Average energy per mode

For molecules with many degrees of freedom, the average energy per mode is small and the harmonic approximation can be made for 3i If these modes are treated as harmonic oscillators and the reaction coordinate approximated by a truncated harmonic oscillator, then (19) becomes... 

Boltzmann statistics imply that the probability of a given cavity mode that lies between hv and hv + hdv is proportional to exp(—hv/k7)hdv. Therefore, the average energy per mode and the average number of photons for each mode (obtained by dividing the former by hv) are given by... 

If another X group is added onto the aniline molecule, the An—X2 dissociation rate is more likely to be determined by the statistical dissociation step because with each additional nonlinear monomer the number of van der Waals modes increases by six. Because the van der Waals modes are extremely anharmonic and coupled to each other, a proper RRKM calculation should use anharmonic densities and sums. However, these are not yet generally available for the systems of interest. In all cases it is best to use the quantum density of states (i.e., RRKM) and not the classical approximation of it (RRK). With a binding energy of say 480 cm and six oscillators, the average energy per van der Waals mode is 60 cm. Since these frequencies typically vary between 20 and about 400 cm, it is evident that the average number of quanta excited per mode is only about 1 or 2, which does not correspond to the classical limit. 

Einstein assumed that there are 3N independent distinguishable oscillators. The vibrational partition function (Equation (11.26)) corrected for the zero-point energy (see Equation (13.21)) is q = 1 - e )" . The average energy per vibrational mode is given by the derivative in Equation (10.34) ... 

Average Energy Intensity per Passenger Mile for Various U.S. Passenger Modes, 1970-1997. 

To determine the average number of photons per mode, we should first determine the mean energy, W, for the particular mode at v = 5 x 10 " s at 300 K. According to the Maxwell-Boltzmann distribution, this is given by... 

Since the energy of a photon is hv = 3.3 x 10 J, the average number of photons for that mode is 1.7 x 10. This indicates that the number of photons per mode in the visible region for a spectral width of 10 s (in the order of the Doppler width) is a very small nnmber compared to unity. This has important conseqnences, as we will see in Section 2.3. 

The Rayleigh-Jeans treatment assigned the classical value kT to the average energy of each mode of oscillation in the cavity jkT for kinetic and kT for potential energy. The number of modes of oscillation dn in the wavelength range from X to X dX per unit... 

The left-hand side of the equation presents the change in average vibrational energy per molecnle, which in one mode approximation can be expressed by the Planck formnla ... 

Weber and Newman do the averaging by using a capillary framework. They assume that the two transport modes (diffusive for a vapor-equilibrated membrane and hydraulic for a liquid-equilibrated one) are assumed to occur in parallel and are switched between in a continuous fashion using the fraction of channels that are expanded by the liquid water. Their model is macroscopic but takes into account microscopic effects such as the channel-size distribution and the surface energy of the pores. Furthermore, they showed excellent agreement with experimental data from various sources and different operating conditions for values of the net water flux per proton flux through the membrane. 

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