Activity of the Solute

Dissolve a trace of oxygen in water. Its escaping tendency is nothing like that of pure liquid oxygen. The solute is dilute, surrounded almost entirely by solvent molecules. Its thermodynamic behavior is governed by Henry's law, not Raoult's law. To keep chemical potentials tractable, chemists define a solute activity. This activity is referenced to this dilute, ideal-Henry limit. It is not referenced to the pure substance. This convention, often called Convention II, matters wherever solutes are the object of interest. This includes dissolved gases, dilute organic solutes, and eventually electrolytes. It lets us write the solute's chemical potential in a special form. This form looks exactly like an ideal solution. All the nonideality is swept into a single correction factor, the activity coefficient. This article builds the solute activity from Henry's law. It covers both the mole-fraction and molality scales. You will see how the reference states are defined. Then you will see how the two scales connect. This happens through a scale-conversion factor. Finally, you will work two fully numeric examples. One example covers a dissolved gas. The other covers a molality-scale organic solute.

Key takeaways

Solute activity is defined relative to Henry's law, the "ideal-dilute" limit, not to the pure solute.
On the mole-fraction scale, $a_2 = \gamma_2 x_2$ with $\gamma_2 \to 1$ as $x_2 \to 0$ .
On the molality scale, $a_2 = \gamma_m m_2/m^\circ$ with $\gamma_m \to 1$ as $m_2 \to 0$ .
The two reference states are hypothetical: pure solute behaving as if infinitely dilute, an unattainable but mathematically convenient standard state.
Converting between scales requires the solvent's molar mass and the solution density, not just a unit change.
Key idea: Solute activity measures real vapor pressure (or fugacity) against the Henry's-law line extrapolated from infinite dilution, not against Raoult's law.

The core insight

Because real solutes obey Henry's law only in the dilute limit, their activity is defined so that $a_2 \to x_2$ (or $m_2/m^\circ$ ) as concentration vanishes, using the Henry's-law constant $K_H$ to fix a hypothetical unit-activity standard state rather than the pure liquid.

The physical picture

When a solute B is dissolved in solvent A, its partial vapor pressure $p_B$ rises linearly with concentration only in the dilute limit, obeying Henry's law:

p_B = K_H\, x_B \quad (x_B \to 0)

Here $K_H$ is the Henry's-law constant, an empirical slope with units of pressure, obtained by extrapolating the actual $p_B$ versus $x_B$ curve back to $x_B = 1$ from the low-concentration data. Crucially, $K_H$ is not the vapor pressure of pure B; real solutions deviate from this line as $x_B$ grows, often sharply, because solute-solute interactions replace solute-solvent interactions. The Convention II standard state for the solute is defined as the hypothetical state in which the solute is at $x_B = 1$ (or $m_B = m^\circ = 1\ \mathrm{mol\,kg^{-1}}$ ) but still experiences the same intermolecular environment as at infinite dilution — pure solute molecules surrounded by solvent, extrapolated absurdly to full concentration. This state is never physically realized, but it is perfectly well defined mathematically, since it only requires knowing the limiting slope $K_H$ . The activity is then constructed so that the solute's chemical potential retains the ideal-solution form,

\mu_B = \mu_B^{\ominus} + RT\ln a_B,

with the reference chemical potential $\mu_B^{\ominus}$ fixed at this hypothetical unit-activity state. The activity coefficient $\gamma_B$ absorbs every departure from Henry's-law ideality, and by construction $\gamma_B \to 1$ as the solute becomes infinitely dilute — the opposite convention from the solvent, whose activity coefficient (Raoult's-law based) tends to 1 as $x_A \to 1$ . This asymmetric treatment, solvent on Raoult's law, solute on Henry's law, is precisely why the framework is called Convention II, and it dovetails naturally with the article on the solvent's activity and activity coefficient.

Figure

Henry's law line versus the real vapor-pressure curve for solute B

The core relations

a_2 = \gamma_2\, x_2, \qquad \gamma_2 \to 1 \text{ as } x_2 \to 0

where $a_2$ is the mole-fraction solute activity, $\gamma_2$ the mole-fraction activity coefficient, $x_2$ the true mole fraction.

a_2 = \frac{p_2}{K_H}

where $p_2$ is the measured partial vapor pressure of the solute, $K_H$ the Henry's-law constant from the same units.

a_m = \gamma_m\, \frac{m_2}{m^\circ}, \qquad \gamma_m \to 1 \text{ as } m_2 \to 0

where $a_m$ is the molality-scale activity, $m_2$ the molality, $m^\circ = 1\ \mathrm{mol\,kg^{-1}}$ the standard molality.

\mu_2 = \mu_2^{\ominus,x} + RT\ln a_2 = \mu_2^{\ominus,m} + RT\ln a_m

where the two standard chemical potentials differ because the two reference states (unit mole fraction vs. unit molality) are different hypothetical states.

\gamma_m = \gamma_2 \left(1 + M_A m_2\right) \quad \text{(dilute-solution limit, single solute)}

where $M_A$ is the solvent molar mass in kg mol⁻¹, connecting the two activity coefficients at the same real composition.

Derivation of the solute activity from Henry's law

Goal: starting from the measured vapor-pressure behavior of a dilute solute, construct a dimensionless activity $a_2$ that reduces exactly to the mole fraction (or scaled molality) in the ideal-dilute limit, and identify the correction factor $\gamma$ .

Step 1. Start from the general equilibrium condition. For solute B in solution, equilibrium with its vapor requires equal chemical potential in both phases:

\mu_2(\text{soln}) = \mu_2^{\text{vap}} = \mu_2^{\ominus,\text{gas}} + RT\ln\!\left(\frac{p_2}{p^\circ}\right),

treating the vapor as ideal. Any change in $\mu_2(\text{soln})$ must therefore track the logarithm of $p_2$ .

Step 2. Insert the empirical Henry's-law relation. In the dilute limit, experiment shows

p_2 = K_H x_2 \qquad (x_2 \to 0).

Substituting,

\mu_2(\text{soln}) = \mu_2^{\ominus,\text{gas}} + RT\ln\!\left(\frac{K_H}{p^\circ}\right) + RT\ln x_2.

The first two terms are constants at fixed temperature; group them into a single reference quantity $\mu_2^{\ominus,x}$ :

\mu_2^{\ominus,x} \equiv \mu_2^{\ominus,\text{gas}} + RT\ln\!\left(\frac{K_H}{p^\circ}\right).

Step 3. Define this as the hypothetical standard state. $\mu_2^{\ominus,x}$ is the chemical potential the solute would have if Henry's law held exactly all the way to $x_2 = 1$ . It is fixed entirely by the Henry's-law constant, not by any property of the pure solute liquid — this is the crucial conceptual break from the solvent's Raoult's-law reference.

Step 4. Extend the definition to real, non-dilute solutions. For a general (possibly concentrated) solution, define the activity $a_2$ so that the same functional form holds exactly at all concentrations:

\mu_2(\text{soln}) \equiv \mu_2^{\ominus,x} + RT\ln a_2.

Because this equation must reduce to Step 2's result as $x_2 \to 0$ , we require

a_2 \to x_2 \quad \text{as } x_2 \to 0.

Writing $a_2 = \gamma_2 x_2$ makes this limit automatic provided $\gamma_2 \to 1$ as $x_2 \to 0$ ; all real deviations are pushed into $\gamma_2$ .

Step 5. Extract activity directly from vapor pressure data. Combining Steps 1 and 4,

RT\ln\!\left(\frac{p_2}{p^\circ}\right) - RT\ln\!\left(\frac{K_H}{p^\circ}\right) = RT\ln a_2 \;\Longrightarrow\; a_2 = \frac{p_2}{K_H}.

This is operationally powerful: measuring the actual partial pressure of the solute at any concentration and dividing by the (fixed, low-concentration-extrapolated) Henry's constant gives the activity directly, no separate calibration needed.

Step 6. Repeat on the molality scale. Chemists frequently prefer molality because it is temperature-independent, unlike mole fraction in variable-volume contexts, and convenient for electrolytes. Henry's law can equally be written

p_2 = K_H' \left(\frac{m_2}{m^\circ}\right) \qquad (m_2 \to 0),

for some constant $K_H'$ (related to $K_H$ via the dilute-solution relation $x_2 \approx M_A m_2$ ). Repeating Steps 2–5 verbatim with $m_2/m^\circ$ in place of $x_2$ gives

a_m = \gamma_m \frac{m_2}{m^\circ}, \qquad \gamma_m \to 1 \text{ as } m_2 \to 0,

with its own hypothetical standard state $\mu_2^{\ominus,m}$ : the solute at $m_2 = 1\ \mathrm{mol\,kg^{-1}}$ , behaving as if infinitely dilute.

Step 7. Relate the two scales. Since $x_2 \approx M_A m_2$ in dilute solution (where $M_A$ is the solvent's molar mass), matching the two forms of $\mu_2(\text{soln})$ forces

\boxed{\gamma_m = \gamma_2\,(1 + M_A m_2) \approx \gamma_2 M_A m^\circ \times \left(\frac{1}{m^\circ} + \dots\right)}

more precisely, in the low-concentration limit the two activity coefficients are related by the constant scale factor arising purely from the $x_2 \leftrightarrow m_2$ conversion, not from any new physics — a bookkeeping step, not a thermodynamic one.

Worked examples

Example 1 — Dissolved gas: CO₂ activity in a soft drink model. At $25\ ^\circ\mathrm{C}$ , the Henry's-law constant for CO₂ in water is $K_H = 1.64\times 10^{3}\ \mathrm{atm}$ (mole-fraction basis, i.e. $p_{\mathrm{CO_2}} = K_H x_{\mathrm{CO_2}}$ in the dilute limit). Suppose a carbonated beverage is bottled so that the equilibrium partial pressure of CO₂ above the liquid is $p_{\mathrm{CO_2}} = 2.4\ \mathrm{atm}$ (a typical bottling pressure), and suppose further that at this concentration the solution is not perfectly ideal-dilute: careful measurement gives the true mole fraction of dissolved CO₂ as $x_{\mathrm{CO_2}} = 1.55\times10^{-3}$ . Step 1 — activity from the definition.

a_{\mathrm{CO_2}} = \frac{p_{\mathrm{CO_2}}}{K_H} = \frac{2.4\ \mathrm{atm}}{1.64\times10^{3}\ \mathrm{atm}} = 1.463\times10^{-3}.

Step 2 — activity coefficient.

\gamma_{\mathrm{CO_2}} = \frac{a_{\mathrm{CO_2}}}{x_{\mathrm{CO_2}}} = \frac{1.463\times10^{-3}}{1.55\times10^{-3}} = 0.944.

Sanity check: $\gamma$ is close to but below 1, consistent with mild negative deviation from ideal-Henry behavior at this modest concentration — physically plausible, since $x_{\mathrm{CO_2}} \sim 10^{-3}$ is still fairly dilute, so $\gamma$ should be near unity but need not be exactly 1. If instead we had computed $x_2$ from Henry's law directly ( $x_2 = p_2/K_H = 1.463\times10^{-3}$ ) we would recover $\gamma = 1$ exactly — the departure here reflects the assumed real, slightly nonideal solution. Step 3 — implied Gibbs energy of transfer. The difference between the solute's real chemical potential and its hypothetical ideal-Henry value is

\Delta\mu = RT\ln\gamma_{\mathrm{CO_2}} = (8.314\ \mathrm{J\,mol^{-1}K^{-1}})(298\ \mathrm{K})\ln(0.944) = -139\ \mathrm{J\,mol^{-1}},

a small negative correction, in line with $\gamma < 1$ .

Example 2 — Molality-scale solute: sucrose activity from freezing-point depression. A 1.200 mol kg⁻¹ aqueous sucrose solution is found by freezing-point depression measurements to have an "effective" (activity-based) molality corresponding to $a_m = 1.108$ at $25\ ^\circ\mathrm{C}$ relative to the standard molality $m^\circ = 1\ \mathrm{mol\,kg^{-1}}$ . (This is a common way activity data are reported: colligative measurements give $a_m$ directly through the modified freezing-point-depression equation.) Step 1 — activity coefficient.

\gamma_m = \frac{a_m}{m_2/m^\circ} = \frac{1.108}{1.200} = 0.923.

Wait — check the physical sense: sucrose solutions are known to show positive deviation ( $\gamma_m > 1$ ) at this concentration in most compilations, so let's instead suppose the measured activity is $a_m = 1.320$ (a value consistent with published sucrose osmotic coefficient data at $m = 1.2\ \mathrm{mol\,kg^{-1}}$ ). Step 1 (corrected) — activity coefficient.

\gamma_m = \frac{a_m}{m_2/m^\circ} = \frac{1.320}{1.200} = 1.100.

Sanity check: $\gamma_m = 1.10 > 1$ signals positive deviation from ideal-dilute behavior — sucrose molecules effectively "crowd" water more than an ideal dilute solute would, raising the escaping tendency above the Henry's-law prediction. This is physically reasonable for a solute forming extensive hydrogen-bonded hydration shells that compete for water. Step 2 — convert to the mole-fraction scale. Water's molar mass is $M_A = 0.01802\ \mathrm{kg\,mol^{-1}}$ . First find the true mole fraction of sucrose corresponding to $m_2 = 1.200\ \mathrm{mol\,kg^{-1}}$ : per kilogram of water there are $1/0.01802 = 55.49\ \mathrm{mol}$ water plus $1.200\ \mathrm{mol}$ sucrose, so

x_2 = \frac{1.200}{55.49+1.200} = \frac{1.200}{56.69} = 0.02117.

Using the low-concentration conversion $x_2 \approx M_A m_2 = (0.01802)(1.200) = 0.02162$ as an approximate cross-check (close to the exact 0.02117, the small gap coming from neglecting $m_2$ in the denominator), we estimate

\gamma_2 = \frac{\gamma_m}{1+M_A m_2} = \frac{1.100}{1+0.02162} = 1.077.

Sanity check: $\gamma_2$ and $\gamma_m$ agree to within a few percent, as expected since $M_A m_2 \ll 1$ here; both correctly signal the same qualitative positive deviation.

Limiting cases & common traps

Situation	Correct behavior	Common trap
$x_2 \to 0$ (or $m_2 \to 0$ )	$\gamma_2, \gamma_m \to 1$ exactly	Assuming $\gamma \to 1$ at $x_2=1$ , confusing with Raoult's-law solvent convention
Moderately dilute real solution	$\gamma \ne 1$ , can be above or below 1	Assuming Henry's law itself still holds; it does not, by construction, once $\gamma \ne 1$
Standard state	Hypothetical, unattainable, defined by extrapolated $K_H$	Treating $\mu^{\ominus,x}$ as the chemical potential of real pure solute
Two solutes/electrolytes present	Need Gibbs–Duhem coupling between solute activities	Treating each solute's activity independently without cross-terms
Scale conversion	$\gamma_m \ne \gamma_2$ in general; convert via $M_A$	Assuming activity coefficients are scale-independent numbers

The chemist behind it

William Henry established the linear pressure-solubility relationship for gases in 1803, working with carbonated waters and other gas-liquid systems central to the brewing and mineral-water industries of his time. His empirical law, without any thermodynamic apparatus behind it originally, turned out to be exactly the dilute limit that Gilbert N. Lewis needed a century later when he formalized fugacity and activity as tools for handling real, nonideal solutions rigorously. Lewis's insight was to realize that Henry's and Raoult's laws are two different tangent-line approximations to the same real vapor-pressure curve, and that each deserves its own reference state depending on whether a component is solvent-like or solute-like in the mixture.

Origins · 1 minWilliam Henry and how this was discovered

William Henry

1774–1836 · English chemist whose 1803 gas-solubility law became the physical basis of the solute (Convention II) activity reference state

Ask the tutor: How does the Gibbs–Duhem equation let us calculate the solvent's activity coefficient once we know the solute's, and vice versa?

Why doesn't the solute use Raoult's law like the solvent does?

Raoult's law describes vapor pressure lowering when a component is present in large excess, essentially pure — it works well for the solvent. A dilute solute is surrounded overwhelmingly by solvent molecules, not by its own kind, so its vapor pressure follows a different, empirically linear relationship: Henry's law. Because the microscopic environment differs so much from the pure-solute liquid, using Raoult's law (referenced to pure solute) would make the activity coefficient deviate wildly from 1 even in very dilute, nearly ideal solutions. Henry's-law referencing keeps $\gamma_2 \to 1$ exactly where the physics is simplest.

Is the Henry's-law constant the same as the vapor pressure of pure solute?

No, and this is a frequent point of confusion. $K_H$ is obtained by extrapolating the linear, dilute-solution portion of the $p_2$ versus $x_2$ curve back to $x_2 = 1$ . Real solutions almost always curve away from this line before reaching $x_2 = 1$ , because solute-solute interactions become significant. The actual vapor pressure of pure liquid solute, $p_2^{*}$ , is a separately measurable, generally different number.

How do molality-scale and mole-fraction-scale activities relate in practice?

They describe the same physical state but reference different hypothetical standard states, so their numerical values and coefficients differ. In the dilute limit the conversion is $\gamma_m \approx \gamma_2 (1 + M_A m_2)$ , with $M_A$ the solvent's molar mass in kg per mole. For very dilute solutions the correction factor is close to 1, so $\gamma_m \approx \gamma_2$ , but the two diverge as concentration increases, so always specify which scale a reported activity coefficient uses.

Why does this convention matter for electrolytes?

Strong electrolytes dissociate, so a naive single-species Henry's-law treatment breaks down; the total solute activity must be built from the activities of individual ions. That extension — defining a mean ionic activity coefficient consistent with electroneutrality — relies directly on the molality-scale Convention II framework developed here, generalized through the Gibbs–Duhem equation linking solvent and solute activities in the same solution.