Activity and Fugacity

Compress a real gas hard enough and it stops obeying $pV = nRT$ . Its chemical potential no longer tracks $\ln p$ cleanly, yet the equilibrium machinery built on chemical potential still needs to work. Chemists rescue the simple ideal-gas formula by inventing two "corrected" pressures and concentrations: fugacity for gases and activity for everything else, including solutions and solids. This matters because real equilibrium constants, real solubilities, and real electrode potentials are governed by activities, not concentrations. Ignoring that gap causes systematic errors in exactly the regimes chemists care about: concentrated electrolytes, high-pressure gases, and non-ideal mixtures. This article builds the activity and fugacity framework from the ideal-gas chemical potential. You will see how fugacity is defined and measured. Then you will see how activity generalizes it to any phase. Finally you will connect both to the equilibrium constant used in the next topic.

Key takeaways

Fugacity $f$ replaces pressure in $\mu = \mu^\ominus + RT\ln(f/p^\ominus)$ so the ideal-gas equation form survives for real gases.
The fugacity coefficient $\phi = f/p$ measures the deviation from ideality; $\phi \to 1$ as $p \to 0$ .
Activity $a$ is the universal generalization: $\mu = \mu^\ominus + RT\ln a$ , valid for gases, solutes, solvents, and pure condensed phases.
Activity coefficients $\gamma$ (solutes) and $\phi$ (gases) absorb all the intermolecular-interaction physics that ideal models ignore.
Standard states are chosen by convention (1 bar for gases, 1 mol/kg or hypothetical ideal 1 M for solutes), and $a$ is always dimensionless.
Key idea: Real chemical potentials keep the ideal-gas logarithmic form if you replace pressure or concentration with an effective "as-if-ideal" quantity, activity, whose deviation from unity is quantified by an activity or fugacity coefficient.

The core insight

Every deviation of a real system from ideal behavior can be absorbed into a single dimensionless correction factor multiplying the ideal variable, so that $\mu = \mu^\ominus + RT\ln a$ remains exactly true by definition, with all the real physics hidden inside $a$ .

The physical picture

For an ideal gas, the chemical potential derivation gives a clean result: as pressure rises, molecules gain chemical potential logarithmically, because compressing an ideal gas costs reversible work with no complications from intermolecular forces. That derivation assumed molecules do not interact except through elastic collisions. Real molecules attract each other at moderate range and repel at short range. At low pressure the average intermolecular spacing is huge, so ideal-gas behavior is an excellent approximation. As pressure climbs, attraction dominates, lowering the energy per particle below the ideal value; at very high pressure, repulsion dominates and raises it. Neither the attraction nor the repulsion is captured by $\ln p$ . Fugacity is built by asking: what pressure would an ideal gas need, at the same temperature, to have exactly the chemical potential this real gas actually has? That "effective pressure" is $f$ , and it absorbs the entire deviation. The same problem appears in solution. An ideal-dilute solute's chemical potential should scale with $\ln(\text{concentration})$ , but real solute molecules interact with each other and with solvent, especially at higher concentration or when charged (ions screen and cluster). Activity plays exactly the same corrective role as fugacity, but for any phase, any standard state, any species. It is the single unifying tool: gases use fugacity as a special case of activity referenced to 1 bar; solutes use activity referenced to a standard molality or concentration; solvents and pure solids use activity referenced to the pure substance.

Figure

Real vs ideal chemical potential as a function of pressure, showing where fugacity coefficient departs from 1

Figure

Activity as a unifying ladder above the ideal chemical potential formula

The core relations

\mu_{\text{real gas}} = \mu^\ominus + RT\ln\left(\frac{f}{p^\ominus}\right), \qquad f = \phi p

where $f$ is fugacity, $\phi$ the fugacity coefficient, $p^\ominus = 1\text{ bar}$ , and $\phi \to 1$ as $p \to 0$ .

\ln\phi = \int_0^{p} \frac{Z - 1}{p'}\, dp' \quad \text{(constant } T\text{)}

where $Z = pV_m/RT$ is the compression factor, obtained from an equation of state or PVT data.

\mu = \mu^\ominus + RT\ln a

where $a$ is the (dimensionless) activity, the single generalized variable valid in any phase.

a_{\text{solute}} = \gamma_m \frac{m}{m^\ominus}, \qquad \gamma_m \to 1 \text{ as } m \to 0

where $\gamma_m$ is the molality-based activity coefficient and $m^\ominus = 1\text{ mol kg}^{-1}$ .

a_{\text{solvent}} = \gamma_x x, \qquad \gamma_x \to 1 \text{ as } x \to 1

where $x$ is mole fraction and the solvent's own pure liquid is the reference state.

\ln\gamma_\pm = -\frac{|z_+z_-| A \sqrt{I}}{1 + B\sqrt{I}} \quad \text{(Debye–Hückel, extended form)}

where $I$ is ionic strength, $z_\pm$ ionic charges, and $A \approx 0.509\ \text{kg}^{1/2}\text{mol}^{-1/2}$ in water at 25 °C.

Derivation AFrom ideal-gas μ to fugacity

Goal. Show that replacing $p$ by fugacity $f = \phi p$ restores the exact ideal-gas-shaped expression for chemical potential of a real gas, and derive the working formula for $\ln\phi$ .

Step 1. Start from the exact thermodynamic identity. At constant temperature, for any pure substance,

d\mu = V_m\, dp .

This follows directly from the Gibbs–Duhem relation applied to one mole; it holds whether or not the gas is ideal.

Step 2. Integrate for the ideal gas (reference calculation). Using $V_m = RT/p$ ,

\mu^{\text{ideal}}(p) = \mu^\ominus + RT\ln\left(\frac{p}{p^\ominus}\right).

This is the familiar chemical-potential result already established for an ideal gas; it is exact only because $V_m = RT/p$ exactly for an ideal gas.

Step 3. Integrate for the real gas using the actual molar volume.

\mu^{\text{real}}(p) - \mu^{\text{real}}(p\to 0) = \int_0^{p} V_m^{\text{real}}\, dp'.

This integral cannot be evaluated in closed form without an equation of state, and it diverges awkwardly at $p \to 0$ if handled naively, so we work with the difference from ideal behavior instead.

Step 4. Define fugacity so the real expression keeps the ideal form. Demand

\mu^{\text{real}}(p) = \mu^\ominus + RT\ln\left(\frac{f}{p^\ominus}\right),

with the boundary condition $f \to p$ as $p \to 0$ (real gases become ideal at vanishing pressure). This is a definition of $f$ , not an approximation — it is always possible because $\mu$ is a monotonic function of $f$ .

Step 5. Extract the fugacity coefficient. Write $f = \phi p$ and subtract Step 2 from Step 4:

RT\ln\phi = \mu^{\text{real}}(p) - \mu^{\text{ideal}}(p) = \int_0^p \left(V_m^{\text{real}} - \frac{RT}{p'}\right)dp'.

Divide through by $RT$ and use $V_m^{\text{real}} = ZRT/p$ where $Z$ is the compression factor:

\ln\phi = \int_0^p \frac{Z-1}{p'}\, dp'.

This integral is finite because $Z - 1 \to 0$ linearly as $p \to 0$ (the second virial coefficient controls the leading deviation), curing the apparent divergence from Step 3.

Step 6. Interpret the boxed result.

\boxed{\ f = \phi\, p, \qquad \ln\phi = \int_0^p \frac{Z(p') - 1}{p'}\, dp' \ }

Given $Z(p)$ from an equation of state (van der Waals, virial, or tabulated PVT data), $\phi$ — and hence $f$ — is computable at any pressure and temperature. The chemical potential $\mu = \mu^\ominus + RT\ln(f/p^\ominus)$ is then exact, not approximate.

Worked examples

Example 1 — Fugacity coefficient of N₂ from the virial equation. At 25 °C (298.15 K), nitrogen's second virial coefficient is $B = -4.5\ \text{cm}^3\text{mol}^{-1}$ (a small, realistic value near room temperature). Using the truncated virial equation,

Z = 1 + \frac{Bp}{RT},

so $Z - 1 = Bp/RT$ , a linear function of $p$ . Substitute into the fugacity-coefficient integral:

\ln\phi = \int_0^p \frac{Bp'/RT}{p'}\, dp' = \frac{Bp}{RT}.

At $p = 50\text{ bar} = 50\times 10^5\text{ Pa}$ , with $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ , $T = 298.15\text{ K}$ :

\frac{Bp}{RT} = \frac{(-4.5\times10^{-6}\text{ m}^3\text{mol}^{-1})(50\times10^5\text{ Pa})}{(8.314)(298.15)} = \frac{-22.5\ \text{J mol}^{-1}}{2478.8\ \text{J mol}^{-1}} = -9.077\times10^{-3}.

So $\ln\phi = -0.00908$ , giving $\phi = e^{-0.00908} = 0.9910$ . Fugacity: $f = \phi p = 0.9910 \times 50\ \text{bar} = 49.55\ \text{bar}$ . Sanity check: at 50 bar, N₂ is only mildly non-ideal (attractive forces slightly dominate, $\phi < 1$ ), consistent with a small negative $B$ and a 0.9% correction. The chemical potential is

\mu - \mu^\ominus = RT\ln\left(\frac{49.55}{1}\right) = (8.314)(298.15)\ln(49.55) = 2478.8 \times 3.903 = 9675\ \text{J mol}^{-1} = 9.68\ \text{kJ mol}^{-1},

compared with the (wrong) ideal-gas estimate $RT\ln(50) = 2478.8 \times 3.912 = 9698\ \text{J mol}^{-1}$ — a 23 J mol⁻¹ overestimate, small but real.

Example 2 — Mean activity coefficient of aqueous CaCl₂ and its effect on an equilibrium calculation. A 0.010 mol/kg solution of CaCl₂ dissociates into Ca²⁺ ( $z_+=2$ ) and 2 Cl⁻ ( $z_-=-1$ ). Compute the ionic strength:

I = \tfrac{1}{2}\sum_i m_i z_i^2 = \tfrac{1}{2}\big[(0.010)(2)^2 + (0.020)(1)^2\big] = \tfrac{1}{2}(0.040 + 0.020) = 0.030\ \text{mol kg}^{-1}.

Apply the Debye–Hückel limiting law (valid for dilute solutions, $I < 0.01$ – $0.1$ as a rough guide):

\log_{10}\gamma_\pm = -|z_+ z_-|\, A\sqrt{I} = -(2)(0.509)\sqrt{0.030}.

Compute $\sqrt{0.030} = 0.1732$ , so

\log_{10}\gamma_\pm = -(2)(0.509)(0.1732) = -0.1763,

\gamma_\pm = 10^{-0.1763} = 0.666.

This is a substantial 33% deviation from ideality even at only 0.01 mol/kg overall salt concentration — the $z_+z_-$ factor of 2 for a 2:1 electrolyte amplifies the effect strongly compared to a 1:1 salt at the same molality. Now use this to correct a solubility-type calculation. Suppose a sparingly soluble salt equilibrium requires the activity of Ca²⁺ to satisfy $a_{\text{Ca}^{2+}} = 0.010$ (from a $K_{sp}$ calculation assuming ideality). The actual molality needed is:

a = \gamma_\pm \frac{m}{m^\ominus} \implies m = \frac{a}{\gamma_\pm} = \frac{0.010}{0.666} = 0.0150\ \text{mol kg}^{-1}.

Ignoring activity coefficients would underestimate the true solubility by 50% (0.0150 vs. 0.0100 mol/kg) — a large, chemically significant error, showing why $K_{sp}$ tables and solubility predictions in ionic solutions demand activity corrections, not raw concentrations.

Limiting cases & common traps

Situation	What happens	Correct handling
$p \to 0$ (dilute gas)	$\phi \to 1$ , $f \to p$	Ideal-gas chemical potential is recovered exactly
$m \to 0$ (dilute solute)	$\gamma \to 1$ , $a \to m/m^\ominus$	Use ideal-dilute (Henry's law) form
Pure solid or liquid in equilibrium	Activity defined as exactly 1	Never write $a =$ mole fraction $= 1$ as an approximation; it's exact by convention
High pressure gas	$Z$ deviates strongly from 1, sign of $\ln\phi$ can flip	Must integrate real $Z(p)$ data or use a full equation of state, not virial truncation
Concentrated electrolyte ( $I > 0.1$ mol/kg)	Debye–Hückel limiting law fails	Use extended Debye–Hückel or Davies equation

The chemist behind it

Gilbert N. Lewis introduced both fugacity and activity in the early twentieth century, driven by a very practical frustration: real gases and real solutions refused to obey the beautifully simple ideal formulas, yet chemists needed a way to keep using logarithmic, additive free-energy relationships for engineering and equilibrium calculations. Rather than abandoning the ideal-gas-shaped equation, Lewis generalized the variable itself — first with fugacity for gases (1901), then more broadly with activity (with Merle Randall, in their influential 1923 textbook Thermodynamics and the Free Energies of Chemical Substances). This move let thermodynamics keep its elegant mathematical skeleton while quietly absorbing all real-world complexity into a single measurable correction factor.

Origins · 1 minGilbert N. Lewis and how this was discovered

Gilbert N. Lewis

1875–1946 · American physical chemist who invented fugacity and activity, and later proposed the electron-pair covalent bond and a generalized acid–base theory

Ask the tutor: How does this activity framework combine with $\Delta_rG^\ominus = -RT\ln K$ to define the true thermodynamic equilibrium constant $K_a$ ?

Why not just use concentration or pressure directly in equilibrium constants?

Because the chemical potential of a real substance is not exactly proportional to $\ln(\text{concentration})$ or $\ln(\text{pressure})$ once intermolecular interactions matter. Using raw concentration works well when solutions are dilute or gases are near-ideal, but breaks down at higher concentrations or pressures, exactly where many industrially and biologically important equilibria occur (concentrated electrolytes, high-pressure synthesis gases, dense polymer solutions). Activity is the corrected variable that keeps the thermodynamic relations exact in all these regimes, so equilibrium constants expressed in activities remain genuinely constant with composition, while "concentration quotients" drift.

Is fugacity just another word for pressure?

No. Fugacity has units of pressure and reduces to pressure in the ideal-gas limit, but it is a distinct thermodynamic quantity: the effective pressure an ideal gas would need to reproduce the real gas's actual chemical potential. At high pressure or low temperature, $f$ and $p$ can differ by tens of percent. Fugacity coefficients can be greater or less than 1 depending on whether repulsive or attractive intermolecular forces dominate at that state point, so fugacity is not simply "corrected pressure in one direction."

Why is the standard state for solutes chosen as a hypothetical 1 mol/kg ideal solution rather than the pure solute?

Because most solutes of interest, especially ions, cannot exist as a stable pure liquid phase at the temperature and pressure of interest, and even when they can, real solute behavior at high concentration is dominated by solute–solute interactions rather than resembling the pure substance. Defining the standard state as the ideal-dilute limit extrapolated to 1 mol/kg keeps the reference state physically meaningful (it corresponds to how the solute behaves when highly diluted) while giving a concrete numerical anchor for tabulating $\mu^\ominus$ values.

How is fugacity actually measured experimentally?

Fugacity is inferred rather than measured directly, since it's a defined thermodynamic construct. The standard route uses PVT (pressure–volume–temperature) data to compute the compression factor $Z(p,T)$ across a pressure range, then numerically or analytically integrates $\ln\phi = \int_0^p (Z-1)/p'\, dp'$ . Alternatively, equations of state (van der Waals, Redlich–Kwong, Peng–Robinson) fitted to a handful of experimental critical constants can generate $\phi$ predictively without needing dense PVT tables for every gas.

Does this framework foreshadow how equilibrium constants are actually defined?

Yes. The true thermodynamic equilibrium constant $K$ is defined entirely in terms of activities of products and reactants raised to their stoichiometric powers, which is exactly why $K$ is truly dimensionless and truly constant at fixed temperature — unlike a "concentration quotient" that drifts with ionic strength or pressure. The next step builds $K$ directly from $\Delta_rG^\ominus = -RT\ln K$ using the activity framework developed here.