Maximum multiplicity, principle

A maximum-multiplicity principle predicts that this inequitable distribution is unlikely. That is, heat will not flow from the cold to the hot object. 

The simplest example is bis-cyclopentadienylco-balt(II), where we add one electron. This may go either into one of the 4p orbitals or into the relatively unstable kag orbital. In either of these cases, of course, it lias one unpaired electron (i.e., it is in a doublet state). For bis-cyclopentadienylnickel-(II),1 on the other hand, two electrons must go into these orbitals. Whether they go, one into kag and the other into some 4p level, or both into the 4p levels, is in many ways immaterial. The proximity of the kag orbital energy to that of the 4p orbitals will ensure that we shall be left with two singly occupied orbitals whose electrons are in a triplet state with their spins parallel (Hund s principle of maximum multiplicity). In bis-eyclopentadienylchromium-(II) two electrons arc removed. If the hag orbital is appreciably more stable than the de2g orbitals, both will come from the latter level, leaving one... 

The ground state of an atom may be chosen by application of Hund s rules. Hund s first rule is that of maximum multiplicity. It states that the ground state will be that having the largest value of 5, in the case of carbon the 3F. Such a system having a maximum number of parallel spins will be stabilized by the exchanf e energy resulting from their more favorable spatial distribution compared with that of paired electrons (see Pauli principle. Chapter 2). 

A second approach to bonding in molecules is known as the molecular orbital (MO) theory. The assumption here is that if two nuclei are positioned at an equilibrium distance, and electrons are added, they will go into molecular orbitals that are in many ways analogous to the atomic orbitals discussed in Chapter 2. In the atom there are s, p, d, f,. . . orbitals determined by various sets of quantum numbers and in the molecule we have a, tt, 5,. . . orbitals determined by quantum numbers. We should expect to find the Pauli exclusion principle and Hund s principle of maximum multiplicity obeyed in these molecular orbitals as well as in the atomic orbitals. 

Be able to distinguish between Pauli s exclusion principle, the Heisenberg uncertainty principle, and Hund s rule of maximum multiplicity. 

Hund s rules Empirical rules in atomic spectra that determine the lowest energy level for a configuration of two equivalent electrons (i.e. electrons with the same n and I quantum numbers), in a many-electron atom. (1) The lowest energy state has the maximum multiplicity consistent with the Pauli exclusion principle. (2) The lowest energy stale has the maximum total electron orbital angular momentum quantum number, consistent with rule (1). These rules were put forward by the German physicist Friedrich Hund (1896-1997) in 1925. 

Predicting Heads and Tails by a Principle of Maximum Multiplicity... 

Now w e will use very simple models to illustrate how this principle of maximum multiplicity also predicts the tendencies of gases to expand and exert pressure, the tendencies of molecules to mix emd diffuse, and the tendency of rubber to retract when it is stretched. 

This simple model shows that a polymer will tend to be found in a partly retracted state = 2, rather than stretched to length = 3, because of the greater number of conformations W = 4) for = 2 than for = 3 W = 1). For the same reason, the polymer also tends not to be flattened on the surface W = 2 when = 1. If you stretch a polymer molecule, it will retract on the basis of the principle that it will tend toward its state of maximum multiplicity. This is the nature of entropy as a force. 

Forces act on atoms and molecules. The degrees of freedom of a system wall change until the system reaches a state in which the net forces are zero. An alternative description is in terms of extremum principles systems tend toward states that have minimal energies or maximal multiplicities, or, as we will see in Chapter 8, a combination of the tw o tendencies. The tendency toward maximum multiplicity or maximum entropy accounts for the pressures of gases, the mixing of fluids, the retraction of rubber, and, as we ll see in the next chapter, the flow of heat from hot objects to cold ones. 

In Example 2.2 we found that gases expand because the multiplicity IV(V) increases with volume V. The dependence of IV on V defines the force called pressure. In Example 2.3 we found that particles mix because the multiplicity W(N) increases as particle segregation decreases. This tendency defines the chemical potential. These are manifestations of the principle that systems tend toward their states of maximum multiplicity, also known as the Second Law of thermodynamics. 

In this and the preceding chapter we have used simple models to predict that gases expand to hll the volume available to them, molecules will mix and diffuse to uniform concentrations, rubber retracts when pulled, and heat flows from hot bodies to colder ones. All these tendencies are predicted by a principle of maximum multiplicity a system will change its degrees of freedom so as to reach the microscopic arrangement with the maximum possible multiplicity. This principle is the Second Law of thermodynamics, and is much broader than the simple models that we have used to illustrate it. 

In Chapters 2 and 3 we used simple models to illustrate that the composition of coin flips, the expansion of gases, the tendency of particles to mix, rubber elasticity, and heat flow can be predicted by the principle that systems tend toward their states of maximum multiplicity W. However, states that maximize IT will also maximize ITor 15 IT + 5 or k In W, where k is any positive constant. Any monotonic function of W will have a maximum where W has a maximum. In particular, states that maximize IT also maximize the entropy, S = fclnlT. Why does this quantity deserve special attention as a prediction principle, and why should it have this particular mathematical form ... 

However, the principle of the flat distribution, or maximum multiplicity, is incomplete. The Principle of Fair Apportionment needs a second clause that describes how probabilities are apportioned between the possible outcomes when there u e constraints or biases. If die rolls give an average score that is convincingly different from 3.5 per roll, then the outcomes are not all equivalent and the probability distribution is not flat. The second clause that completes... 

The following three articles show how the maximum entropy principle follows from the multiplication rule. A... 

Of the pos.sible terms allowed by the Pauli principle, that with the maximum multiplicity lies lowest and its multiplicity is 2S + 1 ... 

The second principle is called Hund s Rule, named for its developer, German physicist Friedrich Hund (1896-1997). Himd s Rule is sometimes called the rule of maximum multiplicity. It says that electrons occupying the same subshell tend to have the same spin. The result is to spread the electrons over as many orbitals in the subshell as possible before filling orbitals. 

Putting electrons into orbitals in multielectron atoms is governed by three rules, the Aufbau principle, the Pauli exclusion principle, and Hund s rule. The Aufbau, or building-up, principle tells us to put the electrons in the lowest-energy orbital that is available. The Pauli principle restricts the contents of the orbital to two electrons, with spins, s, +1/2 and -1/2. Hund s rule of maximum multiplicity (the law of antisocial electrons... ) means that where there is more than one orbital of equivalent energy, the electrons distribute between them in order to keep apart. 

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