Equilibrium Planck function

Thus far we have observed that the Gibbs and Planck functions provide the criteria of spontaneity and equilibrium in isothermal changes of state at constant pressure. If we extend our analysis to systems in which other constraints are placed on the system, and therefore work other than mechanical work can be performed, we find that the Gibbs and Helmholtz functions also supply a means for calculating the maximum magnitude of work obtainable from an isothermal change. 

Equilibrium Constant and Change in Gibbs Functions and Planck Functions for Reactions of Real Gases... 

ISOTORE EXCHANGE AT EQUILIBRIUM MASSIEU FUNCTION HELMHOLTZ ENERGY PLANCK FUNCTION MASS SPECTROMETRY Matrix of biominerals,... 

Consider an enclosure of dimensions large compared with any wavelengths under consideration, which is opaque but otherwise arbitrary in shape and composition (Fig. 4.11). If the enclosure is maintained at a constant absolute temperature T, the equilibrium radiation field will be isotropic, homogeneous, and unpolarized (see Reif, 1965, p. 373 et seq. for a good discussion of equilibrium radiation in an enclosure). At any point the amount of radiant energy per unit frequency interval, confined to a unit solid angle about any direction, which crosses a unit area normal to this direction in unit time is given by the Planck function... 

In the case of local thermal equilibrium B(a, x) is equal to the Planck function. 

ORM assumes that the atmosphere is in local thermodynamic equilibrium this means that the temperature of the Boltzmann distribution is equal to the kinetic temperature and that the source function in Eq. (4) is equal to the Planck function at the local kinetic temperature. This LTE model is expected to be valid at the lower altitudes where kinetic collisions are frequent. In the stratosphere and mesosphere excitation mechanisms such as photochemical processes and solar pumping, combined with the lower collision relaxation rates make possible that many of the vibrational levels of atmospheric constituents responsible for infrared emissions have excitation temperatures which differ from the local kinetic temperature. It has been found [18] that many C02 bands are strongly affected by non-LTE. However, since the handling of Non-LTE would severely increase the retrieval computing time, it was decided to select only microwindows that are in thermodynamic equilibrium to avoid Non-LTE calculations in the forward model. 

Kirchhoff s law States that for an opaque surface the absorpion coefflcient is equal to the emission coefficient. For an extended medium such as the atmosphere, assumed to be in local thermodynamic equilibrium, Kirchhoff s law relates the thermal volume emission coefficient to the Planck function. 

It has recently been pointed out by Gordon1 that the root-mean-square fluctuations in the sampled values of the autocorrelation function of a dynamical variable do not necessarily relax to their equilibrium values at the same rate as the autocorrelation function itself relaxes. It is the purpose of this paper to investigate the relative rates of relaxation of autocorrelation functions and their fluctuations in certain systems that can be described by Smoluchowski equations,2 i.e., Fokker-Planck equations in coordinate space. We exhibit the fluctuation and autocorrelation functions for several simple systems, and show that they usually relax at different rates. 

This equation is identified with the macroscopic equation of motion for the system, which is supposedly known. Thus the function A(y) is obtained from the knowledge of the macroscopic behavior. Subsequently one obtains B(y) by identifying (1.4) with the equilibrium distribution, which at least for closed physical systems is known from ordinary statistical mechanics. Thus the knowledge of the macroscopic law and of equilibrium statistical mechanics suffices to set up the Fokker-Planck equation and therefore to compute the fluctuations. 

Figure 16.8. Relic density of gravitationally-produced WIMPZILLAs as a function of their mass Mx Hi is the Hubble parameter at the end of inflation, 1 i, is the reheating temperature, and Mpi 3 x 1019 GeV is the Planck mass. The dashed and solid lines correspond to inflationary models that smoothly end into a radiation or matter dominated epoch, respectively. The dotted line is a thermal distribution at the temperature indicated. Outside the thermalization region WIMPZILLAs cannot reach thermal equilibrium. (Figure from Chung, Kolb Riotto (1998).)...

Any alteration in AG will thus affect the rate of the reaction. If AG is increased, the reaction rate will decrease. At equilibrium, the cathodic and anodic activation energies are equal (AG 0 = AG 0) and the probability of electron transfer will be the same in both directions. A, known as the frequency factor, is given as a simple function of the Boltzmann constant k and the Planck constant, h ... 

We also need some background material about (19). If m(x) denotes the equilibrium probability density function of x(t), i.e. the probability density to find a trajectory (reactive or not) at position x at time t, m(x) satisfies the (steady) forward Kolmogorov equation (also known as Fokker-Planck equation)... 

We suppose that a small probing held Fj, having been applied to the assembly of dipoles in the distant past (f = —oo) so that equilibrium conditions have been attained at time t = 0, is switched off at t = 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function W(i), cp, t) for normal diffusion of dipole moment orientations on the unit sphere in configuration space (d and (p are the polar and azimuthal angles of the dipole, respectively), where the Fokker-Planck operator LFP for normal rotational diffusion in Eq. (8) is given by l j p — l j /> T L where... 

An interesting problem is the field evolution in a cavity that was initially in the equilibrium state at a finite temperature, when the initial occupation numbers were given by the Planck distribution v = [exp(pw) — 1] 1. Let us consider two limit cases. The first one corresponds to the low-temperature approximation v = exp (—(] ). Then the occupation number of the mth mode is merely the coefficient at vm in the expansion (61) with u = exp( (5). Using the well-known generating function of the Legendre polynomials Pm(z) [Ref. 269, Eq. 10.10(39)], one can obtain the following expression (for y = 0) ... 

Blackbody radiation is achieved in an isothermal enclosure or cavity under thermodynamic equilibrium, as shown in Figure 7.4a. A uniform and isotropic radiation field is formed inside the enclosure. The total or spectral irradiation on any surface inside the enclosure is diffuse and identical to that of the blackbody emissive power. The spectral intensity is the same in all directions and is a function of X and T given by Planck s law. If there is an aperture with an area much smaller compared with that of the cavity (see Figure 7.4b), X the radiation field may be assumed unchanged and the outgoing radiation approximates that of blackbody emission. All radiation incident on the aperture is completely absorbed as a consequence of reflection within the enclosure. Blackbody cavities are used for measurements of radiant power and radiative properties, and for calibration of radiation thermometers (RTs) traceable to the International Temperature Scale of 1990 (ITS-90) [5]. 

Donnan Equilibrium and Electroneutrality Effects for charged membranes are based on the fact that charged functional groups attract counter-ions. This leads to a deficit of co-ions in the membrane and the development of Donnan potential. The membrane rejection increases with increased membrane charge and ion valence. This principle has been incorporated into the extended Nemst-Planck equation, as described in the NF section. This effect is responsible for the shift in pH, which is often observed in RO. Chloride passes through the membrane, while calcium is retained, which means that water has to shift its dissociation equilibrium to provide protons to balance the permeating anions (Mallevialle et al. (1996)). 

Consider evolution of the translational energy distribution function / E) of a small admixture of alkaline atoms in non-equilibrium diatomic molecular gas (TV > Tq). The distribution is determined by competition of fast VT-relaxation energy exchange between the alkaline atoms and diatomic molecules and Maxwelhzation translational-translational (TT) processes in collisions of the same partners. It can be described by the Fokker-Planck kinetic equation for diffusion of the atoms along the translational energy spectrum (Vakar etal., 1981a,b,c,d) ... 

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