Solvation of ionic compounds in water: basic theory and examples

::Topics::

1. Gibbs Free Energy of dissolution

1.1. Entropy of dissolution
1.2. Enthalpy of dissolution

2. Solubility limits and Solubility products

2.1. Hydrolysis and pH effects
2.2. Common ion effect
2.3. Ionic strength effect
2.4. Cation/anion size effects

Coming soon. Suggestions welcome!


[Edited on 11-2-2016 by blogfast25]

1. Gibbs Free Energy of dissolution.

Dissolution of a substance (the solute) into a solvent is a process involving a change of state:

$$\text{solute}+\text{solvent} \to \text{solution}$$


In this change of state, consider the left hand side as State 1 and the right hand side as State 2.

Thermodynamics dictates that for this change of state to be thermodynamically favourable the change in Gibbs Free Energy has to be negative:

$$\Delta G = G_2-G_1<0$$

And with:

$$\Delta G =\Delta H-T\Delta S$$

And:

$$\Delta H=H_2-H_1, \Delta S=S_2-S_1$$

H, S and T are Enthalpy, Entropy and absolute temperature respectively.

Note that a negative predicted ΔG does not imply spontaneity or speed. Add some sugar to your tea and don’t stir. The sugar will basically not dissolve at all (or rather, it would take almost eternity to fully dissolve), until we give the tea a quick stir and the sugar then dissolves very swiftly.

1.1. Entropy of dissolution:

On the concept of Entropy entire libraries have been dedicated and I’m not about to add more to it. Instead let’s apply a more narrow understanding of it to the process of dissolution.

In the schematic below, 8 molecules of solute (black balls) and 8 molecules of solvent (white squares) are in a container, in various degrees of mixing.



State 1: complete separation between solute and solvent (think sugar at the bottom of that teacup, unstirred). It would be a mistake to think that only one such arrangement exists. Not at all: if we label the molecules we can create many swaps that amount to the same arrangement. There are in fact 1225 permutations of the same arrangement.

State 2: a mere 2 out of 8 solute molecules have strayed from their initial positions. The number of permutations possible is now already dramatically higher: about 5600.

State 3: only 3 out of 8 solute molecules are in their original positions: the number of permutations possible is now about 10000.

It’s obvious from this that there are far more disordered arrangements than ordered ones and that disordered ones have far higher numbers of possible permutations.

In short, disordered systems are far more probable than ordered ones. We say they have higher Entropy.

Two important conclusions can be drawn here:

a. Dissolution always increases entropy of the system:

$$\Delta S >0$$

And since as:

$$\Delta G=\Delta H-T\Delta S<0$$

... the Entropy increase always favours dissolution, as T is always positive.

b. Dissolution is essentially irreversible: in a solution left unchanged the solute will not at some time magically rearrange at the bottom of the container because that is simply too improbable to happen.

[Edited on 12-2-2016 by blogfast25]

1.2. Enthalpy of dissolution:

For dissolution of ionic compounds to happen several Enthalpies come into play.

a. Lattice energy:

A solid ionic compound is held together by the Coulombic attractions between cations and anions. To change state from ionic lattice + solvent to solution these Coulombic attractions (SM thread) have to be overcome and that costs energy. Bar any other Enthalpic effects, dissolution would always be endothermic (and frequently it is).

b. Coulombic attraction between dissolved ions and water dipoles:

Due to their molecular structure water molecules are permanent electric dipoles (SM thread), with the O atoms carrying a partial negative charge and the H atoms a partial positive charge. Ions are charged atoms and emit an electric field. Water dipoles orient themselves into that field, as in the schematic for a sodium cation and its water cloud below:



This orientation lowers potential energy of the system somewhat and converts it to Enthalpy.

c. Complexation with solvent:

Many cations form stable coordination complexes with water (SM thread). These processes are exo-energetic (ΔG < 0).
<hr>
So... is a compound soluble or insoluble?
$$\Delta G =\Delta H-T\Delta S<0$$
We’ve already established that the term – TΔS is always negative, so for the above condition to be satisfied we have:

$$\Delta H < T \Delta S$$

Where ΔH is the sum of the Enthalpies discussed above.

Simple example: KCl

For KCl (dissolution) ΔH = + 8.4 kJ/mol and ΔS = 96 J/mol K. At 25 C we get:

ΔG = 8.4 kJ/mol – 298 K x 96 J/mol K = - 20 kJ/mol.

So KCl is water soluble and without addition of heat, dissolving KCl in water cools down that solution (a little).

[Edited on 12-2-2016 by blogfast25]

2. Solubility limits and Solubility products:

Solubility limits of soluble ionic compounds are of course easy to find, even in a broad range of temperatures (see e.g. Wikipedia’s solubility table). Far more interesting in my opinion are those ionic compounds that are poorly soluble or even categorised as ‘insoluble’.

Dissolution then becomes a left leaning equilibrium between two states:

$$\text{solute}+\text{solvent} \leftrightarrow \text{solution}$$

For a generic ionic compound we can specify:

$$\mathrm{M_nA_m}(s) \leftrightarrow n\mathrm{M^{m+}}(aq) +m\mathrm{A^{n-}}(aq)$$

As with all equilibria, an equilibrium constant, here known as the solubility product, can be defined:
$$K_{sp}=[\mathrm{M^{m+}}]^n[\mathrm{A^{n-}}]^m$$

... where the bracketed quantities are concentrations.

Thermodynamically, insoluble compounds correspond to:

$$\Delta G_{solvation}>0$$
Which via Nernst means:

$$K_{sp} \ll 1$$

Relation between K_sp and solubility limit:

Let’s assume our generic ionic compound has a solubility limit of S mol/L.
Each mol of the compound dissociates into n mol of cations and m mol of anions, so:
$$K_{sp}=(nS)^n(mS)^m$$
$$K_{sp}=n^nm^m(S)^{n+m}$$

$$S=\sqrt[n+m]{\frac{K_{sp}}{n^nm^m}}$$

Example: BaF₂

K_sp = 1.0 x 10^-6

$$1.0 \times 10^{-6}=S \times (2S)^2$$

$$1.0 \times 10^{-6}=4S^3$$

$$S=0.006\:\mathrm{mol/L}$$

(Note that hydrolysis of the F^- has been ignored here, see next post)
<hr>

Some more basic free energy considerations:

Below we’ll use the simple example of a binary, ionic compound MX. The suffixes “sp”, “L” and “aq” refer to dissolution, lattice and aquation, respectively.

Nernst, applied to the process of dissolution, tells us:
$$\Delta G_{sp}=-RT\ln K_{sp}=\Delta H-T\Delta S$$
Reworked:
$$\ln K_{sp}=-\frac{\Delta H}{RT}+\frac{\Delta S}{T}$$

Simple examination of the last equation shows that a ten-fold change in K_sp will result in a change of less than 6 kJ/mol in the free energy of solution. This means that energetic differences between soluble and insoluble salts are small. Laboratory experience confirms that dissolution processes are by and large not highly energetic.

To analyze the changes it is useful to think in terms of the following cycle:



where ∆G_sp is the free energy of dissolution and must be equal to ∆G_L, the free energy of vapourisation of MX to non-interacting gaseous cations and anions plus ∆G_a, the free energy of aquation (or solvation if the solvent is something other than water). The latter term is obviously the sum of the separate free energies of aquation of the cation and the anion. ∆G_sp is generally rather small, between -80 and +80 kJ/mol, while both ∆G_L and ∆G_a are both very large and of opposite sign. Remember that ∆G_L is essentially the lattice energy (but not entirely, since the work of expansion and the entropies of the ions themselves must be included), which is always >400 kJ/mol and may be as high as >3000 kJ/mol.

Thus the solubility of a salt is in effect determined by a small difference between two large numbers.


[Edited on 14-2-2016 by blogfast25]

2.1. Hydrolysis and pH effects - worked examples:

a. BaF₂ example:

In the example of the previous post we neglected that fluoride is the conjugated base of a weak acid and thus itself a weak base:

$$\mathrm{F}^- (aq) + \mathrm{H_2O}(l) \leftrightarrow \mathrm{HF}(aq)+\mathrm{OH}^{-}$$
Ignoring the auto-dissociation of water and with the following equations:
$$K_a=\frac{[\mathrm{H_3O^+}][\mathrm{F^-}]}{[\mathrm{HF}]}$$

$$K_{sp}=[\mathrm{Ba^{2+}}][\mathrm{F^-}]^2$$

$$[\mathrm{Ba^{2+}}]=\frac12[\mathrm{F^-}]+\frac12[\mathrm{HF}]$$

$$S=[\mathrm{Ba^{2+}}]$$
With some reworking we get:
$$S=\sqrt[3]{\frac{K_{sp}}{4}\Big(1+\frac{[\mathrm{H_3O^+}]}{K_a}\Big)^2}$$
Closer inspection shows that this is the result obtained when ignoring hydrolysis, multiplied by this pH dependent factor:

$$\sqrt[3]{\Big(1+\frac{[\mathrm{H_3O^+}]}{K_a}\Big)^2}$$

As this is a continuously increasing function of [H₃O⁺], solubility increases with decreasing pH.

b. Calcium carbonate:

Ignoring hydrolysis of the carbonate ions the solubility would simply be:

$$S=\sqrt{K_{sp}}$$
Taking hydrolysis into account, that result has to be multiplied by the following pH dependent correction factor:
$$\sqrt{1+\frac{[\mathrm{H_3O^+}]}{K_{a,2}}}$$

The strong influence of pH on CaCO₃ solubility can be seen from the correction factor:

46 at pH = 7
460 at pH = 5
4600 at pH = 3

c. D-block sulphides (mainly):

Sulphides are the salts of a very weak diprotic acid and their solubility therefore depends on pH.

A few sulphide solubility products:

CdS: 8 x 10^-28
CuS: 6 x 10^-37
MnS: 3 x 10^-14
PbS: 3 x10^-28
NiS: 3 x 10^-19
Ag2S: 6 x 10^-51
ZnS: 2 x 10^-25

Source.

The two deprotonation constants are of H₂S are:
$$K_{a,1}=\frac{[\mathrm{H_3O^+}][\mathrm{HS^-}]}{[\mathrm{H_2S}]}=10^{-7}$$

$$K_{a,2}=\frac{[\mathrm{H_3O^+}][\mathrm{S^{2-}}]}{[\mathrm{HS^-}]}=10^{-19}$$

This graph shows the relative concentrations of H₂S, HS^- and S^2- in function of pH:



Multiplying the first two equations with each other we get:
$$K_{a,1}K_{a,2}=\frac{[\mathrm{H_3O^+}]^2[\mathrm{S^{2-}}]}{[\mathrm{H_2S}]}=10^{-26}$$

$$[\mathrm{S^{2-}}]=10^{-26}\frac{[\mathrm{H_2S}]}{[\mathrm{H_3O^+}]^2}$$

Literature puts the solubility of H₂S at about 0.1 M and quite invariant of pH, so we can write:

$$[\mathrm{S^{2-}}]\approx \frac{10^{-27}}{[\mathrm{H_3O^+}]^2}$$

For a generic sulphide MS:

$$K_{sp}=[\mathrm{M^{2+}}][\mathrm{S^{2-}}]$$

Combining and for [M²⁺] = 0.1 mol/L:

$$[\mathrm{H_3O^+}] \approx \frac{10^{-14}}{\sqrt{K_{sp}}}$$

We can also rewrite it using p-functions (- log) as:

$$pH \approx 14 -\frac12 pK_{sp}$$

Now what does this calculated pH really represent? If we take a solution of approx. 0.1 M [M²⁺] and saturate it with H₂S gas, control the pH (by buffering, e.g.), then the above calculated pH is the maximum value for which no MS precipitate forms. Above that pH the sulphide concentration is too high for M²⁺ to stay in solution.

For a few of the less insoluble sulphides listed above:

MnS: pH > 7 to obtain precipitation.
NiS: pH > 5 to obtain precipitation.
ZnS: pH > 1.7 to obtain precipitation.

It also follows that sulphides with a pK_sp > 26 will always precipitate, regardless of pH.

d. pH Selective precipitation of hydroxides - Fe(+3)/Mn(+2) separation:

A dilute (< 1 M) solution of manganous and ferric salts is first part neutralised with ammonia solution until only the small amount of permanent precipitate forms. About ¾ is then set aside (Part A) and the rest treated with ammonia solution to precipitate the metals as hydroxides (Part B).

Part B is filtered and the filter cake washed free of ammonia and ammonia salts. The washed filter cake is then added to Part A and allowed to stand overnight, occasionally stirring.

After filtration, a solution is obtained that is free of ferric ions, because at the pH of Part A + filter cake B, ferric ions only exists as insoluble ferric hydroxide. Mn(+2) stays in solution. This is due to the immense difference of the solubility products of both hydroxides:

Fe(OH)₃: 4 x 10^-38
Mn(OH)₂: 1.9 x 10^-13

This separation method has been discussed and successfully used by several SM members, myself included. There’s also a ‘Nurdrage’ YouTube demo on it.

[Edited on 15-2-2016 by blogfast25]

2.2. Common ion effect:

Quote: Originally posted by MrHomeScientist

Really great resource here. Thanks for volunteering to share your knowledge! One question on this segment: Quote: The strong influence of pH on CaCO<sub>3</sub> solubility can be seen from the correction factor: 46 at pH = 7 460 at pH = 5 4600 at pH = 3 "Solubility" there is a bit misleading, isn't it? Because at low pH CaCO<sub>3</sub> ceases to be the carbonate, as CO<sub>2</sub> escapes. Making it really only "half soluble" so to speak.

At even lower pH a carbonate dissolves completely to a calcium salt, yet do we not claim "calcium carbonate dissolves in acid"? Same with many metals. It becomes semantics, to me.

Incidentally, even at pH = 7, CaCO₃ solutions contain significant amounts of HCO₃^-, which is precisely what brings about the greater solubility, compared to what to expect from:

$$S=\sqrt{K_{sp}}$$

[Edited on 17-2-2016 by blogfast25]

2.3. Ionic strength effect:

Due the formation (simply put) of so-called ionic atmospheres around cations and anions their interaction is reduced due to shielding.



To quantify this, the concept of chemical activity (which is not limited to solvated species) is introduced:

$$a_C=\gamma [C]$$

From Debeye-Huckel’s theory of electrolytes we have an approximate and workable description of the activity coefficient γ:

$$\log \gamma \approx -Az^2\frac{\sqrt{I}}{1+\sqrt{I}}$$
Where:
$$A=0.509\:\mathrm{mol^{-0.5}kg^{0.5}}$$

Where I is the ionic strength of the solution:

$$I=\frac12\sum_{i=1}^n C_iz_i^2$$

E.g. for a solution 0.3 KNO3 + 0.1 MgSO4, then I = 0.7. Note that for pure water I = 0.

The following tool is useful to estimate ionic strengths and activity coefficients (for mono and divalent ions):

http://www.lenntech.com/calculators/activity/activity-coeffi...

Make sure you select the correct units.

The function:

$$\gamma=10^{-\log \gamma}$$

... is worth exploring briefly. Look at the argument:

$$I \to 0, \frac{\sqrt{I}}{1+\sqrt{I}} \to 0$$

So :

$$I \to 0, \gamma \to 1$$

For infinitely diluted solutions, the activity coefficient tends to unity.

Also:

$$I \to \infty, \frac{\sqrt{I}}{1+\sqrt{I}} \to 1$$
$$I \to \infty, \gamma \to 10^{-Az^2}$$

For a simple, poorly soluble compound MA:

$$K_{sp}=\gamma_M[M^{z+}]\gamma_A[A^{z-}]$$
$$K_{sp}=\gamma^2S^2$$
$$S=\frac{\sqrt{K_{sp}}}{\gamma}$$

a. Case of CaSO4 in pure water:

K_sp = 4.93 x 10^-5, S = 2.1 g/L (Wikipedia)

$$\log \gamma= -Az^2\frac{\sqrt{I}}{1+\sqrt{I}}$$
$$\gamma=10^{-\log \gamma}$$
$$I=\frac12([Ca^{2+}] \times 2^2+[SO_4^{2-}]\times ^2)=4S$$
$$S=\frac{\sqrt{K_{sp}}}{\gamma}$$

The problem is that this is a transcendental equation that has no analytical solution. Instead we need to use a numerical, iterative approach. Creating an Excel spread for this task is easy but not even necessary.

As a starting point, assume γ = 1 and calculate the first approximation of S:
$$S_1=\frac{\sqrt{K_{sp}}}{\gamma}=0.00702\:\mathrm{M}$$
Now insert this value into the tool mentioned above, for calcium and sulphate and obtain a new value for γ. Use this new value to obtain the next approximation of S. Adjust the calcium and sulphate to obtain a new value of γ.

I repeated this process 8 times and S then converged to 0.019 M, more than twice the original value. 0.019 M corresponds to about 2.6 g/L, so not too far from Wiki’s stated 2.1 g/L.

b. With 0.1 KNO3:

The same method can be used to estimate calcium sulphate in (e.g.) 0.1 M potassium nitrate.

We can even use a little shortcut because we already know the solubility in pure water: 0.019 M. Use that value with the tool, adding also 0.1 M potassium and 0.1 M nitrate, this gives a starting point γ = 0.256. Use this to calculate the next approximation of S, adjust the tool for the next γ, etc.

Only 5 iterations were needed to converge on S = 0.0308 M, an increase of about 50 % with respect to pure water.

c. With common ion effect: 0.1 M Na2SO4:

$$K_{sp}=\gamma^2S(S+0.1)$$
$$\gamma^2S^2+0.1\gamma^2 S-K_{sp}=0$$
First, using the tool, estimate γ for S = 0, sodium = 0.2 M and sulphate = 0.1 M.

Using that value for γ, obtain the first estimate of S by finding the positive root of the quadratic equation (I used Wolfram Alpha because the equation’s coefficients can then be adjusted easily for subsequent iterations). Adjust the ionic strength for calcium and sulphate using the tool and obtain the next estimate of γ.

Recalculate S and continue to iterate until S converges. It took about 4 iterations to settle on S = 0.00208 M. Note that this is about 10 x less than the calculated value for pure water.

[Edited on 18-2-2016 by blogfast25]

2.4. Cation/anion size effects: