Ideal-Dilute Solutions

Dissolve a little carbon dioxide in water and the fizz obeys a simple rule. The gas pressure above the liquid is proportional to its tiny mole fraction in solution. Dissolve a lot of ethanol in water, though, and neither component behaves so simply. This holds true across the whole range. Real mixtures are usually messy compromises between two clean limits. This article covers one of those limits: the ideal-dilute solution. Here, the abundant solvent obeys Raoult's law. At the same time, the sparse solute obeys Henry's law. This model matters because almost every solution chemists actually study is dilute. Trace pollutants in lakes, drugs in blood plasma, salts in seawater, gases in fermenting broth — all sit in the same regime. There, solute-solute contacts are rare. Solute molecules are surrounded almost entirely by solvent. Understanding why two different linear laws apply to the two components matters greatly. So does knowing how they connect through the Gibbs–Duhem equation. This is the conceptual bridge to activities and colligative properties. You will see why the solvent and solute obey different proportionality constants. You will derive the chemical potential of each component from first principles. Two fully worked numerical examples follow. Then come the common pitfalls. Finally, there is a look at the chemist whose gas-solubility law makes half of this story possible.

Key takeaways

An ideal-dilute solution is dilute enough that solute molecules interact only with solvent, never with each other.
The solvent obeys Raoult's law: $p_A = x_A p_A^*$ , exactly as in an ideal solution.
The solute obeys Henry's law: $p_B = K_B x_B$ , with $K_B \neq p_B^*$ in general.
Both laws are linear in mole fraction but with different slopes because the local environments of solvent and solute molecules are physically different.
The Gibbs–Duhem equation guarantees that if the solvent is Raoult-ideal, the solute must be Henry-ideal in the same concentration range.
Key idea: dilution makes solute-solvent interactions dominate so completely that solute chemical potential becomes linear in $\ln x_B$ , just with a different reference pressure than the solvent's.

The core insight

In a sufficiently dilute solution the solvent molecules experience almost pure-solvent surroundings while the solute molecules are essentially never in contact with another solute molecule. This asymmetry forces the solvent to follow Raoult's law, referenced to its own pure vapor pressure, and forces the solute to follow a different linear law, Henry's law, referenced to an empirical constant $K_B$ that encodes solute-solvent interaction strength rather than solute-solute interaction strength.

The physical picture

Picture a beaker of liquid water with a small amount of dissolved acetone. Zoom into the liquid at the molecular scale. A water molecule near the surface is surrounded by other water molecules almost all the time, because acetone is scarce. Its escaping tendency into the vapor phase is barely perturbed by the rare acetone neighbor, so it evaporates almost exactly as pure water would, just diluted by the mole fraction factor $x_A$ . That is Raoult's law behavior, and it improves as $x_A \to 1$ . Now follow an acetone molecule. It is never surrounded by other acetone molecules. It is essentially always surrounded by water. Its environment is nothing like pure liquid acetone. So its escaping tendency has nothing to do with the vapor pressure of pure acetone. Instead it responds to a proportionality constant, the Henry's law constant $K_B$ , that reflects how comfortable an acetone molecule is when solvated by water rather than by other acetone molecules. As $x_B \to 0$ , this linear relationship becomes exact. Both statements are limiting laws. They become increasingly exact as the solution becomes more dilute, and the two limits are geometrically the same limit: $x_A \to 1$ while $x_B \to 0$ . This is why we call the whole regime a single model, the ideal-dilute solution, rather than two separate approximations bolted together.

Figure

Partial vapor pressures of solvent A and solute B vs mole fraction of B, showing Raoult's-law and Henry's-law tangent lines

The key graphical fact is the tangent construction. At $x_B = 0$ , the actual $p_B$ curve is tangent to a straight line through the origin with slope $K_B$ . That slope is generally not equal to $p_B^*$ , the vapor pressure of pure liquid B, because the curve bends away from ideality as $x_B$ grows and B-B contacts become common. Meanwhile at $x_A = 1$ (the same point, since $x_A = 1-x_B$ ), the $p_A$ curve is tangent to the Raoult line with slope $p_A^*$ , and it stays close to that line for a wider range because the solvent is always nearly pure locally.

The core relations

p_A = x_A p_A^*

where: $p_A$ is the partial vapor pressure of solvent A, $x_A$ its mole fraction in solution, $p_A^*$ the vapor pressure of pure A at the same temperature.

p_B = K_B x_B

where: $p_B$ is the partial vapor pressure of solute B, $x_B$ its (small) mole fraction, $K_B$ the empirical Henry's law constant for B in this particular solvent.

\mu_A = \mu_A^* + RT\ln x_A

where: $\mu_A^*$ is the chemical potential of pure liquid A, valid as $x_A \to 1$ .

\mu_B = \mu_B^{\ominus} + RT\ln x_B, \qquad \mu_B^{\ominus} \equiv \mu_B^*(g\to l) + RT\ln\!\left(\frac{K_B}{p_B^*}\right)

where: $\mu_B^{\ominus}$ is a solute-specific reference chemical potential built from $K_B$ , not from $p_B^*$ alone; valid as $x_B \to 0$ .

x_A\, d\ln p_A + x_B\, d\ln p_B = 0 \quad \text{(Gibbs–Duhem, constant } T,p\text{)}

where: this constraint is what forces Henry-law linearity in B once Raoult-law linearity in A is assumed over the same range.

Derivation AFrom Raoult's law for the solvent to Henry's law for the solute

Goal. Show that if the solvent obeys Raoult's law exactly over some range of $x_A$ near 1, the Gibbs–Duhem equation forces the solute's partial pressure to be linear in $x_B$ over that same range — that is, Henry's law is not an independent postulate but a consequence.

Step 1. Write the Gibbs–Duhem equation for vapor pressures. At constant temperature and total pressure, the molar Gibbs energies (chemical potentials) of the two components in equilibrium with an ideal vapor mixture satisfy

x_A\, d\mu_A + x_B\, d\mu_B = 0.

Since each vapor-phase component behaves ideally, $\mu_i = \mu_i^{\ominus}(g) + RT\ln p_i$ , so $d\mu_i = RT\, d\ln p_i$ . Substituting,

x_A\, d\ln p_A + x_B\, d\ln p_B = 0.

Step 2. Insert the assumed Raoult behavior of the solvent. If $p_A = x_A p_A^*$ holds exactly (with $p_A^*$ constant at fixed $T$ ), then

d\ln p_A = d\ln x_A = \frac{dx_A}{x_A}.

Because $x_A + x_B = 1$ , we have $dx_A = -dx_B$ , so

d\ln p_A = -\frac{dx_B}{x_A}.

Step 3. Substitute back into Gibbs–Duhem.

x_A\left(-\frac{dx_B}{x_A}\right) + x_B\, d\ln p_B = 0 \quad\Longrightarrow\quad x_B\, d\ln p_B = dx_B.

Divide through by $x_B$ :

d\ln p_B = \frac{dx_B}{x_B} = d\ln x_B.

Step 4. Integrate.

\ln p_B = \ln x_B + \text{constant} \quad\Longrightarrow\quad p_B = (\text{constant}) \times x_B.

Call that constant $K_B$ . The integration constant is fixed by matching to the actual limiting slope of the real $p_B(x_B)$ curve as $x_B \to 0$ ; it need not equal $p_B^*$ , because the derivation only used the functional form of Raoult's law for A, not any assumption about B's pure-liquid behavior.

Step 5. Boxed result.

\boxed{p_B = K_B x_B, \qquad K_B \equiv \lim_{x_B \to 0} \frac{p_B}{x_B}}

Henry's law for the solute is the mathematically necessary partner of Raoult's law for the solvent, whenever both are taken as exact limiting laws over the same dilute range. This is why the two empirical rules, discovered independently by Raoult and Henry decades apart, fit together so neatly into one thermodynamic model.

Worked examples

Example 1 — Foundational: vapor pressures over a dilute solution of naphthalene in benzene. Benzene (A) has $p_A^* = 100.0\ \text{Torr}$ at a chosen temperature. Naphthalene (B) is dissolved in benzene at $x_B = 0.0250$ (dilute; naphthalene is the solute, benzene the solvent). Independent Henry's law data for naphthalene in benzene at this temperature give $K_B = 250\ \text{Torr}$ . Step 1: solvent partial pressure via Raoult's law.

x_A = 1 - x_B = 1 - 0.0250 = 0.9750

p_A = x_A p_A^* = (0.9750)(100.0\ \text{Torr}) = 97.50\ \text{Torr}

Step 2: solute partial pressure via Henry's law.

p_B = K_B x_B = (250\ \text{Torr})(0.0250) = 6.25\ \text{Torr}

Step 3: total vapor pressure and vapor composition.

p_{\text{total}} = p_A + p_B = 97.50 + 6.25 = 103.75\ \text{Torr}

y_B = \frac{p_B}{p_{\text{total}}} = \frac{6.25}{103.75} = 0.0602

Sanity check. The vapor is enriched in naphthalene relative to the liquid ( $y_B = 0.0602 > x_B = 0.0250$ ) because $K_B > p_A^*$ : naphthalene escapes more readily per mole fraction than benzene does, consistent with $K_B$ exceeding the solvent's own vapor pressure. Note this comparison is coincidental to these particular numbers — a solute with $K_B < p_A^*$ would instead be depleted in the vapor. Both partial pressures used the linear laws correctly, each with its own proportionality constant, and $x_B = 0.025$ is safely in the dilute regime where both approximations are trustworthy.

Example 2 — Applied: Henry's law constant and molal solubility of CO₂ in water at 25 °C, and comparison with the ideal-dilute chemical potential. The Henry's law constant for CO₂ in water at 25 °C, expressed as $K_B = 1.64\times 10^{3}\ \text{atm per unit mole fraction}$ (a standard tabulated value), lets us find how much CO₂ dissolves under a CO₂ partial pressure of $p_{\text{CO}_2} = 1.00\ \text{atm}$ (roughly the pressure in a freshly capped carbonated beverage). Step 1: solve Henry's law for the equilibrium mole fraction.

x_{\text{CO}_2} = \frac{p_{\text{CO}_2}}{K_B} = \frac{1.00\ \text{atm}}{1.64\times 10^{3}\ \text{atm}} = 6.10\times 10^{-4}

Step 2: convert mole fraction to molality. In 1 kg of water there are

n_{\text{H}_2\text{O}} = \frac{1000\ \text{g}}{18.02\ \text{g mol}^{-1}} = 55.5\ \text{mol}.

Since $x_{\text{CO}_2}$ is small, $n_{\text{CO}_2} \approx x_{\text{CO}_2} \times n_{\text{H}_2\text{O}}$ :

n_{\text{CO}_2} = (6.10\times 10^{-4})(55.5\ \text{mol}) = 3.39\times 10^{-2}\ \text{mol}.

This is dissolved in 1 kg of solvent, so

b_{\text{CO}_2} = 3.39\times 10^{-2}\ \text{mol kg}^{-1} = 0.0339\ \text{m}.

Step 3: express the solute's chemical potential relative to its Henry-law reference. Using $\mu_B = \mu_B^{\ominus} + RT\ln x_B$ with $T = 298\ \text{K}$ ,

RT\ln x_B = (8.314\ \text{J K}^{-1}\text{mol}^{-1})(298\ \text{K})\ln(6.10\times 10^{-4})

= (2478\ \text{J mol}^{-1})(-7.402) = -1.834\times 10^{4}\ \text{J mol}^{-1} = -18.3\ \text{kJ mol}^{-1}.

So $\mu_{\text{CO}_2} = \mu_{\text{CO}_2}^{\ominus} - 18.3\ \text{kJ mol}^{-1}$ under these conditions: the dissolved CO₂ has lower chemical potential than in its Henry's-law standard state, exactly as expected since $x_B \ll 1$ makes $\ln x_B$ strongly negative. Sanity check. A molality of 0.034 m for dissolved CO₂ at 1 atm partial pressure and 25 °C is the right order of magnitude for carbonated water (commercial soda is bottled at several atm of CO₂ headspace precisely to push far more gas into solution than 1 atm would allow, consistent with $x_{\text{CO}_2} \propto p_{\text{CO}_2}$ ).

Limiting cases & common traps

Situation	What happens	Common error
$x_B \to 0$ exactly	Henry's law becomes exact; solute is "infinitely dilute"	Assuming Henry's law holds at any concentration, even $x_B \sim 0.3$
$x_A \to 1$ exactly	Raoult's law becomes exact for solvent	Forgetting the solvent still needs Raoult's law, not Henry's law
Solute self-associates or dissociates (e.g. weak acid, ionic salt)	Effective $K_B$ or apparent mole fraction shifts; simple Henry's law needs modification	Applying raw Henry's law to electrolytes without accounting for dissociation
$K_B \approx p_B^*$ (rare, chemically similar species)	Solution approaches a fully ideal solution over the whole range, not just dilute limit	Concluding $K_B = p_B^*$ always — it is generically false
Non-dilute intermediate concentrations	Neither law holds; only activities (Article: Activity of the Solute) describe it correctly	Extrapolating dilute-limit slopes into the concentrated regime

The chemist behind it

The solvent side of this story belongs to François-Marie Raoult, whose systematic vapor-pressure measurements on many solvent-solute pairs in the 1880s established that $p_A/p_A^*$ tracks mole fraction for the majority component. The solute side belongs to William Henry, who in 1803 measured how the volume of gas absorbed by water scales linearly with the pressure of gas above it — the earliest quantitative statement of what we now call Henry's law, originally framed for the solubility of gases rather than general solutes. It took nearly a century, and the thermodynamic apparatus of chemical potential built by Gibbs and later systematized by Lewis, to recognize that these two empirical rules are not separate curiosities but two faces of a single dilute-solution limit, linked rigorously by the Gibbs–Duhem equation as shown above.

Origins · 1 minWilliam Henry and how this was discovered

William Henry

1774–1836 · English chemist and physician whose 1803 gas-absorption experiments gave the linear solute law that bears his name

Ask the tutor: How does the ideal-dilute model change when the solute is a strong electrolyte that dissociates into ions?

Is an ideal-dilute solution the same thing as an ideal solution?

No. An ideal solution (Raoult's law for every component over the entire composition range) requires A and B molecules to be so similar that A-A, B-B, and A-B interactions are essentially identical — benzene and toluene are the classic near-example. An ideal-dilute solution only requires that the solute be scarce enough that solute-solute contacts are negligible; the solvent can be chemically very different from the solute. Ideal-dilute behavior is far more common in practice, since most real solutions are not composed of near-identical molecules.

Why does the solute's reference chemical potential

\mu_B^{\ominus}

depend on

K_B

instead of

p_B^*

The standard state for a Henry's-law solute is a hypothetical state at $x_B = 1$ that preserves the dilute-solution proportionality constant $K_B$ , not the real pure liquid. This hypothetical state usually cannot be physically realized — pure liquid B does not behave as if every molecule still saw only solvent neighbors. Building $\mu_B^{\ominus}$ from $K_B$ keeps the reference state consistent with the actual linear law the solute obeys near $x_B = 0$ .

Does temperature affect whether a solution is ideal-dilute?

Temperature changes $p_A^*$ and $K_B$ individually, and can shift how wide the dilute range is, but it does not change the qualitative picture. At any fixed temperature, sufficiently small $x_B$ will always show Raoult behavior for A and Henry behavior for B, because the argument rests on molecular environment statistics, not on any specific numerical value of vapor pressure.

How is this model used later for colligative properties and solute activities?

Colligative properties (boiling-point elevation, freezing-point depression, osmotic pressure) are derived by holding the solvent's Raoult-law chemical potential expression and asking how much solute must be added to shift the solvent's equilibrium. The solute's own activity, needed when Henry's law itself starts failing at higher concentration, is defined so that it reduces exactly to $x_B$ in the ideal-dilute limit — this article's Henry's-law regime is the anchor point for that definition.