### The Gibbs phase rule

The construction of phase diagrams as well as some of the principle is governed by laws of thermodynamics. This rule represents a condition for the number of phases that will coexist within a system at equilibrium. It is expressed by the simple equation,

$$P+F=C+N$$

Where, The term F is the number of degrees of freedom ,P is the number of phases present and the parameter C represents the number of components in the system. Finally, N is the number of non compositional variables. Components are normally elements and, in the case of phase diagrams, are the materials at the two extremities of the horizontal compositional axis. Let us demonstrate the phase rule by applying it to binary temperature - composition phase diagrams, specifically the copper–silver system. Because pressure is constant (1 atm), the parameter N is 1—temperature is the only noncompositional variable.

$$P+F=C+1$$

P+F=2+1

F=3-P

Consider the case of single-phase fields on the phase diagram (e.g., $$\alpha, \beta$$, and liquid regions). Because only one phase is present, P=1 and

F=3-P

F=3-1=2

This implies that to completely describe the characteristics of any alloy that exists within one of these phase fields, we must specify two parameters; these are composition and temperature. Which locate the horizontal and vertical positions of the alloy on the phase diagram.

It is necessary to specify the one of the component from temperature and composition to completely define the system. The compositions of the solid and liquid phases ($$C_\alpha$$and CL) are thus find by the end points of the tie line constructed at T1 across the $$\alpha$$ L field. it says that the overall alloy composition could lie anywhere along this tie line constructed at temperature T1. And it gives $$C_\alpha$$ and CL compositions for the respective $$\alpha$$ and liquid phases. For example, if we specified $$C_\alpha$$ as the composition of the solid phase that is in equilibrium with the liquid, then both the temperature of the alloy (T1) and the composition of the liquid phase (CL) can be showed. By the tie line drawn across the $$\alpha$$ L phase field so as to give this $$C_\alpha$$ composition.

$$F=C-P+2\dotsm(1)$$where F=number of degree of freedom

It gives the relation between number of degree of freedom and number of components in the solution, number of phase present. The relationship between them is given by,

$$F=C-P+2\dotsm(2)$$Where,

F=number of degree of freedom

C=number of component

P=number of phase

In one component phase diagram of water ice and there are two degree of freedom and one phase presence and temperature can be independently vary as shown in figure. Along the lines on boundary between two phases that exists in equilibrium only one variable can be independently varied without upsetting low phase equilibrium. Pressure and temperature are selected by clacious clapeyron equation; there is only one degree of freedom. At triple point where solid, liquid and vapor exists ant change in pressure and temperature would upset three phase of equilibrium. This point corresponds to zero degree of freedom.

From above diagram when we calculate the number of degree of freedom from equation (1) at point A (triple point). We have,

P=3(3-phases)

C=1(component)

F=1-3+2=0(triple point)

At point B (Boundary between two phases)

P=2(two phases)

C=1

F=1-2+2=1(1 degree of freedom)

At point D:

P=1(single phase)

C=1(1-component)

F=1-1+2=2(degree of freedom)

This implies the both P and T varies.

#### Demonstration of phase rule in binary alloy (Cu-Ag)

For binary alloy the known compositional variable such as temperature and pressure is only one in the diagram i.e. temperature. So, according to phase rule,

$$F=C-P+1\dotsm(3)$$Inside any one phase the number of component is,

$$F=3-P\dotsm(4)$$

At point B or D,

F=3-2=1(only one degree of freedom)

At point E, P=1

$$\therefore$$ F=3-1=2

At point A, number of phase p=3

Degree of freedom=3-3=0

This means that the compositions of all three phases as well as the temperature are fixed. This condition is met for a eutectic system by the eutectic isotherm. For the Cu–Ag system, it is the horizontal line that extends between points B and G. At temperature $$779^\circ$$C, all $$\alpha$$,L, and $$\beta$$ phase fields touch the isotherm line correspond to the respective phase compositions. The composition of the $$\alpha$$ phase is fixed at 8.0 wt% Ag, that of the liquid at 71.9 wt% Ag, and that of the $$\beta$$ phase at 91.2 wt% Ag.

Hence, the multiple phase equilibrium cannot be represented by a phase field, but rather by the unique horizontal isotherm line. Additionally, all three phases will be in equilibrium for any alloy composition that lies along the length of the eutectic isotherm (e.g., for the Cu–Ag system at $$779^\circ$$C and compositions between 8.0 and 91.2 wt% Ag). One use of the Gibbs phase rule is in analyzing for nonequilibrium conditions.

#### References:

Callister, W.D and D.G Rethwisch. Material Science and Engineering. 2nd. New Delhi: Wiley India, 2014.

Lindsay, S.M. Introduction of Nanoscience . New York : Oxford University Press, 2010.

Patton, W.J. Materials in industry . New Delhi : Prentice hall of India, 1975.

Poole, C.P. and F.J. Owens. Introduction To Nanotechnology. New Delhi: Wiley India , 2006.

Raghavan, V. Material Science and Engineering. 4th . New Delhi: Pretence-Hall of India, 2003.

Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.

1. Important equation:

$$P+F=C+N$$

2.

F=number of degree of freedom

C=number of component

P=number of phase

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