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Examples of Integrated, Multi-set Concept Schemes (Annex 13)

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See other Examples of Integrated, Multi-set Concept Schemes. The concept scheme described here is discussed in the paper on: Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation. This was prepared for a sub-project meeting of the Forms of Presentation group of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (UNU). The annexes were published in Patterns of Conceptual Integration. Brussels, UIA, 1984, pp. 161-204

This discipline gives rise to explanatory texts which are not ordered as systematically as those of the other annexes. It is therefore useful to explore to what extent its principal features may in fact he considered as falling into sets. The investig- ation was suggested by a diagram by W Byers Brown (reproduced in Number and Time. London, Rider, 1974, p. 133by Marie-Louise von Franz) concerning the thermodynamical trinity and quaternity. Other extracts in the following were obtained from:

0.1 "Thermodynamics is that part of physical science that is concerned with the conditions that material systems may assume and the changes in conditions that may occur either spontaneously or as a result of interactions between systems, including interactions such as heat." (EB, p. 290]

0.2 "The description of physical phenomena is based on the concept of state of a system and the changes of state that occur either spontaneously or because of interactions with other systems." (EB, p. 291)

1.1 "The term system means any identifiable collection of matter that can be separated from everything else by a well-defined surface so that changes in everything else need not affect the condition of the collection." (EB, p. 291)

1.2 "The fundamental equation for a closed system...i.e. one which does not exchange matter with its environment...

dU = TdS - PdV (Denbigh, p. 41) In open systems: dU = TdS - PdV + ?????.dn. (Denbigh, p. 79) (See 4.1)

2.1 Reversible and irreversible processes: "Changes which tale place in a system spontaneously and of their own accord....are said to be irreversible....A rever- sible process will therefore be defined as one which can be reversed without leaving more than a vanishingly small change in any other system." (Denbigh,p 23)

2.2 Heat and work are the two limiting cases of fundamental interactions between systems. Neither is a property of a system. They are modes of transfer of energy between one body and another.

2.3 "Thermodynamics is concerned only with the macroscopic properties of a body and not with its atomic properties...These macroscopic properties....may be divided into two groups as follows:

2.4 "There is clearly nothing in the above treatment of the first law which requires us to think of energy as a 'thing' -- it is the fact of conservation which tempts us to regard it as some kind of indestrutible fluid. In dealing with the second law we meet a second quantity, the entropy, which is also an extensive quantity and a function of state, but is not would be preferable, but for the need for economy of words, to speak always of the 'energy function' and the 'entropy function' rather than of energy and entropy. They are not material entities but are mathematical functions having certain properties." (Denbigh, p. 20)

2.5 "What has just been said, to the effect that two intensive properties of a phase usually determine the values of the rest, applies to mixtures as well as to pure substances." (Denbigh, p. 7)

3.1 "The foundations of thermodynamics are three facts of ordinary experience. These may be expressed roughly as follows:

3.2 "An extensive property of a pure phase is usually determined by the choice of three of its properties, one of which may be conveniently chosen as the mass (thereby determining the quantity of the pure phase in question] and the other two as intensive properties." [Denbigh, p. 8)

3.3 "All systems that consist of a single pure molecular species, such as argon, oxygen, or water, exhibit largely common patterns of coexisting phases (Solid, liquid, vapour)" (EB, p. 299)

3.4 Thermodynamic trinity: temperature, pressure, chemical potential (W Byers Brown). "The chemical potential has an important function analogous to temperature and pressure....The chemical potential is thus a Kind of 'chemical pressure' and an intensive property of a system, like the temperature and pressure themselves." (Denbigh, p. 76)

3.5 "The mathematical relation between the pressure, volume, and temperature for stable equilibrium states of a closed system is called its equation of state. Although it...does not completely specify the nature of the system, the mathem- atical relation is an important one because the three properties it relates are relatively easily measured." (EB, p. 300)

3.6 "Thus, in brief, the whole of the fundamental part of thermodynamics may be regarded as the discovery of the quantities T, U and S. Their importance lies precisely in the fact that they are functions of state. That is to say, they form exact differentials and their changes are independent of the path which is taken between assigned initial and final states. It would not be possible to develop an adequate thermodynamics on the basis of heat and work only, because their magnitudes depend on the details of the path." (Denbigh, p, 46)

3.7 3 Laws of thermodynamics:

4.1 "Fundamental equations....A type of equation for which any of the properties P. T, U, V, S (or any algebraic combination of these] not explicit in it are found by differentiation is called a fundamental equation, and the correspond- ing function has been called a characteristic function." (EB, p. 302) "The equations...(below)...form the basis of chemical thermodynamics. The first of them may be regarded as the fundamental relation (see 1.2, above) which contains the physical' information embodied in the properties of U, T and S, whilst the other three are derived from it by virtue of the definitions of H, F and G and contain no additional information.

(Denbigh, p. 77-783)

4.2 Thermodynamic quaternity: entropy, energy, matter, volume (W Byers Brown).

5.1 [See 4.1) "A type of equation for which any of the properties P, T, U, V, S (or any algebraic combination of these) not explicit in it are found by differentiation is called a fundamental equation..." (EB, p. 302)

6.1 (See 3.B) "The whole of the physical Knowledge on which thermodynamics is based has already been embodied in the properties of T, U and S, and these functions alone form a sufficient basis for the development of chemical thermodynamics. It is a matter of convenience only that we introduce certain additional function; These are defined as follows:

H = U + PV F = U - TS G = U + PV - TS

These new quantities are combinations of the previous functions of state,U, P, V, T and S, and are therefore functions of state themselves. They are also extensive properties. Their value is simply that they are easier to use in certain applications...and in such circumstances they also have an easily visualized meaning." (Denbigh, p. 61)

6.2 "An additional crop of useful identities. Known as Maxwell's relations, is obtained by applying a theorem of the calculus concerning exact differentials...,

7.1 The fundamental relation of thermodynamics [see 1.2 and 4.1 above] is:

It defines the relationship between 7 properties:

U = (internal) energy P = pressure T = temperature V = volume

S = entropy n = number of moles of substance i (mass] chemical potential -

The many other relations and equations of thermodynamics do not seem to be presented as a progressively elaborated pattern. It is possible that both sets of equations and quantities [e.g. heat capacities and others) could be presented in an orderly manner in terms of the increasing number of elements in the set. Such a systematic approach could also be developed by relating different sets (e.g. relating the 3 properties of the 'trinity' (see 3.4 above) to the 4 properties of the 'quaternity* (see 4.2 above) would generate 12 possible "functions' relating two properites in each case, one intensive and one extensive).

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