Deviation function

Almost all definitions of molar properties for mixtures lack an unambiguous definition in the sense that they can be related directly to measurable properties. Therefore, it is common practice to compare an actual mixture property with its corresponding value obtained from an arbitrary model, for instance, an equation of state. This approach leads to the introduction of deviation functions. For a general mixture molar property Mm, the deviation function is defined by  

An important aspect in this definition is the choice of the independent variables. Many analytical equations of state are expressions explicit in pressure that is, temperature, molar volume (or density) and composition x = xi, X2, , x, are the natural independent variables. Therefore, eq 2.32 can be rewritten into  

In this case the value of M obtained from the model is evaluated at the same values of T, p and n as used for the actual mixture property. Both approaches are interrelated as follows  

In this equation, p is the reference pressure at which the molar volume of the mixture obtained from the model is equal to the molar volume of the actual 

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For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

Each range has a specific deviation function to be used in equation 8. For example, over the range 0—961.78°C, the deviation function is... 

The deviation function AW Too) is obtained as a function of r90 for various temperature intervals by calibration of the platinum resistance thermometer, using specified fixed points from Table A2.1. The form of the AW(Too) function is dependent on the temperature range in which the thermometer is being calibrated. For example, in the temperature subrange from 234.3156 to 302.9146 K, the form of the deviation function is... 

Table A2.4 Temperature subranges, deviation functions, and calibration points over the temperature range covered by platinum... 

Calibration (Fixed) points used to determine coefficients in the deviation function... 

In general, a thermometer is called primary if a theoretical reliable relation exists between a measured quantity (e.g. p in constant volume gas thermometer) and the temperature T. The realization and use of a primary thermometer are extremely difficult tasks reserved to metrological institutes. These difficulties have led to the definition of a practical temperature scale, mainly based on reference fixed points, which mimics, as well as possible, the thermodynamic temperature scale, but is easier to realize and disseminate. The main characteristics of a practical temperature scale are both a good reproducibility and a deviation from the thermodynamic temperature T which can be represented by a smooth function of T. In fact, if the deviation function is not smooth, the use of the practical scale would produce steps in the measured quantities as function of T, using the practical scale. The latter is based on ... 

The L, risk of the absolute deviation function is given by Konno and Yamazaki (1991) ... 

The Bias as a Large Deviation Function Glassy Dynamics... 

Section 11 introduces two examples, one from physics and the other from biology, that are paradigms of nonequilibrium behavior. Section in covers most important aspects of fluctuation theorems, whereas Section IV presents applications of fluctuation theorems to physics and biology. Section V presents the discipline of path thermodynamics and briefly discusses large deviation functions. Section VI discusses the topic of glassy dynamics from the perspective of nonequilibrium fluctuations in small cooperatively rearranging regions. We conclude with a brief discussion of future perspectives. 

A large deviation function P x) of a function Pl x) is defined if the following limit exists ... 

The bias defined in Eq. (103) is still another example of a large deviation function. Let us define the variable... 

In the following we show that Vn x) defines a large deviation function in the... 

J. L. Lebowitz and H. Spohn, A GaUavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333-365 (1999). 

B. Derrida, J. L. Lebowitz, andE. R. Speer, Exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process. J. Stat. Phys. 110, 775-810 (2003). 

C. Giardina, J. Kurchan, and L. Peliti, Direct evaluation of large-deviation functions. Phyi. Rev. Utt. 96, 120603 (2006). 

Learn to use the standard deviation function on your calculator and see that you get s = 30.269 6... Do not round oft during a calculation. Retain all the extra digits in your calculator. 

It is noted that an independent value for D is obtained from every fringe numbered / whose deflection Yj is measured. For nonideal diffusion, Ct actually will show a drift with fringe number, and the value Ct corresponding approximately to the middle of the diffusing boundary is obtained by an extrapolation procedure (27). Akeley and Gosting (27) developed the deviation function defined by... 

The major purpose of this study is to investigate the coupling effects between specific ligand binding and the whole-molecule-half-molecule interaction, particularly as they produce nonideality in diffusion. This is best accomplished (27) by examining plots of the fringe deviation function, Qh as defined by Equation 4, as a function of the path difference function f(zj), as defined by Equation 1. 

D) Experimental results are frequently reported in terms of deviation functions. The usefulness of these functions arises from the fact many properties of various systems obey approximate laws. Thus, we speak of the deviations from ideal gas behavior or deviations from the ideal solution laws. The advantage of such deviation functions is that their values are usually much smaller than the whole value, and consequently greater accuracy can be obtained with simpler calculations, either graphically or algebraically. As an example, the molar volume of a mixture of liquids is approximately additive in the mole fractions, so that we may write c... 

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