Fixed point defining

The concept of temperature can be defined operationally that is, in terms of a set of operations or conditions that define the concept. To define a temperature scale operationally we need (1) one particular pure or defined substance (2) a specific property of that substance that changes with a naive sense of degree of hotness (i.e., temperature) (3) an equation relating temperature to the specific property (4) a sufficient number of fixed points (defined as reproducible temperatures) to evaluate the constants in the equation in (3) and (5) the assignment of numerical values to the fixed points. Historically, many different choices have been made with respect to the five conditions listed above, and this, of course, has resulted in many temperature scales. 

Kelvin temperature scale -> Two fixed points define the absolute temperature scale T = 273.16 K The triple point of water, where ice, liquid. 

To sketch the vector field, we first find the fixed points, defined by 0 = 0. These occur at 0 = 0 and 0 = zr. To determine their stability, note that sin0 > 0 on the upper semicircle. Hence 0 > 0, so the flow is counterclockwise. Similarly, the flow is clockwise on the lower semicircle, where 0 < 0. Hence 0 = n is stable and 0 = 0 is unstable, as shown in Figure 4,1.1. 

Suppose for now that there are two fixed points, defined implicitly by O) -02... 

Whatever thermometer is used, in either routine or in standardization applications, it must be calibrated periodically against a primary standard, or at least its calibration must be checked periodically at some temperature fixed points (defining fixed points or well-characterized secondary fixed points) in the range of the thermometer. 

Although it is more complex than the direct method (Equation 48.50), this procedure provides the following advantages (1) multiple alloys can be measured in a single experiment (depending on the number of effusion cells in the furnace) (2) T measurements are calibrated with the Au melting in each experiment (Tmp(Au) = 1337.33 K is a fixed point defining ITS-90)... 

Two fixed points define the absolute temperature scale ... 

At this point the system has throe phases (CUSO4 CuS04,Hj0 HjO vapour) and the number of components is two (anhydrous salt water). Hence by the phase rule, F + F = C + 2, t.e., 3+F = 2 + 2, or F=l. The system is consequently univariant, in other words, only one variable, e.g., temperature, need be fixed to define the system completely the pressure of water vapour in equilibrium with CUSO4 and CuS04,Hj0 should be constant at constant temperature. 

The ultimate definition of thermodynamic temperature is in terms of pV (pressure X volume) in a gas thermometer extrapolated to low pressure. The kelvin (K), the unit of thermodynamic temperature, is defined by specifying the temperature of one fixed point on the scale—the triple point... 

Defining fixed points of the International Temperature Scale of 1990 (ITS-90). Except for the triple points, the assigned values of temperature are for equilibrium states at a pressure of one standard atmosphere (101 325 Pa). 

Pressure. Standard atmospheric pressure is defined to be the force exerted by a column of mercury 760-mm high at 0°C. This corresponds to 0.101325 MPa (14.695 psi). Reference or fixed points for pressure caUbration exist and are analogous to the temperature standards cited (23). These points are based on phase changes or resistance jumps in selected materials. For the highest pressures, the most rehable technique is the correlation of the wavelength shift, /SX with pressure of the mby, R, fluorescence line and is determined by simultaneous specific volume measurements on cubic metals... 

The KTTS depends upon an absolute 2ero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equiUbrium, together with specification of an interpolation instmment and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

Table 1. Values of Defining Fixed Points of lTS-90 and Some Fixed Points of IPTS-68...

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Temperatures on ITS-90, as on earlier scales, are defined in terms of fixed points, interpolating instruments, and equations that relate the measured property of the instrument to temperature. The report on ITS-90 of the Consultative Committee on Thermometry is published in Metrologia and in the Journal of Research of the National Institute of Standards and Technology The description that follows is extracted from those publications.3 Two additional documents by CCT further describe ITS-90 Supplementary Information for the ITS-90, and Techniques for Approximating the ITS-90.4... 

The flux ( J ) is a common measure of the rate of mass transport at a fixed point. It is defined as the number of molecules penetrating a unit area of an imaginary plane in a unit of time, and has the units of mol cm 2 s-1. The flux to the electrode is described mathematically by a differential equation, known as the Nemst-Planck equation, given here for one dimension ... 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

Because energy is not conserved in a time-dependent system, it is not meaningful to ascribe a certain energy to a TS trajectory in the way that the fixed point and the NHIM in an autonomous system exist at different energies. Instead, there is typically a single TS trajectory that is uniquely defined by the... 

Modern temperature scale proposed by G. Fahrenheit, defined by a thermometer, a law and three fixed points. Fahrenheit s thermometer was a mercury-in-glass one. Thermal expansion versus temperature was assumed linear. Three fixed points were defined 0°F temperature of a mixture of water, ice and ammonium chloride 32°F temperature of melting ice 96°F temperature of human body... 

In 1968, an international agreement was reached about the definition of an official (practical) scale of temperature for T> 14 K. This temperature scale IPTS-68, corrected in 1975 [11], was defined by reference fixed points given by transitions of pure substances. To extend the low-temperature range of IPTS-68, the EPT 76 [12-13] gave nine reference temperatures defined by phase transition of pure substances in particular the superconductive transition (between 0.5 and 9K) of five pure metals was introduced. Moreover,... 

The ITS 90 was adopted by the Comite International des Poids et Mesures in September 1989 [14-16], The ITS 90 extends from 0.65 K to the highest temperatures, practicably measurable in terms of the Planck radiation law using monochromatic radiation. The defining fixed points of the ITS 90 are mostly phase transition temperatures of pure substances given in Table 8.2. 

---