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f = c - p + 2 ---------- (5a)

where f is the number of degrees of freedom, c is the number of components in the system, and p is the number of phases in the system. Components denote the different kind of species in the system. Phase means a system with uniform chemical composition and physical properties.

For example, the phase rule indicates that a single component system (c = 1) with only one phase (p = 1), such as liquid water, has 2 degrees of freedom (f = 1 - 1 + 2 = 2). For this case the degrees of freedom correspond to temperature and pressure, indicating that the system can exist in equilibrium for any arbitrary combination of temperature and pressure. However, if we allow the formation of a gas phase (then p = 2), there is only 1 degree of freedom. This means that at a given temperature, water in the gas phase will evaporate or condense until the corresponding equilibrium water vapor pressure is reached. It is no longer possible to arbitrarily fix both the temperature and the pressure, since the system will tend to move toward the equilibrium vapor pressure. For a single component with three phases (p = 3 -- gas, liquid, and solid) there are no degrees of freedom. Such a system is only possible at the temperature and pressure corresponding to the Triple point.

One of the main goals of Thermodynamics is to understand these relationships between the various state properties of a system. Equations of state are examples of some of these relationships. The ideal gas law:

pV = NkT ---------- (5b)

is one of the simplest equations of state. Although reasonably accurate for gases at low pressures and high temperatures, it becomes increasingly inaccurate away from these ideal conditions. The ideal gas law can be derived by assuming that a gas is composed of a large number of small molecules, with no attractive or repulsive forces. In reality gas molecules do interact with attractive and repulsive forces. In fact it is these forces that result in the formation of liquids. By taking into accounts the attraction between molecules and their finite size (total volume of the gas is represented by the red square in Figure 06), a more realistic equation for the real gases known as van der Waals equation was derived way back in 1873: | |

## Figure 06 Gas Law [view large image] |

where a and b are constants depending on the gases as listed in the table below:

It is evident that a increases with the ease of liquefaction of the gas; this is to be expected if it is a measure of the attraction between the molecules. At large volume and low pressure, both correction terms in the van der Waals equation may be neglected and Eq.(5c) is reduced to Eq.(5b). Figure 06 is a plot of pV for samples of H

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