Thermal equilibrium state

Nakazawa, E., Noguchi, S., and Takahashi, J. (1984). Thermal equilibrium state of starch-water mixture studied by differential scanning calorimetry. Agric. Biol. Chem. 48, 2647-2653. 

One can see the physical meaning of the operator Y for the case where the field is in a thermal equilibrium state. Indeed, by taking the ensemble average with a Bose-Einstein distribution of field modes at temperature T,... 

This shows that in this case, (T) is the average number of excitations of the oscillator in the thermal equilibrium state. 

Considering a reversible reaction, one assumes that the chemical species A, B, C, and D are in a thermal equilibrium state according to ... 

In Part II we discussed how to measure the electrical parameters n and pn (and/or p and pp), namely, by means of the conductivity and Hall coefficient. Now we must ask how these parameters relate to the more fundamental quantities of interest, such as impurity concentrations and impurity activation energies. Much can be learned from a consideration of thermal excitation processes only, i.e., processes in which the only variable parameter is temperature. Thus, we are specifically excluding cases involving electron or hole injection by high electric fields or by light. We are also excluding systems that have been perturbed from their thermal equilibrium state and have not yet had sufficient time to return. Some of these nonequilibrium situations will be considered in Part IV. 

Figure 22-(a) shows the DD/MAS spectrum in the resonance range of a-methyl-ene carbon at 0 °C. This spectrum represents the thermal equilibrium state of this sample, because it was obtained by a single pulse sequence with the repetition time of 600 s longer than 5 times the longest Tic in the system. The spectrum (b) is that of the crystalline component, which was obtained with use of Torchia s pulse sequence [53]. In the equilibrium spectrum, the noncrystalline contribution (amorphous plus interfacial) can be seen upfield to the crystalline component. Figure 23 shows the elementary line shapes of the amorphous and crystalline-amorphous interphases that comprise the noncrystalline resonance. 

Molecular Conformation of sPP gel. Figure 27 shows the DD/MAS 13C NMR spectrum of sPP gel. This spectrum was obtained by a single-pulse sequence (tt /2—FIDdd-tt)ii with the repetition time ty more than 5 times the longitudinal relaxation time Tic. Hence, this spectrum reflects the thermal equilibrium state of the gel. For comparison, the spectrum of the bulk ttgg crystal of this sample... 

Another example is the thermal and photochemical cis-trans isomerization of Cp2Fe2(CO)2( -CO)( -Sip-TolH).25 In this case, both cis(H) and trans isomers can be isolated at full purity by flash chromatography. Interconversion between these isomers occurs both thermally and photochemi-cally in cyclohexane-d12, and the composition in the thermal equilibrium state (cis(H) trans = 2 98 at 25°C) is extremely different from that in the photostationary state (cis(H) trans = 70 30). Kinetics of the thermal isomerization in decalin afforded the activation parameters shown in Eq. (58). The large negative activation entropies imply that this reaction also... 

In practice, even the requirement for a "full range of frequencies" is sometimes eased because we can identify the particular features of response important to a particular force computation. The principal limitation to the language of dielectric response is its restriction to electric fields weak enough to provoke only a linear response. This weak-field condition poses no limitation to computing forces between materials in their thermal-equilibrium states. 

Several other sources of external excitation result in metastable defect or dopant creation in a-Si H. Most have the characteristic property that a shift in the Fermi (or quasi-Fermi) energy causes a slow increase in the density of states and that annealing to 150-200 °C reverses the effect. The phenomena are therefore similar in origin to the optically-induced states and fall within the same general description of departures from the thermal equilibrium state induced by excess carriers. 

The amplitude and spatial dimensions of the potential fluctuations are different in undoped, doped, and compensated a-Si H. The band tail electrons and holes provide the mobile screening charge in doped a-Si H and the doping dependence of the potential fluctuations can be estimated from the experimental data. The magnitude of n. depends on the thermal equilibrium state, but is usually about 10 % of the donor and defect density. According to the square root law for doping,... 

The power supplies connected to the cold junction, the air bath, the digital D.C. microvoltmeter, the digital D.C. millivolt recorder and the two-pen strip chart recorder are then switched on, respectively. The Tdijf pen of the two-pen strip chart recorder is, however, left on the short position until the thermal equilibrium state is attained nearly around the reference material confined in the closed cell and inserted into the adiabatic jacket maintained at the nominal T,. of the run. For the term, the Tdiff pen, refer to Subsection 4.5.4. 

In the course of time, both the T pen and the ATdiff pen come to run parallel with the time axis on the strip chart (see Fig. 38). In this thermal equilibrium state, the position of the T pen on the temperature axis perpendicular to the time axis indicates the T, of the run. However, the exact value of the F, is noted down separately as the digital output of the thermoelectromotive force of the CA thermocouple to measure the F-e/ in the following subsection. 

In the case of the digital record of the self-heating process exemplified in Table 5, the digital D.C. microvoltmeter indicates that the exact value of the T, of this run is 3111.5 jUV. This value of T, is determined, based on the fact that the indication of the digital D.C. microvoltmeter has fluctuated constantly between 3111 and 3112 juV for 1 to 2 h after the thermal equilibrium state has been attained around the reference material charged in the open-cup cell and inserted into the adiabatic jacket maintained at the nominal T, of 76 °C in the adiabatic self-heating process recorder. 

The thermal equilibrium state is attained around the reference material confined in the closed cell and inserted into the adiabatic jacket maintained at the nominal T of the run 2 3 h after the insertion of the reference cell... 

As stated above, the thermal equilibrium state comes to be attained again around the substance charged in the cell, 1.5 2 h after the reset of the temperature of the air bath. After the thermal equilibrium state has nearly been attained around the substance, the power supply to the analog D.C. microvoltmeter is switched on in the meantime, the A Tjiff pen of the two-pen strip chart recorder is also switched over from short to on . It is then observed that the indicator of the analog D.C. microvoltmeter enters on to the scale span of the meter from the minus side, and, it comes to stay near the graduation line of zero at the center of the scale span of the meter in the meantime, the A Tdiff pen of the two-pen strip chart recorder also enters on to the strip chart of the two-pen strip chart recorder from the minus side. For the term, the minus side, refer to Subsection 4.5.4. 

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