Chemical Potential and the Theory of Ideal and Real Solutions

A pure substance's Gibbs energy tells you everything about its equilibrium behaviour. The moment a second component is added — a solute dissolved in a solvent, two miscible liquids mixed — that single number is no longer enough: each component now has its own tendency to escape, react or transfer, and that tendency changes as the composition changes. This chapter builds the single concept that tracks it precisely, the chemical potential, and uses it to derive the thermodynamics of mixing, the behaviour of ideal and real solutions, and the colligative properties that follow directly from them.

You should have to hand the Gibbs energy and its spontaneity criterion from Chapter 2. This chapter develops partial molar quantities, the chemical potential and the Gibbs–Duhem equation, the chemical potential of a perfect gas and of a component in an ideal solution (Raoult's law), the thermodynamics of mixing, real solutions and activity (Henry's law), and the colligative properties — boiling-point elevation, freezing-point depression and osmotic pressure. The next chapter uses this same chemical-potential machinery to build phase equilibria in multi-component systems.

1. Partial Molar Quantities

In a mixture, adding a small amount of one component generally changes an extensive property (volume, Gibbs energy, …) by an amount that depends on what else is already present. The partial molar volume of component $i$ is defined as

the rate at which the total volume changes as $n_i$ is added, at constant $T$, $p$, and all other mole numbers. It is not, in general, the molar volume of pure $i$: adding a small amount of water to a large excess of ethanol increases the total volume by less than $1~\text{mol}$ of pure water would occupy, because water molecules fit partly into gaps in the hydrogen-bonded ethanol network. Partial molar quantities are defined for any extensive state function, and the total property is recovered as a mole-weighted sum at fixed composition:

2. The Chemical Potential

The chemical potential of component $i$ is its partial molar Gibbs energy,

For a pure substance, $\mu = G_m$, the molar Gibbs energy — chemical potential generalises molar Gibbs energy to mixtures, where each component's own "molar Gibbs energy" depends on the composition it finds itself in. The fundamental equation of chemical thermodynamics, combining the first and second laws for a system that can also exchange matter, is

At constant $T$ and $p$, this reduces to $dG = \sum_i \mu_i\,dn_i$: the chemical potential is the "price," in Gibbs energy, of adding one more mole of component $i$ to the mixture at fixed composition and conditions. Matter flows spontaneously (at constant $T,p$) from a region of high $\mu_i$ to a region of low $\mu_i$, exactly as heat flows from high to low temperature — $\mu_i$ is the generalised "potential" that drives chemical and physical change.

Key distinction. Temperature is the potential that equalises when two bodies reach thermal equilibrium; chemical potential is the potential that equalises when a species is free to move between phases or react — at equilibrium, $\mu_i$ of every species is the same in every phase (or every side of a reaction) it can access.

3. The Gibbs–Duhem Equation

Since $G = \sum_i n_i\mu_i$ (the Gibbs energy is itself an extensive sum of the chemical potentials, each weighted by its mole number — the same relation as $V=\sum n_iV_i$ applied to $G$), differentiating fully gives

Comparing with the fundamental equation $dG=Vdp-SdT+\sum_i\mu_idn_i$ from Section 2, the $\sum\mu_idn_i$ terms cancel, leaving the Gibbs–Duhem equation:

At constant temperature and pressure, this simplifies to $\sum_i n_i\,d\mu_i = 0$: the chemical potentials of the components in a mixture cannot vary independently — if one rises, the others must fall in a mole-weighted balance. This is the thermodynamic constraint that, among other things, lets the activity of a solvent be calculated from measurements of a solute's activity alone, a technique used throughout later chapters of this block.

4. The Chemical Potential of a Perfect Gas

For a pure perfect gas, $\mu=G_m$, and $dG_m = V_m\,dp$ at constant $T$ (from the fundamental equation with $dn_i=0$, $dT=0$). Substituting $V_m=RT/p$ and integrating from a reference (standard) pressure $p^{\ominus}$ to the pressure of interest $p$,

where $\mu^{\ominus}$ is the chemical potential of the gas in its standard state ($p^{\ominus}=1~\text{bar}$). This logarithmic form — chemical potential rising without bound as $p\to\infty$ and falling without bound as $p\to0$ — recurs throughout this chapter, with $p/p^{\ominus}$ replaced by a mole fraction or an activity as the description becomes more general.

5. The Chemical Potential of a Component in an Ideal Solution

Consider a liquid mixture in equilibrium with its vapour. For each component $i$, equilibrium between the liquid and vapour requires $\mu_i(\text{l}) = \mu_i(\text{g})$. Using the perfect-gas result of Section 4 for the vapour, with partial pressure $p_i$,

For the pure liquid alone (mole fraction $x_i=1$, vapour pressure $p_i^*$), the same relation gives $\mu_i^*(\text{l}) = \mu_i^{\ominus}(\text{g}) + RT\ln(p_i^*/p^{\ominus})$. Subtracting eliminates the (inconvenient, gas-phase) standard chemical potential entirely:

Raoult's law, an empirical observation for mixtures of structurally similar liquids, states that $p_i = x_ip_i^*$: each component's partial vapour pressure is proportional to its mole fraction in the liquid, with the pure-liquid vapour pressure as the constant of proportionality. A liquid mixture that obeys Raoult's law at every composition is called an ideal solution, and substituting Raoult's law into the result above gives its defining chemical potential:

Since $x_i\le1$, $\ln x_i\le0$, so every component's chemical potential in an ideal solution is lower than in the pure state — mixing is always favourable from this perspective, which is the thermodynamic root of why miscible liquids mix spontaneously at all.

Worked example. A solution of benzene and toluene (which, being structurally very similar, closely approximates an ideal solution) has $x_{\text{benzene}}=0.400$ at $25^\circ\text{C}$, where pure benzene has $p^*=12.8~\text{kPa}$. By Raoult's law, the partial pressure of benzene above the solution is

and its chemical potential relative to the pure liquid is $\mu - \mu^* = RT\ln(0.400) = (8.314)(298)\ln(0.400) = -2270~\text{J}\,\text{mol}^{-1}$, lower than in the pure state by $2.27~\text{kJ}\,\text{mol}^{-1}$.

6. The Thermodynamics of Mixing

Consider forming an ideal solution from $n_A$ mol of pure A and $n_B$ mol of pure B (generalising immediately to any number of components). Before mixing, the total Gibbs energy is $n_A\mu_A^* + n_B\mu_B^*$. After mixing, using the boxed result of Section 5,

so the Gibbs energy of mixing is

(writing $n=n_A+n_B$). Since $0<x_A,x_B<1$, both logarithms are negative, so $\Delta_{\text{mix}}G<0$ always — ideal solutions always mix spontaneously, for any composition and at any temperature. Differentiating with respect to temperature at constant composition gives the entropy of mixing,

always positive (more microstates available once the two components can occupy a shared volume). Since $\Delta_{\text{mix}}G = \Delta_{\text{mix}}H - T\Delta_{\text{mix}}S$ and the boxed formula shows $\Delta_{\text{mix}}G$ is exactly $-T\Delta_{\text{mix}}S$ with no additional term, the enthalpy of mixing of an ideal solution is exactly zero:

Mixing is driven entirely by the entropy increase, with no energetic preference between A–A, B–B and A–B interactions — the defining microscopic assumption, now shown to be thermodynamically equivalent to Raoult's law holding at every composition.

7. Real Solutions: Activity and Henry's Law

Real mixtures generally depart from Raoult's law, especially away from $x_i\to1$. The departure is absorbed into an empirical activity coefficient, $\gamma_i$, by defining the activity $a_i \equiv \gamma_ix_i$ and writing the chemical potential in exactly the ideal-solution form with $x_i$ replaced by $a_i$:

with $\gamma_i\to1$ as $x_i\to1$ (Raoult's law is recovered exactly for the major, solvent-like component as it approaches purity — the dilute-solution limit for the solvent). For a dilute solute, however, the solute molecules are overwhelmingly surrounded by solvent, not by other solute molecules, and Raoult's law (built on A–A-like interactions in the pure liquid) is the wrong reference state. Instead, dilute solutes obey Henry's law,

with an empirical Henry's law constant $K_{H,i}$ that is generally not equal to the pure-solute vapour pressure $p_i^*$ — reflecting the different (solute-in-solvent) environment the solute molecules actually experience. A solution that obeys Raoult's law for the solvent and Henry's law for the solute, each in its own appropriate limit, is called an ideal-dilute solution, the standard working model for dilute real solutions throughout chemistry.

Key distinction. Raoult's law is the correct reference for a component approaching purity ($x_i\to1$); Henry's law is the correct reference for a component present only as a trace ($x_i\to0$). Applying Raoult's law to a dilute solute (or Henry's law to the solvent) is a common and serious error.

8. Colligative Properties

Colligative properties depend only on the number of solute particles present, not on their chemical identity — they arise purely from the solvent's chemical potential being lowered by the presence of solute, $\mu_{\text{solvent}} = \mu_{\text{solvent}}^* + RT\ln x_{\text{solvent}}$, with $x_{\text{solvent}}<1$.

Freezing-point depression and boiling-point elevation both follow from equating $\mu_{\text{solvent}}(\text{solution})$ to the chemical potential of the pure solid (freezing) or pure vapour (boiling) phase, and using the Gibbs–Helmholtz relation to track how that equilibrium temperature shifts as $x_{\text{solvent}}$ departs from 1. The results, to leading order in the solute mole fraction (equivalently, molality $b$ for a dilute solution), are:

with the cryoscopic and ebullioscopic constants $K_f = \dfrac{RT_f^{*2}M_{\text{solvent}}}{\Delta_{\text{fus}}H}$ and $K_b = \dfrac{RT_b^{*2}M_{\text{solvent}}}{\Delta_{\text{vap}}H}$ — both depending only on properties of the pure solvent, consistent with colligative properties being insensitive to the solute's identity.

Osmotic pressure, $\Pi$, is the excess pressure that must be applied to a solution, separated from pure solvent by a semipermeable membrane, to stop the net flow of solvent across the membrane (driven by the solvent's lower chemical potential on the solution side). Equating chemical potentials across the membrane and expanding for a dilute solution gives the van 't Hoff equation:

strikingly identical in form to the ideal gas law, reflecting that both describe a dilute population of independent, non-interacting particles.

Worked example. A solution contains $18.0~\text{g}$ of glucose ($M=180~\text{g}\,\text{mol}^{-1}$, a non-electrolyte) in $1.00~\text{kg}$ of water ($K_f=1.86~\text{K}\,\text{kg}\,\text{mol}^{-1}$). The molality is

the solution freezes at $-0.186^\circ\text{C}$ instead of $0^\circ\text{C}$ — a measurement used in practice to determine unknown molar masses, by solving the same equation for $M$ given an experimentally measured $\Delta T_f$.

9. Common Exam Pitfalls

• Treating the partial molar volume as the pure-substance molar volume. They coincide only in the limit of a pure substance; in a mixture, $V_i$ reflects the local packing environment and can even be negative in extreme cases (e.g. some electrolytes in water).
• Applying Raoult's law to a dilute solute. The correct reference for a dilute solute is Henry's law, with its own empirically fitted constant $K_H$, not the solute's pure-liquid vapour pressure $p^*$.
• Forgetting that $\Delta_{\text{mix}}H=0$ is specific to IDEAL solutions. Real mixing generally has $\Delta_{\text{mix}}H\ne0$ (positive for endothermic mixing, e.g. many organic liquid pairs; negative when favourable new interactions form, e.g. strong hydrogen bonding), captured by non-unit activity coefficients.
• Mixing up freezing-point depression and boiling-point elevation signs. Dissolved solute always lowers the freezing point and raises the boiling point — both consequences of the solvent's chemical potential being lowered, which stabilises the liquid phase relative to both the solid and the vapour.
• Assuming the two shifts are the same size. They are not: the size of each shift is set by how steeply $\mu$ changes with $T$ on either side of the transition (the slope of $\mu$ vs $T$ is $-S_m$), and since $\Delta_{\text{vap}}H \gg \Delta_{\text{fus}}H$ for most solvents, the entropy change — and hence the slope — at the boiling point is much larger than at the freezing point. The same solute-induced shift in $\mu_{\text{solvent}}$ therefore produces a smaller temperature shift at the boiling point than at the freezing point, which is exactly why $K_b$ (built from $\Delta_{\text{vap}}H$ in its denominator) comes out smaller than $K_f$ (built from $\Delta_{\text{fus}}H$) — for water, $K_b=0.512~\text{K}\,\text{kg}\,\text{mol}^{-1}$ against $K_f=1.86~\text{K}\,\text{kg}\,\text{mol}^{-1}$.
• Using molarity instead of molality for freezing-point/boiling-point calculations. $K_f$ and $K_b$ are defined per unit molality (mol solute per kg solvent), which, unlike molarity, is temperature-independent — essential for a constant that should not itself depend on the small temperature shift being calculated.
• Forgetting the van 't Hoff factor for electrolytes. An ionic solute that dissociates (e.g. $\text{NaCl}\to\text{Na}^++\text{Cl}^-$) contributes multiple colligative "particles" per formula unit; all colligative formulas above require an extra factor $i$ (the number of particles produced per formula unit) multiplying the nominal concentration.

Summary

• The partial molar property $X_i=(\partial X/\partial n_i)_{T,p,n_{j\ne i}}$ generalises a molar property to mixtures; $X=\sum_in_iX_i$.
• The chemical potential $\mu_i=(\partial G/\partial n_i)_{T,p,n_{j\ne i}}$ is the partial molar Gibbs energy; matter flows from high $\mu_i$ to low $\mu_i$ at constant $T,p$, and equilibrium requires equal $\mu_i$ in every accessible phase.
• The Gibbs–Duhem equation, $\sum_in_id\mu_i=Vdp-SdT$ (or $\sum_in_id\mu_i=0$ at constant $T,p$), constrains the chemical potentials of a mixture's components to vary together, not independently.
• A perfect gas has $\mu=\mu^{\ominus}+RT\ln(p/p^{\ominus})$; a component of an ideal solution obeying Raoult's law ($p_i=x_ip_i^*$) has $\mu_i=\mu_i^*+RT\ln x_i$.
• Ideal mixing: $\Delta_{\text{mix}}G=nRT(x_A\ln x_A+x_B\ln x_B)<0$ always; $\Delta_{\text{mix}}S>0$ always; $\Delta_{\text{mix}}H=0$ exactly.
• Real solutions use the activity $a_i=\gamma_ix_i$; Raoult's law is the correct reference as $x_i\to1$ (solvent), Henry's law ($p_i=K_{H,i}x_i$) as $x_i\to0$ (dilute solute).
• Colligative properties — freezing-point depression, boiling-point elevation, osmotic pressure ($\Pi=cRT$) — depend only on solute particle number, arising from the solvent's chemical potential being lowered by dilution.
• Phase equilibria built from chemical potential equality are developed systematically, for one- and multi-component systems, in the next chapter.