Posts Tagged ‘constraint’

gibbs

It was the American mathematical physicist Josiah Willard Gibbs who introduced the concepts of phase and chemical potential in his milestone monograph On the Equilibrium of Heterogeneous Substances (1876-1878) with which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter” Gibbs mentions – in passing – that for a system of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential μ of any component must have the same value in every phase.

This simple statement turns out to have considerable practical value as we shall see. But first, let’s go through the formal proof of Gibbs’ assertion.

An important result

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Consider a system of two phases, each containing the same components, in equilibrium at constant temperature and pressure. Suppose a small quantity dni moles of any component i is transferred from phase A in which its chemical potential is μ’i to phase B in which its chemical potential is μ”i. The Gibbs free energy of phase A changes by –μ’idni while that of phase B changes by +μ”idni. Since the system is in equilibrium at constant temperature and pressure, the net change in Gibbs free energy for this process is zero and we can write

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hence

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This result can be generalized for any number of phases: for a system in equilibrium at constant temperature and pressure, the chemical potential of any given component has the same value in every phase.

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Visualizing variance

The equality of pressure P, temperature T and component chemical potentials μn between coexistent phases in equilibrium provides a convenient way to visualize variance, or the number of degrees of freedom a system possesses. For example, the triple point of a single component system can be visualized as the array

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where the solid, liquid and vapor phases are indicated by one, two and three primes respectively.

Each row represents a single variable, so the number of rows equates to the total number of variables. Each column lists the variables in a single phase. All but one of these may be independently varied; the last is determined by the Gibbs-Duhem relation

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There are one of these for each phase, so the number of columns equates to the number of relations (=constraints) to which the system variables are subject. The variance, or number of degrees of freedom (f) of the system is defined

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For arrays of the kind presented above, this transposes into

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For the triple point of a single component system, there are three rows and three columns, so f =0. With zero degrees of freedom, the triple point is not subject to independent variation and is represented by a fixed point in the PT plane.

The above rule implies that a system of coexistent phases in equilibrium cannot have more phases than intensive system variables.

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Generating useful equations

For a component present in any pair of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential of that component has the same value in both phases

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From this general relation, equations may be deduced for computing various properties of thermodynamic systems such as ideal solutions, for example the elevation of boiling point, the depression of freezing point, and the variation of the solubility of a solute with temperature.

The key point to grasp is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

dce12 … (1)

where for a given phase, μi is the arbitrary chemical potential of i in the mixture, μ°i is the chemical potential of the pure substance, and xi is the mole fraction of the component.

As an example, let us take the relation

pe08 …(2)

where the chemical potential of the solid solvent is necessarily the standard potential because the mole fraction x is unity. The above relation will generate an equation for the depression of the solvent freezing point in a solution at a fixed pressure (p).

Substituting (1) for the liquid phase in (2) gives

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where by convention the subscript 1 refers to the solvent. Differentiating with respect to T at constant pressure

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using the quotient rule for ΔG/T gives

pe12 … (3)

Now since

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equation (3) simplifies to

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Integrating from the pure solvent state, where the mole fraction x1=1 and T0fus is the freezing point of the pure solvent, to the solution state where the mole fraction x1= x1 and Tfus is the freezing point of the solvent in the solution

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yields the equation for the depression of the solvent freezing point in a solution at a fixed pressure (p)

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Since x1<1 in a solution, the logarithm is negative and therefore the freezing point of the solvent in the solution must be lower than the freezing point of the pure solvent.

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Ok, so maybe that wasn’t the simplest procedure for generating a useful thermodynamic equation. But the point to be made here is that the same procedure applies in the other cases, so you only need to understand the principle once.

For example, the equation for elevation of solvent boiling point in solution with a non-volatile solute at a fixed pressure (p) is

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The similarity to the previous equation is evident.

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P Mander February 2015

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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

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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

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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

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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

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

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

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Why don’t we use Gibbs’ original derivation of the Phase Rule?

This is a question for science historians with better information resources at their disposal than mine. But I can offer a couple of indicators.

The first point to make is that although Gibbs was undoubtedly the first to discover that a coexistent r-phase system containing n independent components has n + 2 – r degrees of freedom, he did not draw any attention to it; in fact his derivation is almost ‘hidden away’ in the text of his milestone monograph.

Nor did Gibbs apply the sobriquet ‘phase rule’; this seems to have originated in Europe. The Dutch chemist Hendrik Roozeboom, who in the 1880s began research into verifying Gibbs’ theoretical predictions of phase equilibria at the University of Amsterdam, certainly introduced the term “Phasenlehre”. But the earliest dated literature reference I can find is from 1893 when the German chemist Wilhelm Meyerhoffer, who was also working at the University of Amsterdam, published a paper entitled “Die Phasenregel”.

The second point concerns the two books which established chemical thermodynamics as a modern, practical science, and set the study curriculum at countless colleges around the world. One was American – the famous Thermodynamics by G.N. Lewis and Merle Randall, published in 1923. The other was European – Modern Thermodynamics by Edward Guggenheim, published in 1933.

The extraordinary thing about Lewis and Randall’s 600+ page book, the pivotal work which first made Gibbs’ powerful ideas accessible to students of physical science, is that it devotes barely a page to the Phase Rule and – crucially – does not even state the equation, let alone its derivation.

That job was left to Edward Guggenheim in Europe. In Chapter 1 of Modern Thermodynamics – Introduction and Fundamental Laws – he states the Phase Rule and gives the derivation.

But it is not Gibbs’ derivation based on a single intensive factor relation. Guggenheim’s method involves counting component concentrations, which are related by an equation of condition within each phase, and are also subject to individual constraints between phases since the chemical potential of any component is the same in all phases at equilibrium.

Two separate sets of constraints are thus imported into Guggenheim’s calculation of degrees of freedom

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which to this writer’s eyes, lacks the simplicity of Gibbs’ approach.

Nevertheless, Guggenheim’s method is the one which has been taught to generations of students (including me), and will in all likelihood be taught for generations to come…

… unless CarnotCycle succeeds with this post in awakening interest in Gibbs’ own derivation, which is surely the original and arguably the best!

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P Mander February 2015