/ ~ Q = Q in Software

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3 ~ osystems more complex than a simple homogeneous pure substance, the number of properties or state functions r that must be arbitrarily specified in order to define the state of the system may be different from two The method of determining this number is the subject of Sec 27

For a closed system undergoing the same change in state by several processes, experiment shows that the amounts of heat and work required differ for different processes, but that the sum Q W is the same for all processes This is the basis for identification of internal energy as a state function The same value of AU' is given by Eq (23) regardless of the process, provided only that the change in the system is between the same initial and final states

Equilibrium is a word denoting a static condition, the absence of change In thermodynamics it means not only the absence of change but the absence of any tendency toward change on a macroscopic scale Thus a system at equilibrium exists under conditions such that no change in state can occur Since any tendency toward change is caused by a driving force of one kind or another, the absence of such a tendency indicates also the absence of any driving force Hence for a system at equilibrium all forces are in exact balance Whether a change actually occurs in a system not at equilibrium depends on resistance as well as on driving force Many systems undergo no measurable change even under the influence of large driving forces, because the resistance to change is very large Different kinds of driving forces tend to bring about different kinds of change For example, imbalance of mechanical forces such as pressure on a piston tend to cause energy transfer as work; temperature differences tend to cause the flow of heat; gradients in chemical potential tend to cause substances to be transferred from one phase to another At equilibrium all such forces are in balance In many applications of thermodynamics, chemical reactions are of no concern For example, a mixture of hydrogen and oxygen at ordinary conditions is not in chemical equilibrium, because of the large driving force for the formation of water However, if chemical reaction is not initiated, this system can exist in long-term thermal and mechanical equilibrium, and purely physical processes may be analyzed without regard to possible chemical reaction This is an example of the fact that systems existing at partial equilibrium are often amenable to thermodynamic analysis

As indicated earlier, the state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values In contrast, when two phases are in equilibrium, the state of the system is fixed when only a single property is specified For example, a mixture of steam and liquid water in equilibrium at 101325 kPa can exist only at 37315 K (100 C) It is impossible to change the temperature without also changing the pressure if vapor and liquid are to continue to exist in equilibrium For any system at equilibrium, the number of independent variables that must be arbitrarily fixed to establish its intensive state is given by the celebrated phase rule of J Willard

~ i b b s ; who deduced it by theoretical reasoning in 1875 It is presented here without proof in the form applicable to nonreacting systems:5

(27)

where n is the number of phases, N is the number of chemical species, and F is called the degrees of freedom of the system The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed These are therefore phase-rule variables, but they are not all independent The phase rule gives the number of variables from this set which must be arbitrarily specified to fix all remaining phase-rule variables A phase is a homogeneous region of matter A gas or a mixture of gases, a liquid or a liquid solution, and a crystalline solid are examples of phases A phase need not be continuous; examples of discontinuous phases are a gas dispersed as bubbles in a liquid, a liquid dispersed as droplets in another liquid with which it is immiscible, and solid crystals dispersed in either a gas or liquid In each case a dispersed phase is distributed throughout a continuous phase An abrupt change in properties always occurs at the boundary between phases Various phases can coexist, but they must be in equilibrium for the phase rule to apply An example of a three-phase system at equilibrium is a saturated aqueous salt solution at its boiling point with excess salt crystals present The three phases (n= 3) are crystalline salt, the saturated aqueous solution, and vapor generated at the boiling point The two chemical species ( N = 2) are water and salt For this system, F = 1 The phase-rule variables are intensive properties, which are independent of the extent of the system and of the individual phases Thus the phase rule gives the same information for a large system as for a small one and for different relative amounts of the phases present Moreover, only the compositions of the individual phases are phase-rule variables Overall or total compositions are not phase-rule variables when more than one phase is present The minimum number of degrees of freedom for any system is zero When F = 0 , the system is invariant; Eq (27) becomes n = 2 N This value of n is the maximum number of phases which can coexist at equilibrium for a system containing N chemical species When N = 1, this number is 3, characteristic of a triple point (Sec 31) For example, the triple point of water, where liquid, vapor, and the common form of ice exist together in equilibrium, occurs at 27316 K (OOl C) and 00061 bar Any change from these conditions causes at least one phase to disappear

4~osiah Willard Gibbs (1839-1903), American mathematical physicist he justification of the phase rule for nonreacting systems is given in Sec 102, and the phase rule for reacting systems is considered in Sec 138