Gibbs equations phase rule

Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

The state variables (intensive, i.e., independent of the amount of matter, and extensive that are proportional to it) are these quantities that describe a system, for example, by means of the equation of state. For the discussions of phase diagrams it is important to know how many of the state variables one may change without going through a phase transition. The total number of variables required to describe a system by Eq 2.14 is N + 2, where N is number of components and 2 stands, e.g., for V and T. For a closed system the number of intensive variables (also known as a degree of freedom), /, is given by the Gibbs the phase rule ... 

Since we have briefly introduced the consideration of mixed systems, we will mention, in passing, Gibbs famed Phase Rule. Its basic equation, which is generally applicable, is ... 

It is enlightening to examine the number of degree of freedom using Gibbs s phase rule (f=C-P + 2-r) for the following discussion. The total number of components (C) is i + 3 (solvent, G, S, M,..., Mj), the number of phases (P) is one (surfactant solution), and the number of equilibrium equations (r) is / + 1 [Eqs. (4.11) and (4.12)]. Therefore, the number of degrees of freedom is three, and the total surfactant concentration C, determines the concentrations of every chemical species at constant temperature and pressure. The mass balances for counterions and surfactant ions, respectively, are expressed as... 

Phase equilibria in ceramic and refractory systems can be studied with the help of phase diagrams. Phase diagrams are based on Gibb s phase rule, which can be mathematically represented by the following equation ... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

We will be looking at first-order phase transitions in a mixture so that the Clapeyron equation, as well as the Gibbs phase rule, apply. We will describe mostly binary systems so that C = 2 and the phase rule becomes... 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

Equation (2.23) is a very important result. It is known as the Gibbs Phase Rule, or simply the phase rule, and relates the number of components and phases to the number of degrees of freedom in a system. It is a more specific case of the general case for N independent, noncompositional variables... 

This observation is the simplest case of the Gibbs phase rule (to be discussed in Section 7.1). It implies, for example, that pressure P = P(V, T) is uniquely specified when V and T are chosen, and similarly, that V = V(P, T) or T = T(P, V) are uniquely determined when the remaining two independent variables are specified. Such functional relationships between PVT properties are called equations of state. We can also include the quantity of gas (as measured, for example, in moles n) to express the equation of state more generally as... 

Turning now to adsorption equilibrium, let us apply algebraic methods to a two component 1,2 phase system. From the phase rule there will be two degrees of freedom, but we shall reduce this to one by maintaining the temperature constant. Then for the total system there exists a Gibbs-Duhem equation... 

Let the number of basic overall equations be Q it is evident that Q< P. Let us denote the number of substances participating in the reaction as M and the number of independent components, in the sense this notion is used in the Gibbs phase rule, as C then... 

For solid-liquid equilibrium in a quaternary system, the Gibbs phase rule allows four degrees of freedom. If T, P, xc, and xD (in which x is the mole fraction of component i in liquid solution) are specified, then xA, x, t/, and xAC (in which x is the mole fraction of component ij in solid solution) are determined, and the system is invariant. These variables are defined by the following equations ... 

One essential question is, How many variables must be specified to obtain a solution unique to the phase equilibrium calculation It is possible to have an infinite number of solutions to a problem if too few variables are specified—or no solution if too many variables are specified. One answer to this question is provided by Gibbs Phase Rule (Gibbs, 1928, p. 96), simply stated for nonreacting systems by the equation ... 

In equation (11.31), AHm and A Vm are the molar enthalpy and molar volume changes associated with the change in phase, and dp/dT gives the slope of the equilibrium lines. Point b is the triple point for CO2. It is an invariant point, since, according to the Gibbs phase rule... 

The Gibbs phase rule relates the number of equations that are required to describe a system at equilibrium to the number of variables necessary to describe the system. The number of degrees of freedom is the number of variables that can be changed independently without affecting the number of phases in equilibrium. It is the difference between the number of variables and the number of equations describing equilibrium. 

Many choices of independent variables such as the energy, volume, temperature, or pressure (and others still to be defined) may be used. However, only a certain number may be independent. For example, the pressure, volume, temperature, and amount of substance are all variables of a single-phase system. However, there is one equation expressing the value of one of these variables in terms of the other three, and consequently only three of the four variables are independent. Such an equation is called a condition equation. The general case involves the Gibbs phase rule, which is discussed in Chapter 5. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

This is the usual relation given for the phase rule. The difference V is called the number of variances or the number of degrees of freedom. (Note that some texts use F = C — P + 2, where F is the same as V.) These are the number of intensive variables to which values may be assigned arbitrarily but within the limits of the condition that the original number of P phases exist. It is evident from Equations (5.63) and (5.56) that the intensive variables are the temperature, pressure, and mole fractions or other composition variables in one or more phases. Note that the Gibbs phase rule applies to intensive variables and is not concerned with the amount of each component in each phase. 

The Gibbs phase rule provides the necessary information to determine when intensive variables may be used in place of extensive variables. We consider the extensive variables to be the entropy, the volume, and the mole numbers, and the intensive variables to be the temperature, the pressure, and the chemical potentials. Each of the intensive variables is a function of the extensive variables based on Equation (5.66). We may then write (on these equations and all similar ones we use n to denote all of the mole numbers)... 

Applications of the Gibbs-Duhem equation and the Gibbs phase rule... 

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