Kirchhoff relation

Demanding continuity of n at r = rg, using the Kirchhoff relation I — l = 1, and identifying (for unit input current) as the escape probability 0, we get... 

We introduce a temperature dependence of Lf in first order through use of the Kirchhoff relation, Section 1.18, (3Lf/3T)P - (Cp)i - (Cp)A ACp, in which ACP represents the difference in molar heat capacity of pure liquid A and pure solid A. Under the assumption that the variation of ACP with T may be ignored, we find... 

Before concluding three items should be noted First, AHd values may be correlated with equilibrium constants and activity coefficients of solutions as shown later in Eq. (3.10.3), for example. Second, to convert AHd values from one temperature to another one uses the Kirchhoff relation, Eq. (1.18.30) (see also Exercise 3.8.3). Third, to convert AHd values from one pressure to another, the integrated form of Eq. (1.18.13b) may be used. 

Determination of ACp. As indicated above, determination of ACp from the difference in the extrapolated DSC baselines at the is unreliable and is never done. The most common method takes advantage of the linkage of pH to stability (see Linkage section). Decreasing the T by decreasing pH (typically between pH 2 and 4) leads to a decrease in AH (see Fig. 1) according to the Kirchhoff relation" ... 

This relation constitutes what we call the Kirchhoff relation. An equivalent relation would give the variation of the heat of transformation at constant volume with the temperature as a function of the molar specific heat capacity at constant volume associated with the transformation. 

The emissivity, S, is the ratio of the radiant emittance of a body to that of a blackbody at the same temperature. Kirchhoff s law requires that a = e for aH bodies at thermal equHibrium. For a blackbody, a = e = 1. Near room temperature, most clean metals have emissivities below 0.1, and most nonmetals have emissivities above 0.9. This description is of the spectraHy integrated (or total) absorptivity, reflectivity, transmissivity, and emissivity. These terms can also be defined as spectral properties, functions of wavelength or wavenumber, and the relations hold for the spectral properties as weH (71,74—76). 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

Usually the electrical resistance of a separator is quoted in relation to area in the above case it is 57 mil cm2. In order to quote it for other areas, due to the parallel connection of individual separator areas, Kirchhoff s law has to be taken into account ... 

Previous to the researches of Ivonowalow, the vapour pressures of mixtures had been investigated theoretically by G. Kirchhoff (Pogg. Ann. (1858), 103, 104 Ostw. Klass. No. 101), and by Gibbs Scientific Papers, Yol. I.). The latter had established the theorem relating to mixtures with stationary vapour pressures. 

Kirchhoff s law The relation between the standard reaction enthalpies at two temperatures in terms of the temperature difference and the difference in heat capacities (at constant pressure) of the products and reactants. 

The temperature dependence of the standard enthalpy is related by Kirchhoff s law ... 

Be warned that a Web search for Kirchhoff will yield dozens of pages on Kirch-hoff s rules, which relate to electronic circuits. 

Zhang, Z., Tillekeratne, L.M.V., Kirchhoff, J.R., and Hudson, R.A., High performance liquid chromatographic separation and pH-dependent electrochemical properties of pyrroloquinoline quinone and three closely related isomeric analogs, Biochem. Biophys. Res. Commun., 212, 41, 1995. 

Absorption. The absorption coefficient, a(v), is related to the power due to spontaneous emission by KirchhofFs law,... 

The classical emission profile, Eq. 5.72, may be converted to an absorption profile with the help of Kirchhoff s law, Eq. 2.70, which relates the absorption coefficient a to the emitted power per unit frequency interval per unit volume, with the help of Planck s law, Eq. 2.71, according to... 

In order to combine the transfer functions, the relation between V (s), the summing resistors, and the other voltages is required. From Kirchhoff s current law, this equation is... 

This relation is called Kirchhoff s law. To use it, we need to know ACP, the difference between the molar constant-pressure heat capacities of the products and reactants ... 

In emission spectrometry, the sample is the infrared source. Materials emit infrared radiation by virtue of their temperature. KirchhofF s law states that the amounts of infrared radiation emitted and absorbed by a body in thermal equilibrium must be equal at each wavelength. A blackbody, which is a body having infinite absorptivity, must therefore produce a smooth emission spectrum that has the maximum possible emission intensity of any body at the same temperature. The emissivity, 8, of a sample is the ratio of its emission to that of a blackbody at the same temperature. Infrared-opaque bodies have the same emissivity at all wavelengths so they emit smooth, blackbody-like spectra. On the other hand, any sample dilute or thin enough for transmission spectrometry produces a structured emission spectrum that is analogous to its transmission spectrum because the emissivity is proportional to the absorptivity at each wavelength. The emissivity is calculated from the sample emission spectrum, E, by the relation... 

Thus, it is very reasonable that the other factors involved in the excited state decay of [Ru(trpy)2]2+ include dissociation of at least one pyridyl ligator. Kirchhoff et al.258) have used an argument based on a kinetic scheme involving photolysis to rationalize inefficient luminescence in [Ru(trpy)2]2+ and related compounds however, they do not observe extensive photolysis in this system. 

In complexes of the quinoline based ligands we saw room-temperature emission which was weak or absent and could be related to unfavorable steric factors. Most likely [Ru(dpt)2]2+ is non-luminescent for the same reason. Kirchhoff et al.258) have argued that steric repulsions may cause a 3MC state to lie at lower energy than the 3CT state so that no CT emission occurs. A metal centered state should be more photoactive and [Ru(dpt)2]2+ does indeed undergo photolysis in the presence of nucleophiles. The photolysis product has been formulated as containing a bidentate dpt ligand. 

In his classic paper on electric networks, G. Kirchhoff[38] (1847) implicitly established the celebrated Matrix-Tree-Theorem which, in modern terminology, expresses the complexity (i.e., the number of spanning trees) of any finite graph G as the determinant of a matrix which can easily be obtained from the adjacency matrix of G. Simple proofs were given by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte [39] (1940), H. Trent [40] (1954), and H. Hutschenreuther [41] (1967) (for relations between the complexity and the spectrum of a graph see Ref. [36] pp 38, 39, 49, 50). 

On the basis of initial calorimetric measurements (Gill et al., 1976 Olofsson et al., 1984 Dec and Gill, 1984, 1985), one can represent the enthalpy of transfer of hydrocarbons from the gaseous phase to water by a linear function of temperature in the temperature range 15-35°C. Bearing in mind Kirchhoff s relation between enthalpy and heat capacity change in the reactions, one can conclude that the transfer of nonpolar molecules to water leads to an increase of heat capacity by a value that is independent of temperature in the mentioned temperature range. 

A further equation (Frame 11) involving an integration is Kirchhoff s equation which relates the enthalpy changes, AH% and AHj for a chemical reaction taking place at temperatures, T and T2, to an integral involving the heat capacity difference, ACp, between the products and reactants in the reaction ... 

Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calculated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be preferable owing to the known low sensitivity of the extremal methods to variation of the space of variables. 

The absorptivity and the emissivity of a body can be related by Kirchhoff s law of radiation, Planck, 1959 [1]. Consider a body inside a black, closed container whose walls are kept at a uniform absolute temperature T and has reached thermal equilibrium with the walls of the container. If flux qx(T) is the spectral radiative heat flux from the walls at temperature T incident on the body and ax(T) is the spectral absorptivity of the body, then the spectral radiative heat flux qx(T) absorbed by the body at the wavelength X is... 

Fresnel-Kirchhoff theory of diffraction discussed in Section 1.3, the diffracted wave is x/2 out of phase with the incident wave. Thus, the twice-diffracted wave 2 is x out of phase with T, and the two waves interfere destructively. Consequently, in a perfect crystal we should expect the intensities of both the transmitted and the diffracted waves to decrease very rapidly as they penetrate the crystal. This phenomenon is observed and is known as primary extinction. The degree of primary extinction is clearly related to the thickness of the crystal and to the crystal perfection. 

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