From WolfWikis

LECTURE 16

Ionic solutions

A solution of a strong electrolyte such as NaCl in water is perhaps one of the most obvious systems to consider but unfortunately it is also one of the more difficult ones. The reason is that the salt produces two charged solutes Na⁺ and Cl⁺ (both in hydrated form) in solution.

First of all we need to consider the dissociation process and its stoichiometry as we are bringing more than one solute species into solution.

Secondly, we need to consider electrostatic interactions between solutes. The charges introduce a strong interaction that only falls off with r^-1 as opposed to r^-6 or so if only neutral species are present. This causes very serious divergence from ideality even at very low concentrations

Consider a salt going into solution:

C_ν+A_ν- =>ν₊C^z₊ + ν_-A^z_-

where: ν₊ and ν_- are the stoichiometric coefficients and z₊ and z_- the formal charges of the cations and anions resp.

As we shall see the stoichiometric coefficients involved in the dissociation proces are important for a proper description of the thermodynamics of strong electrolytes.

Charge neutrality demands:

ν₊z₊ + ν_-z_- = 0

Thermodynamic potentials versus the dissociation

For the salt we can write:

μ₂ =μ₂^o + RTlna₂

However we need to take into account the dissociation, to do so we write:

μ₂ = ν₊μ₊ + ν_-μ_-

Obviously this implies:

μ₂^o = ν₊μ₊^o + ν_-μ_-^o

where

μ₊ =μ_-^o + RTlna₊

μ_- =μ_-^o + RTlna_-

Usually Henry's law is taken as standard state for both type of ions. Admittedly, these equations are rather formal. We cannot measure the activities of the ions separately because it is impossible to add one without adding the other, nevertheless we can derive a useful formulism with them that takes into account the dissociation process.

If we substitute the last two equations in the ones above we get:

ν₊lna₊ + ν_-lna_-=lna₂

Taking the exponent we get:

a₂ =a₊^ν₊a_- ^ν_-

Notice that the stoichimetric coeffients generate exponents in the expression.

Mean ionic activity

We now introduce the sum of the stoichiometric coefficients:

ν₊+ν_- = ν

and define the mean ionic activity a_± as:

a_± ^ν ≡ a₂ =a₊^ν₊a_- ^ν_-

Activity coefficients

All this remains a formality unless we find a way to relate it back to the concentration of the salt. Usually molality is used as a convenient concentration measure rather than molarity, because we are dealing with pretty strong deviations from ideal behavior and that implies that volume may not be an additive quantity. Molality does not involve volume in contrast to molarity.

Working with molalities we can define activity coefficients for both ions, even though we have no hope to determine them separately

a₊ =γ₊ m₊

a_- =γ_- m_-

Stoichiometry dictates the molalities of the individual ions must be related to the molality of the salt m by:

m_-=ν_-m

m₊=ν₊m

Mean ionic molality

Analogous to the mean ionic activity we can define a mean ionic molality as:

m_± ^ν ≡ m₊^ν₊m_- ^ν_-

Mean ionic activity coefficient

We can do the same for the mean ionic activity coefficient

γ_± ^ν ≡ γ₊^ν₊γ _- ^ν_-

Using this definitions we can rewrite

a₂=a_± ^ν=a₊^ν₊a_- ^ν_-

as

a₂=a_± ^ν =γ_± ^ν m_± ^ν

Measuring mean ionic activity coefficients

In contrast to the individual coefficients the mean ionic activity coefficient γ± is a quantity that can be determined. In fact we can use the same Gibbs-Duhem trick we did for the sucrose problem to do so. We simply measure the water vapor pressure above the salt solution and use

lnγ_± = φ -1 + ∫ [ φ -1 ]m'dm' (from m'=0 to the m of interest)

The fact that the salt itself has a negligible vapor pressure does not matter. Particularly for ions with high charges the deviations from ideality are very strong even at tiny concentrations.

Admittedly doing these vapor pressure measurements in pretty tedious, there are some other procedures, e.g. involving electrochemical potentials. They too are tedious.

Debye-Hückel theory

As ionic solutions are very common in chemistry (and often quite complex!), having to measure all γ_± values for all possible (multiple!) solute-solvent combinations is a pretty daunting task, even though in times past extensive tabulation has taken place. We should be grateful for the rich legacy that our predecessors have left us in this respect. (It would be hard to get any funding to do such tedious work today).

Of course it would be very desirable to be able to calculate γ_± values from first principles or if that fails by semi-empirical means. Fortunately considerable progress has been made on this front as well. We can only scratch the surface of that topic in this course and will briefly discuss the simplest approach due to Debye and Hückel

Debye length

Debye and Hückel came up with a theoretical expression that makes is possible to predict mean ionic activity coefficients as sufficiently dilute concentrations. (Unfortunately sufficient usually means tiny..) The theory considers the vicinity of each ion as an atmosphere-like cloud of charges of opposite sign that cancels out the charge of the central ion. From a distance the cloud looks neutral. The quantity 1/κ is a measure for the size of this cloud and kappa is known as the Debye-length. Its size depends on the concentration of all other ions.

Ionic strength

To take the effect from all other ions into account it is useful to define the ionic strength I as:

I =½ Σ m_iz_i²

where m_i is de molality (sometimes molarity c is used instead, as in S&McQ) of ion i and z_i its charge coefficient. Note that highly charged ions (e.g. z=3+) contribute strongly (nine times more than +1 ions!) but the formula is linear in the molality.

Using the ionic strength the Debye-length becomes:

κ ² = constant. I

The constant contains kT and ε_rε_o in the denominator and the number of Avogadro N_A and the square of the charge of the electron e in the numerator:

constant= 2000e²N_A/ε_rε_okT

The Debye length and the logarithmic mean ionic activity coefficient are proportional:

lnγ_± ~ κ

Again there are anumber of factors in the poportionality constant:

lnγ_± = -|q₊q_-| κ /8πε_rε_okT

If there is only one salt being dissolved the ionic strength depends linearly on its concentration The Debye length κ and lnγ_± therefore go as the square root of concentration (usually molality):

lnγ_± ~ √m

Ionic replacement

If there are other ions present the ionic strength involves all of them. This fact is sometimes used to keep ionic strength constant while changing the concentration of one particular ion. Say we wish to lower the concentration of Cu²⁺ in a redox reaction but we want to keep activity coefficients the same as much as possible. We could then replace it by an ion of the same charge say Zn²⁺ that does not partake in the reaction. A good way to do that is to dilute the copper solution with a zinc solution of the same concentration instead of with just solvent

Extensions of the Debye theory

Unfortunately the theory only works at very low concentrations. It is therefore also known as the Debye limiting law.

There are a number of refinements that aim at extending the range of validity of the theory to be able to work at somewhat higher concentrations. The book gives a number of examples such as:

lnγ_± = -1.173 |z₊z_-| √I /[1+√I]

Importance for colloids

When a solid is formed by a reaction from solution it is sometimes possible that it remains dispersed as very small particles in the solvent. The sizes typically range in the nanometers This is why it has become fashionable to call them nanoparticles, although they had been known as colloidal paricles since the mid nineteenth century. They are smaller than the wavelength of the visible reason. This causes liquids that contain them to remain clear, although they can at times be beautifully colored. A good example is the reduction of AuCl₄^- with citrate to metallic gold. This produces clear wine red solutions, even at tiny gold concentrations.

The reason the gold does not precipitate completely is typically that the nanoparticle (AuNP) formed during the reaction are charged by the attachment of some of the ionic species in solution to its surface. This results in an charged particle with an atmosphere with a certain Debye length around it. This charged cloud prevents the particle form coalescing with other particles by electrostatic replusion.

Such a system is called a colloid Of course these systems are metastable. Often they have a pretty small threshold to crashing to a real precipitate under influence of the strong van der Waals interactions that the particles experience once they manage to get in close contact. Under the right conditions colloids can survive for a long time. Some gold colloids prepared by Faraday in the 1850's are still stable today.

It will be clear from the above that addition of a salt -particularly containing highly charged ions like 3+ or 3-- may destabilize the colloid because the ionic strength will changed drastically and this will affect the Debye length.