Posts Tagged ‘Gibbs-Duhem equation’

Dmitiri Konovalov (1856-1929) was a Russian chemist who made important contributions to the theory of solutions. He studied the vapor pressure of solutions of liquids in liquids and in 1884 published a book on the subject which gave a scientific foundation to the distillation of solutions and led to the development of industrial distillation processes.

On the subject of partially miscible liquids forming conjugate solutions, Konovalov in 1881 established the following fact: “If two liquid solutions are in equilibrium with each other, their vapor pressures, and the partial pressures of the components in the vapor, are equal.”

J. Willard Gibbs in America had already developed the concept of chemical potential to explain the behavior of coexistent phases in his monumental treatise On the Equilibrium of Heterogeneous Substances (1875-1878). Konovalov was unaware of this work, and independently found a proof on the basis of this astutely reasoned thought experiment:

«Consider Figure 77 shown above. Two liquid layers α and β in coexistent equilibrium are contained in a ring-shaped tube, and above them is vapor. If the pressure of either component in the vapor were greater over α than over β, diffusion of vapor would cause that part lying over β to have a higher partial pressure of the given component than is compatible with equilibrium. Condensation occurs and β is enriched in the specified component. By reason of the changed composition of β however, the equilibrium across the interface of the liquid layers is disturbed and the component deposited by the vapor will pass into the liquid α. The whole process now commences anew and the result is a never-ending circulation of matter round the tube i.e. a perpetual motion, which is impossible. Hence the partial pressures of both components are equal over α and β and therefore also their sum i.e. the total vapor pressure.»

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Equivalence of vapor pressure and chemical potential

Konovalov showed that the condition of equilibrium in coexistent phases was equality of vapor pressure p for each component. This is consistent with the concept of ‘generalized forces’, a set of intensive variables which drive a thermodynamic system spontaneously from one state to another in the direction of equilibrium. Vapor pressure is one such variable, and chemical potential is another. Hence Gibbs showed that chemical potential μ is a driver of compositional change between coexistent phases and that equilibrium is reached when the chemical potential of each component in each phase is equal. In shorthand the equilibrium position for partially miscible liquids containing components 1 and 2 in coexistent phases α, β and vapor can be stated as:

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P Mander, July 2020

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The Phase Rule formula was first stated by the American mathematical physicist Josiah Willard Gibbs in his monumental masterwork On the Equilibrium of Heterogeneous Substances (1875-1878), in which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter”, Gibbs gives the derivation of his famous formula in just 77 words. Of all the many Phase Rule proofs in the thermodynamic literature, it is one of the simplest and shortest. And yet textbooks of physical science have consistently overlooked it in favor of more complicated, less lucid derivations.

To redress this long-standing discourtesy to Gibbs, CarnotCycle here presents Gibbs’ original derivation of the Phase Rule in an up-to-date form. His purely prose description has been supplemented with clarifying mathematical content, and the outmoded symbols used in the single equation to which he refers have been replaced with their modern equivalents.

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Gibbs’ derivation

Gibbs begins by introducing the term phase to refer solely to the thermodynamic state and composition of a body (solid, liquid or vapor) without regard to its quantity. So defined, a phase cannot be described in terms of extensive variables like volume and mass, since these vary with quantity. A phase can only be described in terms of intensive variables like temperature and pressure, which do not vary with quantity.

To derive the Phase Rule, Gibbs chooses as his starting point equation 97 from his treatise, now known as the Gibbs-Duhem equation

This general thermodynamic equation, which relates to a single phase, connects the intensive variables temperature T, pressure P, and chemical potential μ where μn is the potential of the nth component substance. Any possible variations of these quantities sum to zero, indicating a phase in internal equilibrium.

If there are n independent component substances, the phase has a total of n+2 variables

These quantities are not all independently variable however, because they are related by the Gibbs-Duhem equation. If all but one of the quantities are varied, the variation of the last is given by the equation. A single-phase system is thus capable of (n+2) – 1 independent variations.

Now suppose we have two phases, each containing the same n components, in coexistent equilibrium with each other. Signifying one phase by a single prime and the other by a double prime we may write

since this is the definition of equilibrium between phases. So in the two-phase system the total number of variables remains unchanged at n+2, but there are now two Gibbs-Duhem equations, one for each phase. It follows that if all but two of the quantities are varied, the variations of the last two are given by the two equations. A two-phase system is thus capable of (n+2) – 2 independent variations.

It is evident from the foregoing that regardless of the number of coexistent phases in equilibrium, the total number of variables will still be n+2 while the number subtracted (called the number of constraints) will be equal to the number of Gibbs-Duhem equations i.e. one for each phase.

A system of r coexistent phases is thus capable of n+2 – r independent variations, which are also called degrees of freedom (F). Therefore

This is the Phase Rule as derived by Gibbs himself. In contemporary textbooks it is usually written

where C is the number of independent components and P is the number of phases in coexistent equilibrium.

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Update

gibbs twitter

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P Mander February 2015