Curie equation

Equations ol states of various types of systems are numerous. The Curie equation is the equation of state of a paramagnetic solid. The Beattie and Bridge-man equation. Berthelol equation. Clausius equation. Dieterice equation. Keyes equation, and Van der Waals equation are other examples in this category. 

Note that equation (2) is of the algebraic form xm A + C/T (the Curie equation), where A and C are constants (for a given substance) and xm and T may be measured. For substances obeying the Curie relationship, it is possible to calculate C (and hence p) by measuring the molar susceptibility at two or more temperatures. 

Although there are many substances that, within the limits of experimental error, do show this behavior, there are also many others for which the line does not go through the origin, but instead looks somewhat like one of those shown in Fig. 19-2, cutting the T axis at a temperature below 0°K as in (a) or above 0°K as in (b). Obviously, such a line can be represented by a slight modification of the Curie equation,... 

Finally, the magnetic properties reveal clearly that the triiodide derivatives are species that contain one unpaired electron per dimer as required by oxidation to the cations [M02(O2CR)4] Magnetic susceptibility data for [Mo2(02CC(CH3)3)4]l3 are presented in Table II. These data provide an excellent fit to the Curie equation and, from the slope, the magnetic moment fi = 1.66 BM was found. The crystalline triiodide... 

This has the same form as the Curie equation, and the Curie constant is given by ... 

At a given external pressure, p, the canonical form of the characteristic function of a system having a continuous tension distribution can be given by the Curie equation ... 

Equation (A2.5.20) is the Curie-Weiss law, and the temperature at which the magnetic susceptibility becomes infinite, is the Curie temperature. Below this temperature the substance shows spontaneous magnetization and is ferromagnetic. Nonnally the Curie temperature lies between 1 and 10 K. However, typical ferromagnetic materials like iron have very much larger values for quantum-mechanical reasons that will not be pursued here. 

The lack of correlation between the flucUiating stress tensor and the flucUiating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for flucUiating hydrodynamics are arrived at by a procedure very similar to that exliibited in the preceding section for difllisioii. A crucial ingredient is the equation for entropy production in a fluid... 

These coefficients must be multiplied by the density of air and tissue, respectively. Figure 15.7.2-1 depicts a radiation fallout field. Let C be the curie activity/m. The radiation into a unit area receptor at z = 1 m above the ground. The area is emitting C r dr d0 gammas/s. These are attenuated in the air as exp(-p, R) and geometrically as l/(4 7t R). The radiation received by the receptor is given by equation 15.7.2-1 which becomes 15.7.2-2 by a change of... 

The ligand exchange reaction of tin halides with bis(trifliioroinethyl)iner-cury has been used to prepare trifluoromethyltin halides [6 7] (equation 5)... 

A decarboxylation route provides an alternative entry to perfluorovinylmer-cury compounds [182] (equation 129). 

Suppose the external field H = 0. We see from the above equation that when the temperature lies below the Curie temperature = Jq, there will exist a nonzero spontaneous magnetization, Mq, given implicitly by Mq = ta,nh JqMo/kBT). As for the behavior near Tc, we first set t = T — so that t 0 as T T. ... 

The function of I2g> (T) in the vicinity of the phase transition to centrosymmetric conditions usually has a linear character. Such behavior corresponds to ferroelectrics that undergo type II phase transitions and for which the SHG signal, l2Curie temperature is described by the Curie - Weiss Equation ... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

Table I reports the observed NMR linewidths for the H/3 protons of the coordinating cysteines in a series of iron-sulfur proteins with increasing nuclearity of the cluster, and in different oxidation states. We have attempted to rationalize the linewidths on the basis of the equations describing the Solomon and Curie contributions to the nuclear transverse relaxation rate [Eqs. (1) and (2)]. When dealing with polymetallic systems, the S value of the ground state has been used in the equations. When the ground state had S = 0, reference was made to the S of the first excited state and the results were scaled for the partial population of the state. In addition, in polymetallic systems it is also important to account for the fact that the orbitals of each iron atom contribute differently to the populated levels. For each level, the enhancement of nuclear relaxation induced by each iron is proportional to the square of the contribution of its orbitals (54). In practice, one has to calculate the following coefficient for each iron atom ...

This equation is called the Curie law and relates the equilibrium magnetization M0 to the strength of the magnetic field B0. The constants have the following meaning I is the nuclear spin quantum number (see below), y is the gyromagnetic ratio specific for a given isotope, h is Planck s constant, kB is Boltzmann s constant, N is the number of nuclei and T is the temperature. 

A plot of the reciprocal of the measured susceptibility 1 /imol v.v. T is a straight line with slope 1/C, and which crosses the abscissa at T = 0 (Fig. 19.6). For 0 = 0 the equation is simplified to the classic Curie law mol = C/T. Generally, values of 0 (j arc found when cooperative effects arise at low temperatures (ferro-, ferri- or antiferromagnetism). The straight line then has to be extrapolated from high to low temperatures (dashed lines in Fig. 19.6). 

Gibbs-Curie-Wulff equation for equilibrium form of crystal of constant volume ... 

S.4.3.2 Model of Hillert and Jarl. In his original treatment, Inden (1976) used a complicated but closed expression for the enthalpy, but had to use a series expansion in order to calculate the entropy. Hillert and Jarl (1978) therefore decided to convert the Cp expression directly through a series expansion which substantially simplifies the overall calculation and leads to a maximum error of only 1-2 J/mol at the Curie temperature of Fe. The equivalent equations to those used by Inden (1976) are given by... 

The electric susceptibility and dielectric constant of ferroelectric substances obey a Curie law dependence on temperature (Equation (9.13) ... 

One of the important characteristics of ferroelectrics is that the dielectric constant obeys the Curie- Weiss law (equation 6.48), similar to the equation relating magnetic susceptibility with temperature in ferromagnetic materials. In Fig. 6.55 the temperature variation of dielectric constant of a single crystal of BaTiOj is shown to illustrate the behaviour. Above 393 K, BaTiOj becomes paraelectric (dipoles are randomized). Polycrystalline samples show less-marked changes at the transition temperature. 

In an external magnetic field, Bq, the magnetization, Mq, of an ensemble of spins I is given by Curie s equation (10-13) ... 

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