Hand’s rule

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hand s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins. 

Atomic States, Term Symbols, and Hand s Rule... 

The remarkable agreement indicates that the one-electron approximation is capable of a very complete and adequate description of these systems if carefully carried through. Even the irregularities in the cohesion near the center of the two scries, which arise from spin polarizations associated with Hand s rule, are rather well given. The only significant discrepancies are in the bulk modulus of the strongly magnetic metals at the center of the iron series. 

The final step in this procedure is to determine which term has the lowest energy. This can be done by using two of Hand s rules ... 

The allocation electrons among degenerate orbitals can be formalized as Hand s rule For an atom in its ground state, the term with the highest multiplicity has the lowest energy. 

Rule 3 If two or more empty orbitals of equal energy are available, one electron occupies each with spins parallel until all orbitals are half-full, a statement called Hand s rule. 

The phase behavior of microemulsions is complex and depends on a number of parameters, including the types and concentrations of surfactants, cosolvents, hydrocarbons, brine salinity, temperature, and to a much lesser degree, pressure. There is no universal equation of state even for a simple microemulsion. Therefore, phase behavior for a particular microemulsion system has to be measured experimentally. The phase behavior of microemulsions is typically presented using a ternary diagram and empirical correlations such as Hand s rule. 

This section describes how to use Hand s rule to represent binodal curves and tie lines. The surfactant-oil-water phase behavior can be represented as a function of effective salinity after the binodal curves and tie lines are described. Binodal curves and tie lines can be described by Hand s rule (Hand, 1939), which is based on the empirical observation that equilibrium phase concentration ratios are straight lines on a log-log scale. Figures 7.15a and 7.15b show the ternary diagram for a type II(-) environment with equilibrium phases numbered 2 and 3 and the corresponding Hand plot, respectively. The line segments AP and PB represent the binodal curve portions for phase 2 and phase 3, respectively, and the curve CP represents the tie line (distribntion cnrve) of the indicated components between the two phases. Cy is the concentration (volnme fraction) of component i in phase) (i or j = 1, 2, or 3), and 1, 2, and 3 represent water, oil, and microemulsion, respectively. As the salinity is increased, the type of microemulsion is changed from type II(-) to type III to type II(-i-). C, represents the total amount of composition i. 

To use Hand s rule for phase behavior calculation, we need the values of the Hand parameter Ah. Although Ah is defined in Figme 7.16, UCHEM does not use Ah. Instead, C33maxo. C33maxi, and C33niax2 are the required input parameters, and they may be used to back-calculate Aho, Ahi, and Ah2, respectively, in UTCHEM. It would be less confusing had Am, instead of C33maxi, been used directly in UTCHEM. 

In snmmary, this section has introduced Hand s rule to describe phase compositions and discnssed how to estimate Hand parameters. From the preceding discnssion, we know that seven parameters are needed to describe phase... 

To follow the preceding section s discussion of the ternary diagram and Hand s rule, this section discnsses how to calculate phase compositions Cy. The general approach is to represent the binodal and distribution curves (tie lines) as a fnnction of the total concentration of water, oil, and surfactant (i.e., Ci, C2, C3—of which only two are independent) with the electrolyte concentration as a parameter, based on an idea outlined by Lake (1989). A similar approach was proposed by Pope and Nelson (1978) and Camilleri (1983). 

For N, the 2p is the only incomplete subshell. Following Hand s rule the three electrons in this subshell will singly occupy the three available orbitals there will be 3 unpaired electrons. 

When several molecular orbitals of equal or almost equal energy exist, electrons tend to fill these orbitals so as to maximize the number of unpaired electron spins. The more nearly equal the energy levels, the greater is this tendency. Hand s rule)... 

None of the three arrangements violates the PauU exclusion principle, so we must determine which one will give the greatest stability. The answer is provided by Hand s rule," which states that the most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins. The arrangement shown in (c) satisfies this condition. In both (a) and (b) the two spins cancel each other. Thus the orbital diagram for carbon is... 

When this condition is satisfied, the electron spins will spontaneously align as Hand s rule indicates they do in atoms. If Z < 5, all Z cl electrons will be of the same spin if Z > 5, all holes will be of the same spin, giving a net moment of 10 — Zj per ion, as illustrated in Fig. 20-18. 

An additional phenomenon is that electrons show a strong tendency to occupy as many states with the same spin as possible. Cr and Mn, for instance, both have five electrons with spin j in the 3d orbital. Electrons tend to occupy states with maximum spin angular moment, or at least with maximum total angular moment. In free atoms, the occupancy of states can be determined by Hand s rules (Hund, 1927 Condon Shortley, 1935) ... 

In our example of p, the ground term is the P. This term can be identifed as having the configuration in the margin. This is sometimes called Hand s rule of maximum multiplicity, introduced in Section 2.2.3. 

This behavior is summarized by Hand s rule (named for German physicist F. H. Hund), which states that the lowest-energy configuration for an atom is the one having the maximum number of unpaired electrons allowed by the Pauli principle in a particular set of degenerate orbitals. 

This is known as Hand s rule. An atom adopts the electronic configuration that has the greatest number of unpaired electrons in degenerate orbitals. Whilst this is all a bit theoretical in that isolated atoms are not found very often, the same rule applies to electrons in degenerate orbitals in molecules, as you will see soon. 

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