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##### 1.3 Gibbs phase rule

The degrees of freedom of a system indicate the number of variables that can be arbitrarily specified without modifying the system, hence:

DEGREES OF FREEDOM = NUMBER OF VARIABLES - NUMBER OF COSTRAINTS

For nonreacting systems, this relationship can be translated into the following equation, known as the Gibbs phase rule:

F = C - P + 2

with
F = degree of freedom;
C = number of components;
P = number of phases.

For a system with 2 components (see binary systems) and 2 phases (e.g. vapor and liquid), the number of degrees of freedom is 2. This means that 2 variables can be freely set (i.e. pressure and temperature, pressure and composition, temperature and composition), to uniquely identify the system or, in other words, that the number of independent variables is 2, while the third is dependent on these two.

To note: the Gibbs phase rule refers to the so-called intensive variables. Pressure, temperature and composition are intensive variables since they do not depend on the amount of present matter. Flow rate, volume and number of moles are examples of extensive variables.

Binary systems with two phases can be easily graphically represented using planar diagrams for two variables while one is constant. In this way, since one variable has been already set, the system has only one left degree of freedom. Hence it is possible to select a different variable among all the intensives ones and plot this versus any other as the dependent variable.

##### Single component systems

Let's take in consideration a simple system where only one component is present, i.e. water, and let's have a look at the here below pressure-temperature diagram. Depending on temperature and pressure, water has three different states: ice, liquid and vapor, each of them existing in specific zones of the diagram.
The red lines, delimitating the one phase regions, are the equilibrium lines between two different phases. The equilibrium lines are two phase regions since they are the "loco" of points of pressure and temperature, at which the two phases are in equilibrium.

Let's apply Gibbs phase rule to point A:

F = C - P + 2 = 1 - 1 + 2 = 2

since we have
C = 1 (only water)
P = 1 (only vapor phase)

This system has 2 degrees of freedom. This means that two variables, temperature and pressure, can be changed independently without changing the phase.
The same would happen for point B.

Let's now take in consideration point C on the equilibrium line between the liquid and the vapor phase. In this case it is:

F = C - P + 2 = 1 - 2 + 2 = 1

since we have
C = 1 (only water)
P = 2 (liquid in equilibrium with vapor)

Only 1 degree of freedom is now present, thus meaning that only one variable can be freely changed in order to still keep the same state for the system. For example, if the temperature decreases (going from point C to point D), also the pressure must decrease in order that point C stays always on the equilibrium line.

Finally let's consider point T:

F = C - P + 2 = 1 - 3 + 2 = 0

since we have
C = 1 (only water)
P = 3 (ice, liquid and vapor are all coexisting in equilibrium)

##### Binary systems

Here below a typical diagram for a two components system representation is shown.

Three different zones can be distinguished: a region at higher temperatures where only vapor is present; a region at lower temperatures where only liquid is present; and a biphasic region where both vapor and liquid are present at equilibrium. Applying Gibbs phase rule to the monophasic region, like point A, it is:

F = C - P + 2 = 2 - 1 + 2 = 3

since we have
C = 2
P = 1 (vapor)

Three variables are independent. The pressure is already fixed, since the diagram is drawn at constant pressure, but it is still possible to change both temperature and composition to keep the system in the vapor phase.

Moving into the biphasic region (point C), Gibbs phase rule gives:

F = C - P + 2 = 2 - 2 + 2 = 2

since we have
C = 2
P = 2 (liquid and vapor)

In point C, 2 degrees of freedom are available. And since 1 is already taken by the pressure, now temperature and composition must change according to each other in order to stay within the biphasic region. The same happens for points B or D. Again the system has 2 degrees of freedom.

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