The Phase Rule

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The Phase Rule describes the possible number of degrees of freedom in a (closed) system at equilibrium, in terms of the number of separate phases and the number of chemical constituents in the system. The Phase Rule was an incidental offshoot of the classic investigations in which J. Willard Gibbs laid out the foundations of chemical thermodynamics between 1875 and 1878. A reprint exists (Gibbs, 1961) together with two helpful commentaries (Donnan and Haas, 1936; Seeger, 1974).

Phase Rule

The phase rule is a method to count the number of degrees of freedom (how many independent variables are sufficient to specify a multi-component, multi-phase system.

The idea is from Gibbs and the derivation of the equation is:

Consider the general case: <math>C\text{ components and }P\text{ phases}\text{.}\,\!</math>
At equilibrium all pressures, temperatures, and each chemical potential is constant: <math>\begin{align}
 & p_{a}=p_{b}=\ldots =p_{P} \\ 
& T_{a}=T_{b}=\ldots =T_{P} \\ 
& \mu _{ia}=\mu _{ib}=\ldots =\mu _{iP};\text{    }i=1,2,\ldots C \\ 


The number of unknowns in each phase is (C+1): <math>p,\text{ }T,\text{ and }(C-1)\text{ mole fractions}\,\!</math>
times the number of phases <math>P\,\!</math>
or <math>=\left( C+1 \right)P</math>
The number of equations: <math>=\left( P-1 \right)\left( C+2 \right)</math>
Therefore the degrees of freedom are: <math>F=\left( C+1 \right)P-\left( P-1 \right)\left( C+2 \right)=C+2-P</math>
The phase rule: the number of degrees of freedom is: <math>F=\left( C+2-P \right) </math>

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Applications of the phase rule to soil systems

Gibbs' Phase rule provides a theoretical basis for considering problems concerning the mineralogy of soils. In a deductive way, it serves as a check and a balance to the science of pedology, which like all the earth sciences is fundamentally inductive in nature. Since the Phase Rule has its derivation in classical thermodynamics it deals with systems at equilibrium. As soils are manifestly in a state of disequilibrium, application of the Phase Rule (or thermodynamics in general) needs some initial justification. First, the equilibrium state represents the state a system would achieve given the time and energy to get there. As such the equilibrium model lays fundamental constraints on any hypothesis of mineral genesis in soils. It indicates the direction of change, and places an end bracket on all states the system might pass through. In some cases, an equilibrium mineralogy may be approached closely, for example in microscopic systems (Chesworth and Dejou, 1980) and as a result of long term weathering in the humid tropics.

The basic problem in using the Phase Rule to set up an equilibrium model for weathering systems and soils in particular, is that such systems are of great compositional complexity. Consequently, there is a need to simplify as much as possible, while still retaining sufficient complexity to enable reasonable statements to be made about real systems. In particular, we need to know what components and what phases we should consider, and what range of environmental conditions are appropriate.

The pe-pH framework of mineral genesis in soils.

What are the important components? Over wide areas, the average composition of the earth's continental surface is andesitic. Since the crust is also essentially a close packed framework of oxygens it can be considered initially as being made up of oxide components such as SiO2 , A12O3, Fe2O3, CaO, MgO, Na2O and K2O as majors, and TiO2, MnO2 and P2O5 as minors. In addition important components from the atmosphere and hydrosphere are H2O, CO2 and O2. The biosphere provides further complications in terms of organic components.

What are the important phases? The important phases that need to be modeled in a soil system are the ones that form there. These include the following: oxides and hydroxides (e.g. quartz, goethite, hematite, gibbsite and boehmite), 1:1 sheet silicates (e.g. kaolinite and halloysite), 2:3 sheet silicates (e.g. illite, smectite and vermiculate), 2:1:1 sheet silicates (e.g. hydroxy-interlayered vermiculite), framework silicates (e.g. zeolites and possibly feldspars), carbonates (e.g. calcite, siderite) and other minerals such as gypsum and halite. No single system contains all of these phases. Generally no more than two to four need be considered together, the specific soil environment under consideration dictating the choice.

What are the appropriate environ-mental conditions? The most generally useful master variables of the weathering environment are pe (or Eh) and pH. The spread of pe-pH conditions in the stability field of water, and found in nature is approximately as shown in the image displayed on the right hand side. Ignoring a number of rather rare environments at the surface of the earth (e.g. acid sulphate soils, weathering vanadium deposits), the normally expected conditions cover a pe-pH field which shows three salients, each of which corresponds to one of the three lines of chemical evolution shown by soils in weathering.

The behavior of water dictates which trend is followed. The acid and alkaline trends are in the oxidizing (water-unsaturated) zone of weathering. The acid trend requires an excess of water with a net leaching or downward movement. The alkaline trend is found in dryer environments with a net loss of water by evapo-transpiration. A reduced trend is found in water-saturated conditions in a weathering profile.

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Application on Water

A body of water under normal atmospheric pressure can be heated up and cooled down to any desired temperature between 0 and 100 degrees Celsius without restriction. This enables us to change either the pressure or temperature giving us two degrees of freedom. Therefore, the phase rule tells us that in order to change the temperature without losing one component of the mixture, we have to simultaneously change both temperature and pressure.

There are several specific cases in the phase diagram of water we should consider:

- When we have three phases in equilibrium, π = 3, and Gibbs phase rule states that: F = 2 − 3 + 1 = 0. Thus, there can be no variation of temperature or pressure - we must be at exactly one point, the triple point (experimentally found to be at 0.01 degree Celsius and a pressure of 611.73 pascals). Only at the triple point can three phases of water coexist.

- When two phases are in equilibrium (along the melting or boiling boundaries) π = 2, and phase rule states F = 2 − 2 + 1 = 1. This means that one variable can be varied independently.

- When only one phase exists away from boundaries (gas,liquid, or solid), π = 1. At these points, Gibbs rule states: F = 1 + 2 − 1 = 2 and only two of the variables are independent.

If we wanted to study the phase diagram of a two component system (e.g. water and another component), we need to find out the composition of a mixture at various temperatures. For this purpose we use a thermal analysis technique where solids of different compositions are separately heated above their melting points. The resultant liquids are cooled slowly and cooling curves are obtained by plotting temperature vs. time. From these information, it is possible to determine the Eutectic point, point where solidification of second component starts.

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