Solutions

In the earlier topic about equilibrium, we saw that solid $\mathrm{NaCl}$ in equilibrium with a solvent (such as water) sets a fixed $V_{\mathrm{Na}^+} - V_{\mathrm{Cl}^-} = \tfrac{1}{F} \mu_{\mathrm{NaCl}} = -3.9813~\mathrm{V}$. But that is for a fully saturated solution. What happens when we only have a tiny amount dissolved? How does $V_{\mathrm{Na}^+} - V_{\mathrm{Cl}^-}$ depend on the concentration of dissolved $\mathrm{NaCl}$?

To anchor the discussion, here are some fairly accurate modelled curves:

Note the roughly linear dependence on the logarithm of concentration (the "ideal slope" in the figure). This is guaranteed in the dilute limit, for a fundamental entropic reason: each solute particle's position becomes independent of the others.

The saturation behaviour is worth noticing too. As more salt dissolves, $V_{\mathrm{Na}^+} - V_{\mathrm{Cl}^-}$ climbs, and a solution pushed above the solid's fixed value is supersaturated: it has become favourable for the excess $\mathrm{NaCl}$ to precipitate back out. Saturation is precisely where the solution's value meets the solid's.

Although the dilute limit is only approximate, it gives us a starting point for more complex questions (what happens when many species of ion are present?) without first laboriously measuring thermodynamic data over every composition.

Ideal ionic voltage in dilute solutions

Statistical mechanics tells us that a dilute solute's electrochemical potential breaks into three parts:

$$\bar\mu_i = \underbrace{z_i F \phi}_{\text{electrostatic}} + \underbrace{\mu^\circ_{\mathrm{int},i}}_{\text{offset}} + \underbrace{RT \ln(c_i/c^\circ)}_{\text{entropic}}.$$

The first term is written as the electrostatic energy of the ion's charge at a local potential $\phi$. The second, the standard internal chemical potential $\mu^\circ_{\mathrm{int},i}$, is fixed by the ion's chemical identity and the way it disturbs its solvent (its solvation shell); it does not depend on concentration.^[1] Only the third, entropic term does, sinking logarithmically as the solute is diluted. Here $c_i$ is the molar concentration (moles per litre, also called molarity), $c^\circ = 1~\mathrm{mol/L}$ is a fixed reference, and the remaining symbols are the molar charge $z_i F$ (coulombs per mole) and the gas constant times temperature $RT$ (joules per mole).

The split between the first two terms is partly conventional, since it turns on how one defines the ambiguous $\phi$, so rather than carry them separately we fold them together. Dividing through by the molar charge $z_i F$ to cast everything in volts, and gathering the two concentration-independent terms into one, gives

$$V_i = \underbrace{\Big(\phi + \tfrac{1}{z_i F}\mu^\circ_{\mathrm{int},i}\Big)}_{V^\circ_i} + \frac{RT}{z_i F} \ln(c_i/c^\circ),$$

$$V_i = V^\circ_i + \frac{RT}{z_i F} \ln(c_i/c^\circ).$$

The new quantity $V^\circ_i$ is the standard species voltage. You can think of it as the voltage the ion would carry at the reference concentration $c^\circ$, sort of;^[2] what it marks on the diagram is the reference level that the carrier's $V_i$ floats away from, the gap between the two reading as concentration. Like $\phi$ it is not pinned to an absolute value, floating with the electrical state of the solution, but as we will see its differences from ion to ion are rigid.

Concentration can equally be measured as molality $b_i$ (moles per kilogram of solvent), giving the same ideal form $V_i = V^{\circ(b)}_i + \tfrac{RT}{z_i F}\ln(b_i/b^\circ)$ with $b^\circ = 1~\mathrm{mol/kg}$. The molal and molar standard states differ only through the solvent density,^[3] which for water is negligible. The plot above used molal units; I will mostly prefer molar $c_i$ and switch when convenient.

Ideal-dilute salt water

Returning to our saltwater example, we then have:

$$V_{\mathrm{Na}^+} - V_{\mathrm{Cl}^-} = (V^\circ_{\mathrm{Na}^+} - V^\circ_{\mathrm{Cl}^-}) + 2\frac{RT}{F} \ln(c/c^\circ),$$

where $c = c_{\mathrm{Na}^+} = c_{\mathrm{Cl}^-}$ is the concentration of either ion. This gives that characteristic $2\tfrac{RT}{F}\ln(\mathrm{concentration})$ slope we saw in the plot above. Also note that $V^\circ_{\mathrm{Na}^+} - V^\circ_{\mathrm{Cl}^-} = -4.0746~\mathrm{V}$ is a constant for water in our conditions; $\phi$ cancels out of the difference, so it does not depend on that ambiguous quantity. When discussing solutions we can draw each $V^\circ_i$ as its own line alongside the $V_i$:

This is the first figure to actually draw $V^\circ_i$ lines, so a word about the artwork. Each is a thinner line in its ion's colour, with a faint hatching on its concentrated side, the side the carrier floats toward as the concentration grows (above for the cation, below for the anion, following the sign of $z_i$). A carrier level in the clear is below standard concentration; one that enters the hatching has passed $1~\mathrm{mol/L}$. The look is borrowed deliberately from the band edges of semiconductor band diagrams, where a Fermi level entering the hatched band means degeneracy; the next topic develops that connection properly. One caution, though: a $V^\circ_i$ rung is a more abstract object than a band edge, a pure concentration reference.^[4]

Ionic standard states are a floating ladder

Although the individual $V^\circ_i$ float, the differences $V^\circ_i - V^\circ_j$ are invariant properties of the pure solvent (they also vary with temperature and pressure, but that won't matter for our purposes). Here is a selection of values for water at standard conditions (25 °C, 1 bar):

I call this the standard state ladder for water. As the electrical state or solute composition of the solution changes, the $V^\circ_i$ may ride up and down, but only all together, every rung locked to the others.^[5] The $V_i$ are the mobile part: each carrier hangs its own logarithmic, concentration-set distance off its own rung, so their relative positions shift freely with composition. That is also the answer to the many-ion question above: however crowded the solution, there is one rigid ladder underneath and one floating carrier level per ion. The data page tabulates the rung values used here, along with the procedure for obtaining them from standard ionic Gibbs energies of formation ($\Delta_{\mathrm{f}} G^\circ_i$ values).

In effect, the standard state ladder is a stand-in for the electrostatic potential $\phi$, with each rung also absorbing the ion-specific parts: the local electrostatic environment that particular ion feels, its chemical structure, the way it disturbs the solvent around itself. It also stretches the band-edge analogy in a pleasing direction: where a semiconductor's two band edges frame a single band gap, a solvent offers a whole ladder of "band edges", with a fixed "band gap" between any pair of rungs (the $-4.0746~\mathrm{V}$ above being the one framed by $\mathrm{Na}^+$ and $\mathrm{Cl}^-$).

Spatial variations

So far the ladder has hung in a single well-mixed solution, and it would be easy to walk away thinking of the $V^\circ_i$ as fixed properties of that solution, like its density. They are not. The ladder floats with the local electrical state, and nothing obliges that state to be the same everywhere. The gentlest demonstration: take the ideal-dilute salt water from above and drive a steady current through it.

This uniform tilt is the tamest move in the ladder's repertoire. In later topics it will bend where concentration gradients pile up (basic transport), swing sharply across the few nanometres beside a charged interface (basic electrostatics), and step outright where the medium itself changes, since a different solvent or phase carries its own ladder (charge neutrality and mass action). For now the point is only that the ladder is a living thing rather than a fixed backdrop: rigid in its spacings, free in its motion.

Activities and non-ideality

The ideal-dilute law holds only while each ion can ignore the others. Beyond that, chemistry keeps the logarithmic form and folds every deviation into an effective concentration, the activity $a_i$, defined precisely so that

$$\bar\mu_i = z_i F \phi + \mu^\circ_{\mathrm{int},i} + RT \ln(a_i)$$

holds exactly, with the very same standard $\mu^\circ_{\mathrm{int},i}$ we met above. In $V_i$ terms,

$$V_i = V^\circ_i + \frac{RT}{z_i F} \ln(a_i),$$

with the standard state pinned by requiring $a_i \to c_i/c^\circ$ in the dilute limit.^[6]

There is a catch, and it is a big one. We can measure $V_i$; we cannot directly measure $\phi$, $V^\circ_i$, or $a_i$. In the ideal-dilute regime this didn't matter, since the concentrations pinned every gap. In a non-ideal solution, nothing any longer fixes where the $V^\circ_i$ ladder sits relative to the carriers: its rung spacings are still rigid solvent properties, but its overall position against the $V_i$ becomes a pure convention, settled afresh by every solution of every composition. This is what chemists mean by the dictum that only mean ion activities are measurable: the measurable combinations are differences like $(V_i - V^\circ_i) - (V_j - V^\circ_j)$, from which the ladder's position cancels out. The appendix topic on non-ideality works through the consequences.

What none of these choices can reach is $V_i$ itself. Concentration measure, reference value, single-ion activity convention: each merely shuffles bookkeeping between $V^\circ_i$ and $a_i$, while the physical total $V_i$ (that is, the observable $\bar\mu_i$) stays fixed. Offsets galore keeps the full scoreboard of such conventions, where the activity emerges as the traditional framework's true rival to $V_i$: the dimensionless $a_i$ buys immunity from every global convention at the price of that severe per-composition local one, while $V_i$ makes exactly the opposite trade.

Takeaways

A dilute ion's species voltage sits a logarithmic distance from its standard species voltage $V^\circ_i$, the distance set by concentration through $V_i = V^\circ_i + \tfrac{RT}{z_i F}\ln(c_i/c^\circ)$. The $V^\circ_i$ float with the electrical state of the solution, yet their differences form a rigid ladder fixed by the solvent; once the solution is no longer dilute, the concentration generalizes to an activity, at the price of a convention that unmoors the ladder from the carriers while leaving every $V_i$ untouched. This same structure, a carrier floating logarithmically above a standard-state level, turns out to describe the electronic carriers in a semiconductor every bit as well, and that is where we go next.

NEXT TOPIC: Semiconductors