Real Solutions and Excess Functions

Mix ethanol and water and the total volume shrinks below the sum of the parts. Mix benzene and cyclohexane and the mixture warms less ideally than expected, releasing a nonzero heat of mixing. Real liquids rarely mix the way simple lattice models predict. The excess functions give chemists a precise, measurable language for this imperfection. They isolate exactly how far a real mixture deviates from an ideal solution, at the same temperature, pressure, and composition. This matters because separations, solubility limits, and phase diagrams all hinge on those deviations. This article defines the excess Gibbs energy, enthalpy, entropy, and volume. It shows how they connect to measurable quantities like vapor pressure and mixing volumes. It works two full numerical examples, and previews how excess functions feed directly into activity coefficients and regular-solution theory.

Key takeaways

An excess function $X^E = X_{mix} - X_{mix}^{ideal}$ measures the real deviation from ideal-solution behavior at the same $T$ , $P$ , composition.
$G^E$ is built directly from activity coefficients: $G^E = RT\sum_i n_i \ln\gamma_i$ .
$V^E$ and $H^E$ are directly measurable — from dilatometry and mixing calorimetry, respectively.
Positive $G^E$ signals unfavorable interactions and can drive phase separation or azeotrope formation.
The Gibbs–Duhem equation links the activity coefficients of components in a mixture, giving a consistency test for excess-function data.
Key idea: Excess functions strip away the trivial ideal-mixing entropy and enthalpy, leaving a clean thermodynamic signature of real intermolecular interactions.

The core insight

Every thermodynamic mixing property of a real solution can be split into an ideal-solution part, computed exactly from mole fractions alone, plus an excess part that carries all the chemistry — the extra (or reduced) Gibbs energy, enthalpy, entropy, and volume that arise because real molecules interact differently with like and unlike neighbors.

The physical picture

An ideal solution is defined by one property: every molecule feels the same environment whether surrounded by its own kind or by another component. This is a strong assumption. In an ideal solution, $\Delta_{mix}H = 0$ and $\Delta_{mix}V = 0$ , and the entropy of mixing is purely combinatorial:

\Delta_{mix}S^{ideal} = -R\sum_i n_i \ln x_i

Real solutions almost never satisfy this. When ethanol mixes with water, hydrogen bonding networks reorganize, and the mixture contracts. When benzene mixes with carbon tetrachloride, weak dispersion mismatches release a small amount of heat. When two liquids are chemically similar in size and polarity — benzene and toluene, for instance — real behavior approaches ideal. The excess function machinery captures these deviations without needing a microscopic model. For any extensive mixing property $X$ (with $X = G, H, S, V$ ), define

X^E \equiv \Delta_{mix}X_{real} - \Delta_{mix}X_{ideal}

at the same temperature, pressure, and composition. Because the ideal reference already accounts for the entropy of mixing, $S^E$ measures only the non-ideal part of the entropy — the ordering or disordering caused by preferential interactions, not the trivial mixing entropy itself. $H^E$ is simply the enthalpy of mixing itself, since $H_{mix}^{ideal}=0$ . $V^E$ is the mixing volume itself, since $V_{mix}^{ideal}=0$ . The excess Gibbs energy is the master function. It connects directly to the activity coefficient $\gamma_i$ , the correction factor that makes the real chemical potential match experiment:

\mu_i = \mu_i^{*} + RT\ln x_i + RT\ln\gamma_i

Here $\mu_i^{*}$ is the chemical potential of pure liquid $i$ . The $RT\ln\gamma_i$ term is precisely the partial molar excess Gibbs energy of component $i$ . When $\gamma_i = 1$ for all compositions, the solution is ideal and $G^E = 0$ .

Figure

Excess Gibbs energy across composition for a positively deviating binary mixture

The peak of $G^E(x)$ near equimolar composition, and its sign, encode the strength and type of nonideality. A positive hump (shown above) means the components "dislike" each other relative to ideal mixing — this is the signature that eventually leads to limited miscibility, covered under regular-solution theory.

The core relations

X^E = \Delta_{mix}X_{real} - \Delta_{mix}X_{ideal}

where $X$ stands for $G$ , $H$ , $S$ , or $V$ , all evaluated at identical $T$ , $P$ , composition.

G^E = RT\sum_i n_i \ln \gamma_i

where $\gamma_i$ is the activity coefficient of component $i$ (mole-fraction based, $\gamma_i \to 1$ as $x_i \to 1$ ).

H^E = \Delta_{mix}H \qquad (\text{since } \Delta_{mix}H^{ideal}=0)

where $\Delta_{mix}H$ is measured directly by mixing calorimetry.

V^E = \Delta_{mix}V = V_{mix} - \sum_i x_i V_{m,i}^{*}

where $V_{m,i}^{*}$ is the molar volume of pure component $i$ .

G^E = H^E - TS^E

where $S^E$ is the excess (non-ideal) entropy, separate from the ideal combinatorial mixing entropy.

\left(\frac{\partial (G^E/T)}{\partial T}\right)_{P,x} = -\frac{H^E}{T^2}

where this Gibbs–Helmholtz form lets $H^E$ be extracted from the temperature dependence of activity coefficients.

x_1 d\ln\gamma_1 + x_2 d\ln\gamma_2 = 0 \quad (\text{at constant } T, P)

where this is the binary Gibbs–Duhem constraint linking the two activity coefficients.

Derivation AFrom vapor pressures to the excess Gibbs energy

Goal: show how $G^E$ for a binary liquid mixture is obtained purely from measured total vapor pressure data, without any lattice or interaction model.

Step 1. For a real binary liquid in equilibrium with its vapor (treated as ideal gas), each component obeys

\mu_i(l) = \mu_i^{*}(l) + RT\ln\left(\frac{p_i}{p_i^{*}}\right)

where $p_i$ is the partial vapor pressure of $i$ above the mixture and $p_i^{*}$ is the vapor pressure of pure $i$ . This is the modified Raoult's law route to activity.

Step 2. Define the activity coefficient through

\gamma_i = \frac{p_i}{x_i p_i^{*}}

so that $\gamma_i \to 1$ as $x_i \to 1$ , recovering Raoult's law in the dilute-solvent limit. Any measured deviation of $p_i$ from the Raoult's-law line $x_i p_i^{*}$ is captured entirely inside $\gamma_i$ .

Step 3. The molar excess Gibbs energy of the mixture is the mole-fraction–weighted sum of the two partial molar excess terms:

G_m^E = RT\left(x_1\ln\gamma_1 + x_2\ln\gamma_2\right)

This follows because $G^E = \sum_i n_i RT\ln\gamma_i$ and dividing by total moles gives the molar quantity.

Step 4. Measure total pressure $P = p_1 + p_2$ across the full composition range at fixed $T$ . Using the Gibbs–Duhem relation from Step 1's consequence,

x_1 d\ln\gamma_1 = -x_2 d\ln\gamma_2

one can extract $\gamma_1(x_1)$ and $\gamma_2(x_1)$ self-consistently even if only total pressure (not partial pressures) is measured — this is the basis of the Barker method used in vapor–liquid equilibrium data reduction.

Step 5. Insert the resulting $\gamma_1(x_1)$ , $\gamma_2(x_1)$ into Step 3's formula at each composition to build the full curve $G_m^E(x_1)$ .

\boxed{G_m^E(x_1) = RT\left[x_1\ln\gamma_1(x_1) + x_2\ln\gamma_2(x_1)\right]}

This single curve, built entirely from P–x–y measurements, is the experimental fingerprint of non-ideality for that binary system at that temperature.

Worked examples

Example 1 — Foundational: excess Gibbs energy from activity coefficients At 328 K, a benzene(1)/cyclohexane(2) mixture at $x_1 = 0.40$ has measured activity coefficients $\gamma_1 = 1.30$ and $\gamma_2 = 1.14$ (representative literature values for this classic near-regular system). Step 1 — compute $RT$ :

RT = (8.314\ \text{J K}^{-1}\text{mol}^{-1})(328\ \text{K}) = 2727\ \text{J mol}^{-1}

Step 2 — compute each $\ln\gamma_i$ :

\ln(1.30) = 0.2624, \qquad \ln(1.14) = 0.1310

Step 3 — weight by mole fraction ( $x_2 = 0.60$ ):

x_1\ln\gamma_1 + x_2\ln\gamma_2 = (0.40)(0.2624) + (0.60)(0.1310) = 0.1050 + 0.0786 = 0.1836

Step 4 — multiply by $RT$ :

G_m^E = (2727\ \text{J mol}^{-1})(0.1836) = 501\ \text{J mol}^{-1}

Sanity check: the sign is positive, consistent with benzene/cyclohexane being a positively-deviating (endothermic-leaning) system whose molecules pack less favorably than ideally mixed spheres — matching the well-known partial miscibility tendency of similar systems at lower temperatures. The magnitude, ~500 J/mol, is a typical order for a "mildly nonideal" organic mixture — far smaller than a hydrogen-bond energy (~20 kJ/mol), as expected since this pair has no strong specific interactions.

Example 2 — Applied: extracting $H^E$ and $S^E$ from mixing calorimetry and vapor pressure A calorimeter measures the molar enthalpy of mixing for the same benzene/cyclohexane system at $x_1 = 0.40$ , $T = 328$ K:

\Delta_{mix}H_m = H_m^E = +380\ \text{J mol}^{-1}

Using the $G_m^E = 501$ J/mol found in Example 1 at the same composition and temperature, find $S_m^E$ and interpret the entropy contribution. Step 1 — rearrange $G^E = H^E - TS^E$ :

S_m^E = \frac{H_m^E - G_m^E}{T}

Step 2 — substitute:

S_m^E = \frac{380\ \text{J mol}^{-1} - 501\ \text{J mol}^{-1}}{328\ \text{K}} = \frac{-121\ \text{J mol}^{-1}}{328\ \text{K}} = -0.369\ \text{J K}^{-1}\text{mol}^{-1}

Step 3 — interpret sign and magnitude. A negative $S^E$ means the real mixture is more ordered than an ideal solution at this composition — plausible given that similarly-shaped, weakly-interacting molecules like benzene and cyclohexane pack with some residual local order beyond pure random mixing. Step 4 — compare magnitudes. The excess entropy term $TS^E = (328)(-0.369) = -121$ J/mol is about 24% the size of $H_m^E$ . It is a genuine but secondary contribution to $G^E$ — the enthalpic mismatch dominates, which is typical of "simple" nonpolar mixtures without hydrogen bonding.

Sanity check: $G_m^E = H_m^E - TS_m^E = 380 - (-121) = 501$ J/mol, matching Example 1 exactly — the three excess functions are mutually consistent, as they must be by construction.

Limiting cases & common traps

Situation	Behavior	Common error
$\gamma_i \to 1$ for all $x_i$	$G^E \to 0$ ; solution is ideal	Assuming $\gamma_i=1$ automatically implies $H^E=0$ too (true only if also $S^E=0$ )
$G^E > 0$ everywhere	positive deviation from Raoult's law; can produce a maximum-pressure azeotrope	Confusing sign of $G^E$ with sign of $H^E$ — they can differ
$G^E < 0$ everywhere	negative deviation; can produce a minimum-pressure azeotrope	Forgetting $G^E<0$ still permits $\gamma_i>1$ for one component if $S^E$ dominates
Dilute limit $x_i \to 0$	$\gamma_i \to \gamma_i^{\infty}$ (Henry's law constant regime)	Applying Raoult's-law-based $\gamma_i$ formula for the solute in the very dilute limit without switching reference state
$V^E \ne 0$	volumes are not additive; mixing can expand or contract	Assuming $V^E=0$ whenever $H^E \approx 0$ — these are independent excess functions

The chemist behind it

The systematic use of excess functions to characterize real solutions grew out of early twentieth-century work on non-electrolyte solution theory, most notably by Joel Henry Hildebrand, who spent decades measuring solubility, vapor pressure, and mixing behavior of nonpolar liquids at Berkeley. Hildebrand's insistence on treating deviations from ideality as directly measurable excess quantities — rather than fitting ad hoc empirical constants — set the experimental standard that later fed into regular-solution theory and the solubility-parameter framework still used in polymer and coatings chemistry today.

Origins · 1 minJoel Henry Hildebrand and how this was discovered

Joel Henry Hildebrand

1881–1983 · Pioneered quantitative measurement of nonideal liquid mixing and solubility, laying groundwork for regular-solution theory

Ask the tutor: How does a simple model like $G_m^E = Ax_1x_2$ generate the activity coefficient expressions used in regular-solution theory?

Why is G^E built from activity coefficients rather than measured directly like H^E or V^E?

Gibbs energy itself is not directly measurable — only differences and derivatives are. Activity coefficients are extracted from measurable quantities like vapor pressure or freezing-point depression, and then $G^E = RT\sum n_i\ln\gamma_i$ converts that measured nonideality into an energy. This indirect route is standard: $H^E$ and $V^E$ happen to be directly calorimetric or dilatometric, but $G^E$ almost always comes through equilibrium (activity) measurements instead.

Can V^E be nonzero even when H^E is essentially zero?

Yes, and this happens often. Ethanol–water mixtures show substantial volume contraction ( $V^E < 0$ ) alongside a non-negligible but comparatively modest enthalpy signature at some compositions. Volume and enthalpy excess functions arise from different aspects of molecular packing and interaction energy; there is no thermodynamic law forcing them to vanish together. Each excess function must be measured independently.

How do excess functions relate to activity coefficients that I will study next?

The molar excess Gibbs energy is literally $RT$ times the composition-weighted sum of $\ln\gamma_i$ values, so $G^E$ and the activity coefficients are two sides of one coin. Once you define a model for $G_m^E(x)$ , such as $G_m^E = Ax_1x_2$ , you can differentiate to get closed-form expressions for $\ln\gamma_1$ and $\ln\gamma_2$ . This differentiation step is exactly how regular-solution theory generates its activity coefficient formulas.

Why does the excess entropy S^E exclude the ideal combinatorial mixing entropy?

Because that combinatorial term, $-R\sum x_i\ln x_i$ , is present in every solution regardless of chemistry — it is a purely statistical counting effect. Including it in $S^E$ would swamp the genuinely chemical information about how real molecules order or disorder relative to random mixing. By subtracting it out, $S^E$ isolates the non-ideal ordering effects, such as local clustering or orientational preferences, that vary from system to system.

Is a positive G^E always a sign that a mixture will phase-separate?

Not necessarily. Positive $G^E$ raises the total Gibbs energy of mixing relative to ideal, but phase separation requires the total $\Delta_{mix}G$ curve, ideal plus excess, to develop a region of negative curvature (spinodal region). A small positive $G^E$ merely shifts activity coefficients above one; only sufficiently large positive $G^E$ , typically several kJ/mol, produces the double-well shape needed for a miscibility gap, the subject of regular-solution theory.