Differential molar

Similarly, the differential molar enthalpy of adsorption, Ji is defined as... 

Thus by measuring the small amount of heat 5Q which is evolved when the adsorption increases by the small amount 6n mole at constant temperature, the differential molar energy of adsorption can be evaluated calorimetri-... 

The differential molar entropies can be plotted as a function of the coverage. Adsorption is always exothermic and takes place with a decrease in both free energy (AG < 0) and entropy (AS < 0). With respect to the adsorbate, the gas-solid interaction results in a decrease in entropy of the system. The cooperative orientation of surface-adsorbate bonds provides a further entropy decrease. The integral molar entropy of adsorption 5 and the differential molar entropy are related by the formula = d(n S )ldn for the particular adsorbed amount n. The quantity can be calculated from... 

For all samples, the differential molar heats of adsorption decreased rapidly with SO2 coverage. However the steepness of the curve decreased with the nitrogen enrichment, suggesting greater sites homogeneity. 

Acidic properties of zeolite L were observed to correlate well with its structural disorders. The Si-MAS-NMR spectrum of zeolite L having a Si/AI ratio different from 3 revealed that Al distribution deviated from the ideal and suggested the presence of six different boat-shaped 8-ring patterns. Differential molar heats of adsorption of ammonia changed step-wise with the adsorbed amount, which reflects the difference in the acid strength of protons located in structurally different 8-rings. 

For a given uptake and temperature T, dSs/dT = Cp where Cp is the differential molar heat capacity of sorbed fluid. This expression can be approximated by Tm ASs/AT = Cp where Tm is the mean temperature corresponding with the interval AT over which ASs is the entropy change, and where Cp refers to the temperature Tm. For classical oscillators Cp should be 24.9 J/mole/deg, and thus it is interesting to compare Cp calculated as above with this value. A5S/AT did not vary much with amount sorbed, so that Cp found for one uptake is typical. Several values of Cp are given below. All are near but a little below the classical oscillator value. 

The standard differential molar Gibbs free energy (AG°) for transfer of 1 mole of vapor at 1 atm to Peg, the equilibrium... 

We have introduced integral molar quantities, which indicates that there are corresponding differential quantities. Integral refers to the fact that the total amount of adsorbed gas is involved. In contrast, the differential molar energy of adsorption is determined only by the last infinitesimal amount adsorbed. It is defined as... 

Figure 16 shows the reconstruction of the molar mass distribution from two measurements in Fig. 13 according to Eq. (43) for an exposure time Tp = 0.025 s, which is shorter than the inverse of the fastest rate in the system, and for Tp = 1.5 s, which is longer than the inverse of the slowest rate. The inserts show the respective rate distributions. It is evident that the slow mode is much more pronounced in P(T) for the long exposure time. The peak positions in the differential molar mass distributions c(M) agree reasonably well with the nominal molar masses, which are marked by the dashed vertical lines. The two peaks are somewhat broader and less resolved in the case of the shorter exposure. The peak areas in the molar mass distributions are, contrary to the rate distributions, identical for both measurements, as can be seen from the plateaus at I(M) 0.5 in the cumulative molar mass distribution... 

According to experimental data, and assuming that PSD is Gaussian, Dubinin and Radushkevich obtained an equation, which relates the degree of micropore filling (9) with the differential molar work of adsorption ... 

Electronic CD (ECD), observed most commonly between 180 and 300 nm and also referred to as UV-CD, is a measure of the differential molar absorptivities a of the sample toward left and right circularly polarized light ... 

Since the differential molar absorptivity, m can be related to the theoretically relevant quantity, R,... 

TABLE 2 Temperature dependence of differential molar enthalpy for the... 

Experimentally, the CD intensity is often quantified by the differential molar absorption coefficient Ae = l — r for the absorption of left-handed vs right-handed circular polarized light, where Ae and e are usually in units of L mol 1 cm-1. The conversion from Ae in L moP1 cm-1 to the molar ellipticity in deg cm2 dmoP1 is [0] = (18,0001n(10)/47r)Ae. The connection with quantities that can be calculated from first-principles theory is given by the following equation [35] ... 

Figure 11.4. Differential molar energy of adsorption of Ar and N2 on samples of kaolinite as a function of surface coverage, (a) On sample GB3 (b) On sample FU7 (reproduced courtesy of Cases et al., 1986).

This work is a continuation of our earlier study [1] of the hydrogen interaction with intermetallic compound (IMC) AB2-type Tio.9Zro.1Mn . 3V0.5. The measurements were carried out in twin-cell differential heat-conducting Tian-Calvet type calorimeter connected with the apparatus for gas dose feeding, that permitted us to measure the dependencies of differential molar enthalpy of desorption (AHdes.) and equilibrium hydrogen pressure (P) on hydrogen concentration x (x=[H]/[AB2]) at different temperatures simultaneously. The measurements were carried out at 150°C, 170°C and 190°C and hydrogen pressure up to 60 atm. 

CD spectra are usually measured as the differential absorption of left and right circularly polarized light, i.e. (A -AJ and are usually reported as the differential molar extinction coefficient (As),... 

On the basis of differential molar absorption coefficients, the latter process appears to be quantitative. Indeed, using the rate constant for direct reverse electron transfer (Eq. 18) derived following excitation into the Au porphyrin subunit (Ati8 = 2.0 X 10 s ), oxidation of the central Cu complex ( 19 = 4.8 x lO s ) should account for ca. 95 % of the total decay of the intermediate CT state. The residual Au porphyrin neutral radical decayed on a much longer time scale (Figure 26d) due to electron donation to the Cu + complex ... 

The systems considered in the present context may consist of several phases, some or ail of which may he present as droplets. The differential molar free energy for an n-component system is given by... 

Let us now consider the basic properties of the characteristic Equation 5, which is a Weibull distribution equation 13). Irrespective of the distribution order, n, the curves expressed by it have 2 points in common a point corresponding to maximal adsorption a = Uo 6 = 1) and a characteristic point at a = Oo/e, where e is the natural logarithm base ( = 0.368). If the differential molar work of adsorption for the char-... 

In the case of adsorption in micropores, when the condition of temperature invariance of the characteristic curve is fulfilled, the net differential molar heat of adsorption, q, and the differential molar entropy of adsorption, AS, may be expressed (2) as... 

Figure 4. Dependence of differential molar entropy of adsorption, AS, on degree of filling, 0, for zeolite L (AS, e.u.)...

M. M. Dubinin (Academy of Sciences of the USSR, Moscow, USSR) In the work under discussion, the authors for the first time undertook an extensive and systematic investigation of heats of immersion into water of various zeolites in different cation-exchange forms containing varied amounts of preadsorbed water. On the basis of their experiments, they calculate the dependence of differential molar heats of adsorption on the adsorption values of water. In principle, assuming that equilibrium states are reached, the curves obtained should coincide with similar curves determined calorimetrically in adsorption of water vapors or with... 

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