Activities Versus Concentrations

Introduction

When you first met Kc, it was written with concentrations in square brackets, and it seemed perfectly constant. Careful measurements show otherwise: the "constant" for a weak acid measured in pure water differs slightly from the value measured in a salt solution. The explanation is that ions do not behave as independent particles. Chemists therefore define the true equilibrium constant using activities , effective concentrations that allow for the interactions between ions.

Core explanation

What activity means. The activity of a solute is its effective concentration: how concentrated it behaves thermodynamically, rather than how much of it is actually present. It is defined relative to a standard concentration c° = 1 mol dm⁻³:

a = γ × (c / c°)

where γ is the activity coefficient. Because c is divided by c°, activity is a pure number with no units.

Why effective and actual differ. In a solution, each ion is surrounded, on average, by slightly more ions of opposite charge than of the same charge. This "ionic atmosphere" stabilises the ion and lowers its tendency to react, escape or take part in an equilibrium. The ion behaves as if it were less concentrated than it is, so γ is less than 1 for ions in most dilute electrolyte solutions.

The thermodynamic constant. Thermodynamics links the equilibrium constant to the standard Gibbs energy change: ΔG° = −RT ln K. The logarithm requires a dimensionless quantity, and the derivation uses activities. For a weak acid HA:

Ka = (a(H⁺) × a(A⁻)) / a(HA)

This Ka is truly constant at a fixed temperature, whatever else is dissolved in the solution.

The concentration constant. If we write the expression with concentrations instead, we get a concentration quotient, sometimes written Kc′:

Ka = Kc′ × (γ(H⁺) × γ(A⁻)) / γ(HA)

Since the activity coefficients change as other ions are added, Kc′ changes too. That is why a "constant" in concentrations drifts with salt content.

When concentrations are good enough. In very dilute solutions ions are far apart, their interactions are weak and γ approaches 1. For total ionic concentrations below roughly 0.001 mol dm⁻³, using concentrations introduces errors of only a few per cent. For school-level calculations, concentrations are used throughout, and this is a deliberate approximation. Neutral molecules, such as undissociated ethanoic acid, have activity coefficients close to 1 even at moderate concentrations.

Solids and solvent. The activity of a pure solid or pure liquid is defined as 1. That is why solids do not appear in Ksp expressions and why water does not appear in Ka expressions for dilute solutions: their activities are fixed at essentially 1, not because they are unimportant.

pH is really an activity. The modern definition is pH = −log a(H⁺). A pH electrode responds to hydrogen-ion activity, not concentration, which matters when pH meters are used in salty solutions such as seawater.

Formulae

a = γ × c/c° (c° = 1 mol dm⁻³). Ka (thermodynamic) = a(H⁺)a(A⁻)/a(HA). pH = −log a(H⁺). ΔG° = −RT ln K.

Step-by-step reasoning

To decide whether activities matter in a problem:

1. Estimate the total concentration of ions in the solution, including spectator ions. 2. If it is very low (below about 10⁻³ mol dm⁻³), γ ≈ 1 and concentrations are adequate. 3. If it is higher, expect γ below 1 for ions, more so for highly charged ions. 4. State clearly whether your constant is thermodynamic or a concentration quotient.

Visual explanation

Picture a single cation drawn at the centre of a circle. Around it are scattered ions, with anions slightly more numerous near the centre and cations slightly more numerous further out. This faint cloud of opposite charge partly shields the central ion, so from outside it looks less charged and less reactive than a bare ion would.

Real-world analogy

A celebrity walking alone through a street can greet everyone freely. In a crowd of bodyguards and fans, the same celebrity's effective "reach" is much smaller. The number of celebrities has not changed, but their effective presence has, just as an ion's activity falls when it is crowded by other ions.

Real-world example

Seawater has an ionic strength of about 0.7 mol dm⁻³. Oceanographers cannot use freshwater acid dissociation constants for the carbonate system; they use constants measured specifically in seawater, because the activity coefficients of ions such as CO₃²⁻ are far below 1.

Why?

Why is the activity coefficient of a doubly charged ion affected more than that of a singly charged ion? Electrostatic attraction depends on the product of charges, so a 2+ ion attracts a denser ionic atmosphere and is stabilised more strongly, lowering its activity further.

Common misconception

"Activity is just another word for concentration." Activity is a dimensionless effective concentration. It equals c/c° only in the ideal limit; in real ionic solutions it is usually smaller than the concentration value.

Worked example

Question: In a solution, [H⁺] = 1.00 × 10⁻³ mol dm⁻³ and γ(H⁺) = 0.90. Compare pH calculated from concentration with pH calculated from activity.

Reasoning: From concentration: pH = −log(1.00 × 10⁻³) = 3.00. Activity a = 0.90 × 1.00 × 10⁻³ = 9.0 × 10⁻⁴, so pH = −log(9.0 × 10⁻⁴) = 3.05.

Answer: 3.00 from concentration; 3.05 from activity, a small but measurable difference.

Quick check

1. Why does a pure solid not appear in an equilibrium constant expression? Answer: Its activity is defined as 1 and does not change with the amount of solid present.

Exam focus

Be able to state that thermodynamic equilibrium constants use activities and are dimensionless, and that using concentrations is an approximation valid for dilute solutions. Questions often ask why Ka appears to change when an inert salt is added.

Advanced insight

Activity coefficients can exceed 1 in very concentrated electrolyte solutions, where ions become strongly hydrated and the amount of "free" water falls. In concentrated hydrochloric acid, the mean activity coefficient rises well above 1, which helps explain the surprisingly high reactivity of concentrated acids.

Summary

Activity is an effective, dimensionless concentration, a = γc/c°. Ions interact through their ionic atmospheres, so their activity coefficients are usually below 1. Thermodynamic equilibrium constants are written in activities and are truly constant at fixed temperature; constants written in concentrations drift as the ionic content changes. In dilute solutions γ ≈ 1 and concentrations are an acceptable approximation.

Practice questions

1. Define activity coefficient. Answer: The factor γ that converts concentration into activity, a = γ × c/c°; it measures departure from ideal behaviour. 2. Explain why Ka measured in 0.1 mol dm⁻³ NaCl differs from Ka measured in pure water when concentrations are used. Answer: The added ions lower the activity coefficients of H⁺ and A⁻, so the concentration quotient must increase to keep the activity-based Ka constant. 3. Which is expected to have the lower activity coefficient in the same solution, Na⁺ or Mg²⁺? Explain. Answer: Mg²⁺, because its higher charge attracts a denser ionic atmosphere, stabilising it more. 4. Why must the thermodynamic equilibrium constant be dimensionless? Answer: It appears inside a logarithm in ΔG° = −RT ln K, and only a pure number can have a logarithm; activities are dimensionless.