W. (1945). Entropy in binary liquid mixtures partial molal entropy in dilute solutions structure and thermodynamics in aqueous electrolytes. Journal of Chemical Physics, 13, 507-32. 

Ac° estimated from "correspondence principle". This approach, proposed By Criss and Cobble (12), appears to be the most widely used one. It rests on what has been called the entropy correspondence principle which states that the partial molal entropy of an ion at temperature T is related to its partial molal entropy at 298°K by a relation... 

Figure 2. Standard partial molal entropy for NaCl in aqueous solution (experimental data based on Liu and Lindsay, taken from Ref. ) (--) Cp° = constant...

The coefficients a, and b are dependent on temperature and the type of ion (cations, OH- and simple anions, oxyanions and acid oxyanions) and have been established by Criss and Cobble by analyzing available experimental data. SiggCabsolute) is the 298°K entropy on an absolute sca e corresponding to the partial molal entropy of H (aq) as -5.0 cal deg" mole-1 at 298°K. The absolute and conventional partial molal entropy for any ion are related by... 

Heat Capacity and Standard Partial Molal Entropy... 

Combining the solvation and nonsolvation contributions, the partial molal entropy change between reference state conditions P and 7) and the P and T of interest is given by... 

The Born coefficients of the HKF model for the various organic species are simply related to their standard partial molal entropies at 25 °C and 1 bar through the effective charge (Shock et al., 1989). 

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 ... 

This equation gives the variation of the chemical potential of a constituent in a mixture with respect to temperature at constant pressure and composition of the system. S i is the partial molal entropy of zth component of the mixture. 

If we consider the partial molal entropy of solution with respect to the solute, Equation 2.1 becomes... 

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