Thermodynamic Activity

In chemical thermodynamics, activity is a fundamental concept used to describe the effective concentration of a chemical species in a mixture. It provides a more accurate measure of how a substance behaves in real, non-ideal systems compared with simple concentration terms. The importance of activity lies in the fact that the chemical potential of a species in a real system depends on its activity in the same mathematical way that it depends on concentration in an ideal system. The concept was introduced by the American chemist Gilbert N. Lewis in 1907 and has since become central to modern thermodynamics, solution chemistry, and equilibrium analysis.

Concept and Significance of Activity

Activity is treated by convention as a dimensionless quantity, although its numerical value depends on the chosen standard state for the species under consideration. It reflects not only the amount of substance present but also the effects of intermolecular interactions, which become significant in non-ideal gases and real solutions. These interactions cause deviations from ideal behaviour, making activity a more reliable parameter than concentration alone.
For pure substances in condensed phases, such as pure solids and pure liquids, the activity is defined as unity. This simplifies thermodynamic expressions and provides a convenient reference point. In contrast, for gases and solutions, activity varies with temperature, pressure, and composition, as well as with the nature of interactions between different chemical species.

Relationship Between Activity and Chemical Potential

The formal thermodynamic definition of activity is based on the chemical potential. For a species i, the relative activity aᵢ is defined by the expression:
μᵢ = μᵢ° + RT ln aᵢ
where μᵢ is the molar chemical potential of the species under the given conditions, μᵢ° is the molar chemical potential in the chosen standard state, R is the gas constant, and T is the absolute thermodynamic temperature. This equation highlights that activity quantifies how much the chemical potential deviates from its standard-state value.
Activity therefore depends on any factor that alters the chemical potential, including concentration, pressure, temperature, electrical fields, and intermolecular interactions. Because the standard state is defined by convention, activity is a relative measure, describing how “active” a species is compared with its behaviour in the standard state.

Standard States and Their Role

The choice of standard state is central to the definition of activity. Although arbitrary in principle, standard states are chosen for mathematical simplicity and experimental convenience. Common standard states include pure substances at a specified pressure, or hypothetical ideal solutions at a reference concentration.
An alternative concept is absolute activity, defined directly in terms of the chemical potential without reference to a standard state. While theoretically useful, absolute activity is rarely employed in practical chemical thermodynamics, where relative activities are sufficient and more convenient.

Activity Coefficient and Non-Ideal Behaviour

The activity coefficient, usually denoted by γ, relates activity to a measurable concentration term such as mole fraction, molarity, or molality. In general, activity can be expressed as:
a = γ × (dimensionless concentration term)
The division by a standard concentration, such as 1 mol L⁻¹ for molarity or 1 mol kg⁻¹ for molality, ensures that both activity and the activity coefficient remain dimensionless.
When the activity coefficient is close to unity, the system behaves nearly ideally, and activity can be approximated by the corresponding concentration measure. However, even when γ ≈ 1, the system may obey Henry’s law or Raoult’s law rather than behaving as a truly ideal solution. Significant deviations from unity indicate strong non-ideal behaviour due to intermolecular forces.

Activity in Gaseous Systems

For gases, activity is linked to the concept of fugacity, which represents the effective pressure of a real gas. In real gases, molecular interactions cause the effective pressure to differ from the mechanical pressure. Fugacity has the dimensions of pressure, but activity is made dimensionless by dividing fugacity by a standard pressure, commonly 1 bar or 1 atmosphere.
At low pressures, real gases behave nearly ideally, and fugacity approaches the actual pressure. Under these conditions, activity can be approximated by the ratio of partial pressure to standard pressure. At higher pressures, deviations become significant, and fugacity must be used to obtain accurate thermodynamic descriptions.

Mixtures and Raoult’s Law

In mixtures, composition is often expressed using mole fractions. The standard state of each component is usually taken as the pure substance at the same temperature and pressure, for which the activity is defined as one. When Raoult’s law applies, the activity of a component is proportional to its mole fraction, with the proportionality constant given by the Raoult’s law activity coefficient.
An activity coefficient equal to unity indicates ideal behaviour according to Raoult’s law. Deviations from unity arise due to differences in intermolecular interactions between unlike molecules compared with like molecules in the pure substances.

Dilute Non-Ionic Solutions

For dilute non-ionic solutions, Henry’s law is generally more appropriate than Raoult’s law. In such systems, composition is commonly expressed in terms of molarity or molality rather than mole fraction. The standard state is defined as a hypothetical ideal solution at infinite dilution with a reference concentration of 1 mol L⁻¹ or 1 mol kg⁻¹.
Molality is often preferred over molarity because it is independent of volume and temperature, whereas molarity depends on solution volume, which can change with temperature and non-ideal mixing. In the dilute limit, activity approaches the numerical value of the chosen concentration scale, although the corresponding activity coefficients remain similar across different scales.

Ionic Solutions and Mean Ionic Activity

Ionic solutions exhibit particularly strong non-ideal behaviour due to long-range electrostatic interactions. In such systems, it is impossible to measure the activity of individual ions directly, as cations and anions cannot be introduced independently. Consequently, thermodynamics employs the concept of mean ionic activity and mean ionic activity coefficient.
These quantities are defined using the stoichiometric coefficients of the ions produced during dissociation. Although the activities of individual ions are not accessible experimentally, the mean ionic activity is measurable and plays a crucial role in describing electrolyte solutions.
At low concentrations, the behaviour of ionic solutions can be described using Debye–Hückel theory, which accounts for electrostatic interactions between ions. At higher concentrations, this theory becomes insufficient, and more advanced models, such as the Pitzer equations, are required to describe activity coefficients accurately.

Importance in Equilibria and Kinetics

Thermodynamically rigorous expressions for equilibrium constants are defined in terms of activities rather than concentrations. Similarly, reaction rate equations are strictly valid when activities are used. In many practical cases, concentrations are substituted for activities as an approximation, which is justified only when activity coefficients are close to unity.
There are important situations in which this approximation fails. In solutions of high ionic strength, activity coefficients can differ markedly from one, leading to substantial discrepancies between concentration-based predictions and actual behaviour. Such effects are particularly significant in acid–base equilibria, electrochemistry, and biochemical systems.