Salting-in and Salting-out

Since activity is a measure of the effective concentration of solute in a solution, it can be used to describe solubilities of slightly soluble solutes in ionic solutions. In other words, we can investigate the effects of ionic strength on solubility.

For example, since copper (II) sulfide is only somewhat soluble in water, we can write its dissolution as an equilibrium:

CuS(s) ↔ Cu²⁺(aq) + S^2-(aq)

The thermodynamically correct equilibrium constant is then

K_sp=a_Cua_S=γ_Cuγ_Sm_Cum_S=γ_±²K_sp^obs

The parameter K^obs is the observed (or apparent) equilibrium constant and will be the one that is measured by standard measurements like titration. Since this is a 1:1 ionic compound, K_sp is related to S, the molal solubility, by:

K_sp = S² and K_sp^obs = (S^obs)²

Combining the two equations above, we can solve for the ratio of solubilities:

S/S^obs = γ_±

Therefore, for mean activity coefficients less than one, we see an increase in solubility with respect to water [salting-in] and a corresponding solubility decrease when γ_± is greater than one [salting-out].

Debye-Hückel Theory

One of the problems that arises when treating the theory of ionic solutions is how exactly to deal with the activities of cations versus anions when both are always found together (all solutions are neutral). Recognizing this fact, we can define a mean activity coefficient in terms of the individual ion activity coefficients:

γ_± = [γ₊^ν+γ₊^ν-]^1/ν

But how do we obtain activity coefficients without going through tedious cryoscopic or electrochemical measurements? This is one of the results that the Debye-Hückel theory seeks to uncover. Before going through the details, we need a convenient way to describe how much a given solution can conduct electricity: the ionic strength I.

I = 1/2 Σ z_i² (m_i/m°) = 1/2 Σ z_i² m_i

Ionic strength is unitless so the last term above is basically a unitless molality.

Debye and Hückel proposed a simple model that they hoped would capture most of the behavior of ionic solutions. The assumptions:

1. an ionic atmosphere is established with many ions with valence z and radius a
2. the solvent is structureless with electric permittivity ε
3. the solution is dilute.

Several results were obtained via mind-bleeding statistical mechanics but among them were the Debye length τ_D, which is a measure of the effective range that ionic charges interact in this solution.

Another important result is the extended Debye-Hückel law:

lnγ_± = - |z₊z_-|αI^1/2/(1 + βaI^1/2)

which can also be written in ion activity form:

lnγ_i = -z_i²|αI^1/2/(1 + βaI^1/2)

As mentioned in class, this formula works quite well for dilute to moderately concentrated solutions (and is fairly bad past 0.25 m). For dilute solutions we can consider the limiting Debye-Hückel law:

lnγ_± = - |z₊z_-|αI^1/2

Since α=1.171 for water at 25°C, we can transmogrify any of these formulas to log₁₀ world. For example, the above equation becomes

logγ_± = -0.509 |z₊z_-|I^1/2 (aqueous solutions, 25°C)

Bjerrum, Mayer and Davies extended the Debye-Hückel formula so that it works for a wide range of ionic strengths (and remains the primary equation used in pchem today):

logγ_± = -0.509 |z₊z_-|(I^1/2/1+I^1/2 - 0.30I) (aqueous solutions, 25°C)

At different temperatures, new values for the Debye-Hückel parameter (the 0.509 value) would need to be calculated through α.

Tomorrow, salting-in and salting-out.

hw.1 corrections

For question 1, I mistakenly chose the chemical potential that was the lowest, not the highest.

Also, for question 12, the hydrogens on each benzoic acid are missing.

Activity Gets Promoted

Before grappling with the chemical potentials of ionic solutions, we took a moment to update the colligative property formulas for ionic solutes. Since these properties depend not on what is dissolved in solution but how much, the fact that salts break into ions needs to be addressed, which is accomplished by i, the van't Hoff factor:

• freezing point depression: ∆T_f=iK_fm₂
• boiling point elevation: ∆T_b=iK_bm₂
• osmotic pressure: Π=iM₂RT

where i = (actual number of free particles)/(number before dissociation).

For very dilute solutions, i is approximately equal to the number of ions in the formula (i=2 for KCl, i=3 for Na₂SO₄, etc.). As the concentration increases, the actual van't Hoff factor will be less (sometimes very much so) than these stoichiometrically derived values. We also saw that we could relate i to the degree of dissociation α by the formula α=(i-1)/(ν-1).

Although earlier we defined the chemical potential of a solution in terms of the Raoult's standard state [purity and mole fraction], it is more convenient to use a standard state based on molality:

For a solute in an ideal solution: μ_i = μ°_i + RT lnm_i
For a solute in a real solution: μ_i = μ°_i + RT lna_i

Henceforth the symbol ° will mean 1 bar for gases and 1 m for solutions. Since we now have six different chemical potential expressions [ideal gas, real gas, ideal solution, real solution, liquid, solid] it is customary to promote the idea of activity so that it covers all possibilities. We will then use a general expression μ_i = μ°_i + RT lna_i with the following mappings:

• ideal gas: a_i = P_i [partial pressure]
  
• real gas: a_i = f_i [fugacity]
  
• ideal soln: a_i = m_i [molality]
  
• real soln: a_i = a_i [activity]
  
• liquid: a_i = 1 [assuming normal pressure]
  
• solid: a_i = 1 [assuming normal pressure]

Now, for any reversible chemical reaction aA( )+bB( ) ↔ cC( )+dD( ) we can construct reaction quotient and equilibrium constant expressions immediately in terms of the relevant activities.

Tomorrow we will go through the powerful Debye-Hückel theory, which gives considerable insight into ionic solutions, including a method to calculate activity coefficients. Also, next hw set..

The Born Model of Solvation

Since ionic compounds, when dissolving, produce a cationic and anionic pair, the thermodynamics of ionic solutions becomes complicated in that values for a single ion are unknowable. Conventional values are obtained by assigning ∆_fG°[H⁺]=0 for all temperatures and then calculating every other ion value with respect to that reference. This process works well in Hess' Law-type calculations as long as we don't assign meaning to individual ion values (particularly true for negative S_f° values).

Though ultimately less successful, we can gain a lot of insight into the solvation process by attempting to derive the thermodynamic values from scratch, that is, from first principles. In the Born model, we imagine a hard-sphere of radius r being charged up from 0 to a final value q. The difference between the work of charging this ion in the medium of choice and a vacuum will be taken as ∆_solvG_i° for ion i.

Recalling that the infinitesimal work of charging is dw=φdq and the electric potential is φ=q/4πεr, we obtain for 1 mole of ions, after integrating and substituting q=ze:

∆_solvG_i°=z²e²N_A/8πε_or(1/ε_r-1)

Since the dielectric constant is greater than 1 for all materials, the solvation of any ion is a spontaneous process in any solvent.

Focusing on 298K water as the solvent leads us to gibbs energy of hydration:

∆_hydG°=-68.6z²/r

A test of the Born model can be obtained by plotting experimental ∆_hydG° values versus z²/r for a bunch of ions and see whether or they not they track along a line of slope –68.6. As seen in class, most univalent ions generally agree with the Born results but polyvalent ions deviate appreciably, especially as the valence z gets large.

This is a good example of how physical chemistry works: Propose a model that is as simple as possible and see how much of the chemical behavior is captured. When it fails for some cases, go back and examine the assumptions (and tweak and repeat). In this model, the assumption of the aqueous solvent being structureless (that is, a continuum) is likely the culprit. Indeed, we saw that many of the high-valence ions seem to be tracking along a line in which the dielectric constant was approximately 2 (indicative of a case in which the mobility of water has been decreased to the point that it acts effectively as a nonpolar substance). An additional deviation from the Born model, not discussed much in class, arises from ions that are highly polarizable and would therefore not be adequately treated as a hard-sphere.

Many models exists treating the solvent as a molecular entity exist and experiments and simulations support the theory that an ion in solution produces a hydration sphere encompassing 4-8 water molecules (the actual number being a function of ionic size). Indeed, femtosecond laser studies provide hydration lifetimes for water molecules in the presence of a particular ion. Some ions with high charge densities have very long lifetimes, supporting the idea that water in those cases has been effectively locked immobilized.

Though the underlying explanation has not been fully developed, we can experimentally verify that some ions in solution increase the viscosity when compared to water. These kosmotropic ["structure-making"] ions include Li⁺, Na⁺ and F^-. Analogously, viscosity is decreased by chaotropic ["structure-breaking"] ions like Rb⁺, Cs⁺ and NO₃^-.

On Wednesday, a brief return to colligative properties (to adjust them for ionic solutions) before tackling chemical potentials of ionic solutions, activity and the Debye-Huckel theory.

Chemical Potential Leads to More Goodies

Colligative properties, those that depend not on the identity of a solute but only on how much is dissolved, had been known experimentally for years before an underlying theory was established. We typically discuss four such properties of solutions (and are compared to the corresponding pure solvent): vapor pressure lowering, freezing point depression, boiling point elevation and osmotic pressure.

It should be apparent from Monday's lecture that vapor pressure lowering for ideal solutions is an obvious consequence of Raoult's Law, although all solutions experience the phenomenon (which is due to the fact that the chemical potential is lower for a solution than it is for the solvent: μ₁(soln)=μ*₁(l)+RTlnX₁). But all four of these phenomena are explained by setting the appropriate chemical potentials equal (where we designate 1 as the solvent and 2 as the solute):

vp lowering: μ₁(soln) = μ₁(g)
fp depression: μ₁(soln) = μ₁(s)
bp elevation: μ₁(soln) = μ₁(g)
osmotic pressure: μ₁(l) = μ₁(soln) + ΠV_m

Using some math elegance with a few key assumptions (like ∆_transH being relatively constant and the solution dilute), we are able to obtain the experimentally obtained formulae familiar to students of general chemistry:

bp elevation: ∆T_b=K_bm where [ebullioscopic constant] K_b=RM₁T_b²/∆_vapH
fp elevation: ∆T_f=K_fm where [cryoscopic constant] K_f=RM₁T_f²/∆_fusH
osmotic pressure: Π=MRT

Since we have thus far dealt only with molecular [nonionic] solutions, we needed to refresh some concepts from basic electrostatics [coulomb's law, electric permittivity, dielectric constants] before moving onto ionic solutions, which can be radically different (and never truly ideal).

Ideal Solutions, Raoult's Law and Azeotropes

We begin pchem II by examining the behavior of solutions. In Chapter 9 we briefly study nonelelectrolyte solutions and then, in Chapter 10, focus on the more common ionic solutions and activity.

Ideal solutions are those solutions in which components A and B attract each other to the same degree that they attract themselves and would thus posses zero values for ∆_mixV and ∆_mixH. Examples of solutions that approximate ideality are hexane:heptane and CCl₄:SiCl₄ (approximately the same structure and intermolecular forces).

Though originally obtained experimentally, Raoult's Law can be derived from chemical potentials and describes how the vapor pressure of a solution compares to that of the pure solvent: P_i=X_iP*_i. Since this relationship only works for ideal solutions, some have argued that it could just as easily be used as a second definition of what constitutes ideality in a solution.

In a binary ideal solution, if both components are volatile they will both contribute to the overall vapor pressure: P=X₁P*₁+X₂P*₂. Using the fact that mole fractions add up to 1, it is easy to obtain the equation P=P*₂+(P*₁-P*₂)X₁. Clearly, if we were to plot the vapor pressure versus composition, we obtain a line connecting the pure vapor pressures (which represents the liquid-vapor coexistence curve for this P-X phase diagram).

With a little bit of work, we also obtain an expression for the total vapor pressure in terms of Y_i, the vapor-phase mole fraction, whose plot is squashed-hyperbolic and also shown above. Both are typically superimposed together to form a P-Z (pressure-composition) diagram.

A generally more useful diagram however is one that plots the boiling temperature rather than the vapor pressure on the vertical axis. Such plots are useful to explain the physical chemistry of simple and fractional distillation. For example, this T-Z diagram shows the analysis of an ideal benzene:toluene solution in which X_benzene=0.20 initially. As the solution is slowly heated, the temperature eventually reaches the coexistence curve at about 375°C. The corresponding vapor-phase mole fraction is found by drawing a straight line across to the other coexistence curve (which represents Y values): Here we find a value of 0.36 which is clearly richer in benzene than the original solution. If we were to simply remove this vapor and condense it, we'd get a solution in which X_benzene=0.36 and we could start the process again. By using a Vigreaux condensor, however, we can do it all in one process.

When solutions possess considerable nonideality, kinks in the T-Z diagram can occur, leading to azeotropes (solutions that boil at constant temperature). Positive deviations from Raoult's Law lead to minimum boiling point azeotropes while negative deviations lead to maximum boiling point azeotropes. Azeotropic pairs, like the famous ethanol:water solution, can not be completely separated by distillation methods.