Ideal-Dilute Solutions and Henry's Law

Introduction

The solvent in a dilute solution often behaves nearly like its pure liquid, but a dissolved solute occupies a very different environment from a pure liquid of that solute. Raoult's law is therefore usually the natural limiting description of the solvent, while Henry's law describes the dilute solute. This distinction matters especially for a gas dissolving into a liquid and for thermodynamic standard states.

Core explanation

Choose a clear Henry-law convention. Let x B be the liquid mole fraction of dilute solute B and p B its equilibrium partial pressure in a gas phase. Write p B = K H,x x B as x B → 0 at fixed temperature, where K H,x has pressure units. Under this convention a larger K H,x means that a given dissolved mole fraction requires a larger gas pressure; it corresponds to less favourable dissolution. Some references instead write dissolved concentration c B = k H p B, where k H has units such as mol L⁻¹ bar⁻¹. These constants are related but not numerically identical, and verbal claims that “a larger Henry constant means more soluble” depend on which convention is used.

For the solvent A, the high-x A limit is p A ≈ x A p A , with p A the pure-solvent vapour pressure. For the dilute solute B, p B ≈ K H,x x B, and K H,x need not equal the vapour pressure of pure B. This is the ideal-dilute model. Raoult and Henry lines can have different slopes on a pressure–composition graph because unlike-molecule interactions affect a trace solute differently from the solvent in its nearly pure environment.

The corresponding solute chemical potential may be written μ B = μ B,H° + RT ln x B in the Henry-standard convention for an ideal-dilute solution. The reference μ B,H° is obtained by extrapolating the dilute linear law to a hypothetical state with x B = 1. That state need not be physically realisable; its role is to set a consistent zero for the logarithmic activity. For a real solute, use activity a B,H = γ B,H x B, with γ B,H → 1 as x B → 0 under this convention.

Henry behaviour holds only over a concentration range where proportionality is a good approximation. Strong association, chemical reaction with the solvent, ionisation, or high solute concentration can change the relationship. Temperature affects the constant, so a value quoted without temperature is incomplete. Pressure may also affect the solvent and gas nonideality, requiring fugacity instead of plain p B in careful high-pressure work.

Step-by-step reasoning

State whether the question uses p B = K H,x x B or c B = k H p B and record the constant's units. At a fixed stated temperature, substitute measured pressure or composition into the chosen equation. Keep solvent Raoult behaviour separate from solute Henry behaviour. Before extending the result, check whether the solution is actually dilute and whether the solute reacts or dissociates.

Visual explanation

Draw partial pressure against liquid mole fraction for a binary mixture. Near x A = 1, the solvent curve follows a tangent of slope p A . Near x B = 0, the solute curve has tangent slope K H,x. Extend the solute tangent to x B = 1 with a dashed line to show the hypothetical Henry standard, which need not equal pure B's real vapour pressure.

Real-world analogy

A visitor joining a crowded group of residents experiences a different setting from residents among themselves. The visitor's first few additions can follow a predictable limiting trend, but extrapolating that trend to a room full of visitors may be unrealistic. Dilute solute and nearly pure solvent similarly have distinct limiting reference behaviours.

Real-world example

Carbon dioxide dissolves in water in response to its partial pressure above the liquid. At sufficiently low concentration and fixed temperature, dissolved amount is approximately proportional to gas pressure. In real water, hydration and acid–base equilibria complicate the exact species counted, so a practical Henry constant must specify whether it refers to molecular CO₂ or an analytical total.

Why?

At equilibrium, solute chemical potential in the liquid equals that in the gas. In the dilute limit, the liquid chemical potential has a logarithmic dependence on x B, while an ideal gas has a logarithmic dependence on p B. Equating them yields p B proportional to x B, with the proportionality constant encoding the difference between chosen standard chemical potentials.

Common misconception

Henry's law is not a universal law saying all dissolved gases obey one constant at every pressure. Its form and constant units must be specified. Another error is to use pure B's vapour pressure in place of K H,x for a dilute nonideal solute. The solvent may obey Raoult's limiting law at the same time the solute obeys Henry's limiting law.

Worked example

At 298 K, suppose the convention p B = K H,x x B uses K H,x = 500 bar for a dilute gas in a solvent. With gas partial pressure p B = 0.050 bar, the limiting dissolved mole fraction is x B ≈ p B/K H,x = 0.050/500 = 1.0 × 10⁻⁴. If p B doubles while the dilute approximation remains valid, x B doubles. The numerical constant is pressure per mole fraction, not the coefficient in c B = k Hp B.

Quick check

1. In p B = K H,x x B, what are the units of K H,x? Answer: Pressure units, because x B is dimensionless and p B is a pressure. A constant in the alternative c B = k Hp B form has different units and reciprocal-style solubility interpretation.

Exam focus

State the Henry convention before calculations and keep its temperature fixed. Use x B for the dilute liquid solute and p B for its gas partial pressure, not total gas pressure unless B is the only gas. Identify the solvent's Raoult limit separately. Mention that a hypothetical standard-state extrapolation need not correspond to pure solute.

Advanced insight

Gibbs–Duhem consistency links solvent and solute behaviours: as the solute approaches infinite dilution and follows Henry's limiting law, the solvent approaches Raoult's limiting law. This does not imply a fully ideal solution at intermediate composition; deviations can be substantial away from the endpoints.

Summary

An ideal-dilute solution uses Henry's law for trace solute and Raoult's law for nearly pure solvent. With the explicit convention p B = K H,x x B, the constant has pressure units. A Henry-standard solute chemical potential uses an extrapolated reference, and real solutions need activities when concentration moves beyond the dilute range.

Practice questions

1. Under p B = K H,x x B, K H,x = 200 bar and x B = 2.0 × 10⁻³. Find p B. Answer: p B = 200 × 2.0 × 10⁻³ = 0.40 bar, assuming the dilute law is valid at that composition and temperature. 2. Under the same convention, which gas is more soluble at the same partial pressure: one with K H,x = 100 bar or one with K H,x = 1000 bar? Answer: The 100-bar gas, because x B = p B/K H,x is larger for the smaller constant. The conclusion would be described differently under an alternative Henry-constant definition. 3. Why may the Henry line extrapolated to x B = 1 fail to describe pure liquid B? Answer: The dilute law reflects B surrounded mainly by solvent A. At x B = 1, B is surrounded by B, so interactions and chemical potential differ; the extrapolated reference is hypothetical.