The Susceptibility Isotherm

From the susceptibility isotherms it may be expected that three kinds of information may be obtained first, the oxidation state of the paramagnetic ion second, evidence of intercation covalence and third, the effectiveness of dispersion of the paramagnetic ions. There will now be presented specific applications of the method. 

An attempt will now be made to explain the shape of the susceptibility isotherm for the chromia-alumina system in terms of the previous discussion. Provided that the Curie-Weiss law holds for this system it will be possible to determine whether the great increase of susceptibility at low concentration is due to change of oxidation state, the relaxation of... 

Chromia-alumina catalysts are often prepared by procedures other than the method of impregnation. A precipitated chromia was prepared as follows y-alumina was suspended in 25 per cent ammonium hydroxide solution. The mixture was stirred rapidly while chromic nitrate solution was added from a buret. The resulting mixture was then dried, ignited, and reduced in the same manner as for impregnated samples. A total of four samples was prepared. The susceptibility isotherm for this series is of the same general form as for the impregnation series except that point I is virtually absent. But the most striking... 

The general form of the susceptibility isotherm is not unlike that obtained for supported chromia. The magnetic moment is very close... 

The situation with respect to the low-ignition manganese on alumina is very different. Figure 17 shows the susceptibility isotherms for this series. Fig. 18 shows the magnetic moment and the chemical oxidation state, and Fig. 19 shows the Weiss constant. The susceptibility isotherm differs from all those previously presented in the sharpness of point 1. 

The expected result was obtained, namely, all the manganese remained in the -1-4 state even at the lowest manganese concentration investigated, 1.3 per cent. The susceptibility isotherm was a typical example of the chromia-alumina type in which practically the whole change is due to a change of ionic environment, and not to a change of oxidation state. The magnetic moment corresponded satisfactorily with the theoretical value for Mn+ . 

Below 6 per cent nickel the first observation is that the Weiss constant is zero. The form of the susceptibility isotherm is thus in the case of nickel in no way related to the exchange interaction between adjacent nickel ions. This is not to say that the nickel ions are at infinite magnetic dilution. For nickel in massive nickel oxide the exchange integral, the paramagnetic neighborhood (z), and the number of unpaired electrons are smaller than they are for chromium ions in massive chromia. The quantity. A, is understandably smaller for the case of nickel, and it... 

Preliminary data on a few other systems have been obtained in the writer s laboratory and as they show the general scope of the susceptibility isotherm method they will be described briefly. 

All explosives undergo thermal decomposition at temperatures far below those at which explosions take place. These reactions are important in determining the stability and shelf life of the explosive, The reactions also provide useful information on the susceptibility of explosives to heat. The kinetic data are normally determined under isothermal condi-... 

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

The line forms, described by Eqs. (371) and (373), are illustrated in Figs. 45 and 46. In the first one (Fig. 45) we compare the loss (a) and absorption (b) for the isothermal, Gross, and Lorentz lines see solid, dashed, and dashed-and-dotted curves, respectively. These curves are calculated in a vicinity of the resonance point x = 1. In Figure 45c we show the frequency dependences of a real part of the susceptibility the three curves are extended also to a low-frequency region. The collisions frequency y and the correlation factor g are fixed in Fig. 45 (y = 0.4, g = 2.5). 

Figure 45. Frequency dependence of an imaginary (a) and of real (c) parts of the susceptibility (b) absorption coefficient versus x. All quantities are nondimensional. Calculation for the isothermal (solid curves), Gross (dashed curves), and Lorentz (dashed-and-dotted curves) lines. The normalized collision frequency, y, is 0.4, and the correlation factor, g, is 2.5.

Figure 46. Evolution of frequency dependencies of loss (a, c) and of real part of the susceptibility (b, d) stipulated by change of the correlation factor g (nondimensional quantities). In Figs, (a, b) g = 2.5 and in Figs, (c, d) g — 2. Isothermal line (solid curves) and Gross line (dashed curves). Curves 1, 3 for y = 0.4 and curves 2, 4 for y = 0.8.

Extrapolating each isotherm to infinite concentration, gives the susceptibility of the full monolayer (which is the value of 5 on an arbitrary scale). The ratio Lang/ Frum = 2 is very suggestive of a packing of the dye molecules that is twice as dense in the phase in equilibrium with the bulk above a concentration of 6 pM. This suggests that in keeping with previous observations of dimer formation at the interface, an alternative way to view such a pair of ordered layers may be as adsorbed dimers. 

Although these chemical analyses of propane oxidation [13] are among the best that have been reported, it is rather difficult to make a clear judgement from them on the applicability of the alkylperoxy radical mechanism of oxidation. There is reasonable evidence in support of enhanced yields of propene at higher temperatures, especially if allowance is made for the non-isothermal effects in cool-flames. The susceptibility of... 

The critical isotherm is characterized by t = 0 and H = 2brj. This immediately shows that the critical exponent for the magnetic field variation with r = M is 5 = 3, in conformity with the standard mean field value encountered in the literature. Returning to Eq. (7.5.10) and differentiating with respect to M one obtains the magnetic susceptibility as... 

Early studies were carried out at the liquid gas interface [22, 23]. Castro et al. [24] studied the adsorption of / -propyl-phenol from aqueous solutions at the air interface as a function of phenol concentration in the bulk. They showed that the square root of the second-harmonic intensity plotted against bulk phenol concentration followed a Langmuir isotherm with a standard Gibbs energy of adsorption equal to -24.3 kJmol Similar results were obtained for other alkylphenols and alkylanilines. In other work with phenols, the orientation of phenol at the water air interface was determined by studying the phase of the xfl component of the susceptibility. As expected, the OH was oriented toward the water phase [25] so that it could participate in the hydrogen-bonded structure of water. The same conclusion was reached for / -bromophenol and -nitrophenol. 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what in pure-fluid systems is the isothermal compressibility Kj, and what in mixtures is the osmotic compressibility, and determines how fast these quantities diverge as the critical point is approached (i.e. as 7. —> 1). 

The coexistence curve is nearly flat at its top, with an exponent P = 1/8, instead of the mean-field value of 1/2. The critical isotherm is also nearly flat at T the exponent 6 (determined later) is 15 rather than the 3 of the analytic theories. The susceptibility diverges with an exponent y = 7/4, a much stronger divergence than that predicted by the mean-field value of 1. 

Evidently, the susceptibility subduced from such experiments is the mean mass susceptibility. This quantity, for linear magnetics, is close to the isothermal susceptibility. However, deviations from linear behaviour increase with... 

The molar isothermal magnetic susceptibility becomes expressed as follows... 

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