Mathematical variables

We have implied in Section 1.1 that certain properties of a thermodynamic system can be used as mathematical variables. Several independent and different classifications of these variables may be made. In the first place there are many variables that can be evaluated by experimental measurement. Such quantities are the temperature, pressure, volume, the amount of substance of the components (i.e., the mole numbers), and the position of the system in some potential field. There are other properties or variables of a thermodynamic system that can be evaluated only by means of mathematical calculations in terms of the measurable variables. Such quantities may be called derived quantities. Of the many variables, those that can be measured experimentally as well as those that must be calculated, some will be considered as independent and the others are dependent. The choice of which variables are independent for a given thermodynamic problem is rather arbitrary and a matter of convenience, dictated somewhat by the system itself. 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

The symbols used to denote units are printed in roman font those denoting physical quantities or mathematical variables are printed in italics and should generally be single letters that may be further specified by subscripts and superscripts, if required. The unit of any physical quantity can be expressed as a product of the SI base units, the exponents of which are integer numbers, e.g. [ ] = m2 kg s 2. Dimensionless physical quantities, more properly called quantities of dimension one, are purely numerical physical quantities such as the refractive index n of a solvent. A physical quantity being the product of a number and a unit, the unit of a dimensionless quantity is also one, because the neutral element of multiplication is one, not zero. 

Two dependent mathematical variables are needed for a network to be capable of periodic behavior. A third is required to permit chaos. It has become quite a sport among the apostles of nonlinear chemical dynamics to invent ever new simple, if not exactly realistic networks that can admit chaos. A classical example, and one of the simplest, is the Hudson-Rossler model [45]. The core of the network is... 

Oscillations may involve periods with more than a single wave, and may be chaotic (with no recurring periods at all). Aperiodic behavior is called chaos, but is not random The seeming randomness results from the fact that minutes difference in starting conditions can lead to drastically different behavior (butterfly effect of meteorology). Even relatively simple hypothetical networks can produce chaotic behavior. However, while periodic oscillation are possible in networks with only two independent mathematical variables, chaos requires at least three. 

We have mentioned several times now that the.rmodynamics is best viewed as a model, rather than some kind of description of natural processes. In this section we take a closer look at why this must be so. One of the main reasons is that we use physical properties as mathematical variables. 

TAe subject matter of chemistry tends to hold still, making mathematical description somewhat easier than in the case of chemistry education. Chemists operate under the assumption that a collection of hydrogen molecules today is indistinguishable from one assembled 50 years ago. However, student bodies change from semester to semester, and students are exposed to countless influences that are difficult to describe as mathematical variables f 54, p. 851)... 

If you find yourself wondmng how one could lock and unlock the piston if the system is truly isolated, you have not yet fully grasped the fact that thermodynamics deals with mathematical models, not real things. In the model, the position of the piston is, or could be, a mathematical variable. 

Now we are going to apply the very powerful little big trick 4 When no amount (mass, or number of moles, or concentration) of product is given you express it as a mathematical variable, x, and use algebra to find its value. 

These relationships can also be presented in a mathematical form external variables of a system can be treated as mathematical variables. The reduction of the total number of all conceivable parameters by subtracting the number of all laws (equations of state) that relate them yields the number of independent variables degrees cffreedom) of a system. Which of the state variables can then be regarded as independent and which are to be treated as dependent according to the indicated laws is a matter of free choice the decision must be based on the expediency of the... 

These attributes are reduced to mathematical variables that are used in equations to calculate the performance of a particular machine. However, the assembly process for electronic circuit boards is comprised of several machines or cells that are in series. Therefore, once the... 

Experimental studies on gels (Tokita, 1984 Adam, 1985) have been analyzed using this analogy. The most important difficulty arises from the choice of the physical variable which must be related to the unknown mathematical variable (P - Pq). Several variables, such as the concentration of monomer or density, have been proposed, assmning implicitly a proportionality between the variables and the P scale. 

As a matter of definition a transcendental function is a function for which the value of the function can not be obtained by a finite number of additions, subtractions, multiplications or divisions. Exponential, trigonometric, logarithmic and hyperbolic functions are all examples of transcendental functions. Such functions play extremely important roles in engineering problems and are the source of many of the nonlinear equations of interest in this book. For engineering models an important feature of transcendental functions is that their argument must be a dimensionless mathematical variable. 

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