Wien equation

For short wavelengths XT —> 0, the asymptotic form of Eq. (5-102) is known as the Wien equation... 

The error introduced by use of the Wien equation is less than 1 percent when XT < 3000 pm K. The Wien equation has significant practical value in optical pyrometry for T < 4600 K when a red filter (X = 0.65 pm) is employed. The long-wavelength asymptotic approximation for Eq. (5-102) is known as the Rayleigh-Jeans formula, which is accurate to within 1 percent for XT > 778,000 pm-K. The Raleigh-Jeans formula is of limited engineering utility since a blackbody emits over 99.9 percent of its total energy below the value of XT = 53,000 pm-K. 

In the late nineteenth century, Wien analyzed experimental data on blackbody radiation and found that the maximum of the blackbody radiation specti um shifts with the temperature according to the equation... 

Procedure. One approach to the problem is to select a value for a that is obviously too small and to increment it iteratively until the equation is satisfied. This is the method of program WIEN, where the initial value of x is taken as 1 (clearly, -E i < 1 as you can show with a hand calculator). 

Change Program QWIEN so that the second term on the right of the line below statement 10 is x instead of x/5. Solve for this new equation. Change the line below statement 10 so that the second term on the left is xj2. Repeat with j/3, xj, etc. Tabulate the values of a and the values of the denominator. Is a a sensitive function of the denominator in the second term of Program WIEN ... 

Duhem s equation was integrated by Margules (Sitzungsber. Wien Akad.j 1895) by means of the substitutions ... 

The total emission of radiant energy from a black body takes place at a rate expressed by the Stefan-Boltzmann (fourth-power) lav/ while its spectral energy distribution is described by Wien slaws, ormore accurately by Planck s equation, as well as by a n umber of oilier empirical laws and formulas, See also Thermal Radiation,... 

The derivation of this equation is somewhat lengthy, and need not be discussed here. An excellent treatment of Wien s law will be found in. Planck s V(yrl. uber d. Theorie der WdrmestraJdung, 2nd ed. p. 68 et seq. 

The form of the function F cannot be determined in this way. Equation (5) is the most general form of Wien s displacement law. 

For short waves and high temperatures an empirical equation due to W. Wien has been found to represent the facts remarkably well. It is... 

The hydroesterification of dienes gave both the unsaturated monoesters and saturated diesters. In some cases, y-ketoesters were obtained and carbonylation of 1,5-cyclooctadiene in absence of alcohol gave a ketone. [Pdl2(PBu3)2] was used as catalyst. If the catalyst contained a halide anion, butadiene underwent normal hydroesterification. Wien halide-free catalysts were used, the reaction took a different course. Dimerization of the diene occurred to give the ester of 3,8-nonadienoic acid as the major product (equation 128). " ... 

Optical Pyrometer Temperature Scale.—The temperature scale for the optical pyrometer is based upon Wien s law for the distribution, in the spectrum, of the energy of a black body. This law may be stated by the following equation where X denotes the wave length in microns, C2 a constant = 14,350, d the absolute temperature of the black body, J the intensity at the wave length X i.e. at a particular color such as red), and Ci a constant, the value of which is of no moment in pyrometry since, as will be seen, it disappears from the actual working equation... 

Wilhelm Carl Werner Otto Fritz Franz Wien (1864-1928) became an assistant to Hermann v. Helmholtz at the Physikalisch-Technische Reichsanstalt in Berlin in 1890. It was there that he discovered the displacement law in 1893, and also published an equation for M s in 1896, that only slightly differed from Planck s law. Wien became Professor of Physics at the TH in Aachen in 1896, moved in 1899 to become a professor in Wurzburg, and once again changed to the University of Munich in 1920. In 1911 he was awarded the Nobel prize for Physics as an acknowledgement of his work on thermal radiation. 

Without going beyond thermodynamics and the electromagnetic theory of light, we can deduce two laws regarding the way in which black body radiation (or, as it is also called, cavity radiation) depends on the temperature. Stefan s law (1879) states that the total emitted radiation is proportional to the fourth power of the temperature of the radiator the hotter the body, the more it radiates. Proceeding a step further, W. Wien found the displacement law (1893) which bears his name, and which states that the spectral distribution of the energy density is given by an equation of the form... 

Wien s approximate formula, which yields a relative error of less than 0.7% up to X. If XT Cj, the RHS of Equation (7.5) is approximately equal to c,T /, which is known as the Rayleigh-... 

Figure 8.6 shows Ebx/T5 versus XT, and Fig. 8.7 gives Ef,x versus A, with T as a parameter. Note from Fig. 8.7 that, for higher temperatures, the maximum of Et is shifted to shorter wavelengths. Equating the derivative of Eq. (8.36) with respect to XT to zero gives the special value of the constant in Wien s law,... 

Without breaking with classical physics, Wien had found a theoretical equation that fit the Lummer-Pringsheim curve at relatively short wavelengths, and Rayleigh and Jeans one that fit at relatively long wavelengths. Max Planck adopted a different approach he found, in 1900, a purely empirical equation t/F/rfX = /(A.) that fit the facts, and then tried to interpret the equation theoretically. To do this he had to make two assumptions ... 

Wien s law also can be derived from Planck s law [Eq. (14.6)] by differentiating with respect to A, equating the derivative to zero, and solving for... 

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