Temperature and pressure are often taken as independent intensive variables.(see P-T diagram) This is because it is usually easy to experimentally vary and measure these two properties directly. However, any intensive property can be choosen to be one of the independent variables. If the intensive enthalpy (J/mol) and intensive entropy (J/mol K) of the water in the glass were specified, its temperature and pressure could then be found using the steam tables or a Mollier diagram.

The Gibbs phase rule tells how many independent intensive properties, F, can be chosen. This will depend on the number of chemical species, N, and number of phases, pi, present. In the absence of chemical reaction, the Gibbs phase rule is simply:

F=2+N-pi

For a pure substance (N=1), the Gibbs phase rule can be applied as follows:

single phase (pi=1) | F=2+1-1=2 |

two phases (pi=2) | F=2+1-2=1 |

three phase (pi=3) | F=2+1-3=0 |

**An example showing that for a single phase of a pure substance, F=2:**

For a glass of liquid water, specify one of the independent intensive variables to be pressure. Choose this pressure to be 1 atm. If liquid is in the glass, the temperature can take any value between 0 'C and 100 'C. Within this range, the temperature can be choosen independently of the pressure. Thus, both T and P are independent. After choosing a (T, P) pair, any other property, such as volume or entropy, can be found using the steam tables or a Mollier diagram. Therefore the remaining properties can not be independently choosen after T and P are specified.

**Example showing that for two coexistant phases of a pure substance,
F=1.**

For a glass of boiling water (also call saturated liquid water) in equilibrium with saturated steam, specify the one of the independent intensive variables to be pressure. Choose this pressure to be 1 atm. In order for the water to boil (or be saturated) at this pressure, the temperature must be 100 'C. Thus, the temperature can not be choosen independently of the pressure when both liquid and vapor water are present. Thus, only the P is independent. The temperature required at other pressures, as well as the values of all remaining thermodynamic properties, can be found in the tables for saturated steam or on a Mollier diagram.

**Example showing that for three coexistant phases of a pure substance,
F=0.**

At the triple point; vapor, liquid and solid all coexist. For any given
substance, the triple point occurs only at one a specific pair of temperature
and pressure. Once it is stated the substance at the triple point, the
values of this temperature and pressure pair as well as the values of all
other thermodynamic properties can be found in a table or graph. Thus,
no thermodynamic property can be choosen independently.

The Gibbs Phase rule can be applied to a P-T thermodynamic diagrams for pure substances. On any such of diagram, each point represents a state of the material. Once the point has been located, all other thermodynamic properties can be read off.

For a single phase, the required point can fall anywhere on the region of the plane representing this phase. Thus, two coordinates are needed to specify a point within that part of the plane. For example the horizontal and vertical axis values, T and P, can be given. (F=2)

A two phase region on a P-T is represented by a curved line. Thus, to specify a point on this curve only one additional coordinate, either T or P is needed. (F=1)

The triple point is represented by a single point. No additional information,
neither T nor P is needed. (F=0)

FOR SOLVING THERMODYNAMIC PROBLEMS

DETERMINE F PROPERTIES WITH KNOWN VALUES. & TAKE THESE AS THE INDEPENDENT INTENSIVE PROPERTIES EVALUATE ANY REMAINING DEPENDENT PROPERTIES FOR WHICH VALUES ARE DESIRED. IT IS QUICK TO DO THIS USING TABLES OR GRAPHS. ALTERNAT IVELY, EQUATIONS OF STATE AND HEAT CAPACITY DATA CAN BE USED TO CALCULATE THE DEPENDENT PROPERTIES.