Orientational entropy

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

Clearly, proximity and orientation play a role in enzyme catalysis, but there is a problem with each of the above comparisons. In both cases, it is impossible to separate true proximity and orientation effects from the effects of entropy loss when molecules are brought together (described the Section 16.4). The actual rate accelerations afforded by proximity and orientation effects in Figures 16.14 and 16.15, respectively, are much smaller than the values given in these figures. Simple theories based on probability and nearest-neighbor models, for example, predict that proximity effects may actually provide rate increases of only 5- to 10-fold. For any real case of enzymatic catalysis, it is nonetheless important to remember that proximity and orientation effects are significant. 

In either case each dimer has two possible orientations, and random disorder between these accounts for the residual entropy of the crystal (6.3JmoH of dimer). More recently ii)... 

Consider an ionic crystal in which the anion is a molecular ion. The orientation of this anion in the crystal is completely determined, or determined to a large extent, by the crystal structure and furthermore, its freedom of libration is severely limited by the intense fields of the adjacent ions. When this ion goes into solution, it will have a greater number of possible orientations, and its freedom of libration will be greater. Hence the AS0 for a molecular anion will contain a considerable increment in entropy over and above the cratic term (which is all that we subtract in the case of an atomic ion). This additional increment in entropy is likely to be somewhat different for different species of anion. The best we can do at present is to subtract an amount that is of the right order of magnitude. The question is whether we can, by sub-... 

The entropy discrepancy resulting from mixing the two different orientations of the dimer is 5 ft In 2 = 2.88 J-K -mol"1. The factor of results from the fact that only j mole of N2O2 is present. The experimental difference of 3.2 J-Kr -mol-1 is within experimental error of the expected value. 

Paramagnetic substances have a magnetic entropy associated with the random orientation of the spin of the unpaired electrons as shown in (a) below ... 

The temperature coefficient of the potential of zero charge has often been suggested to indicate the orientation of solvent molecules at the met-al/solution interface. However, this view is based only on the response of a simple two-state model for the interfacial solvent, and on neglecting any contribution from the electronic entropy.76,77 This is in fact not the case. The temperature coefficient of 0in many instances is negative and of the... 

That is, S —> 0 as T - 0. The perfect crystal part of this statement of the third law refers to a substance in which all the atoms are in a perfectly orderly array, and so there is no positional disorder. The T— 0 part of the statement implies the absence of thermal motion-—thermal disorder vanishes as the temperature approaches zero. As the temperature of a substance is raised from zero, more orientations become available to the molecules and their thermal disorder increases. Thus we can expect the entropy of any substance to he greater than zero above T = 0. 

Calculate the entropy of a tiny solid made up of four diatomic molecules of a compound such as carbon monoxide, CO, at T = 0 when (a) the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left (top-left image in Fig. 7.7) and (b) the four molecules lie in random orientations (but parallel, any of the images in Fig. 7.7). 

Self-Test 7.9A Calculate the entropy of a sample of a solid in which the molecules can take any one of three orientations with the same energy. Suppose there are 30... 

For a more realistic sample size than that in Example 7.7, one that contains 1.00 mol CO, corresponding to 6.02 x 1023 CO molecules, each of which could be oriented in either of two ways, there are 2602x10 (an astronomically large number) different microstates, and a chance of only 1 in 2< 02x l0" of drawing a given microstate in a blind selection. We can expect the entropy of the solid to be high and calculate that... 

STRATEGY The existence of residual entropy at T = 0 suggests that the molecules are disordered. From the shape of the molecule (which can be obtained by using VSEPR theory), we need to determine how many orientations, W, it is likely to be able to adopt in a crystal then we can use the Boltzmann formula to see whether that number of orientations leads to the observed value of S. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

OB Nonzero entropy at T = 0 indicates disorder. This disorder results when a molecule can orient itself more than one way in the crystal. In ice, each O atom is surrounded by four H atoms, of which there are two types. Two of the H atoms are covalently bonded to the O atom and the other two H atoms, which belong to neighboring water molecules, are interacting with the central O atom through hydrogen bonds. Thus, more than one orientation is possible in the crystal and entropy will nor be zero at T = 0. 

Chapter 9, on entropy and molecular rotation in crystals and liquids, is concerned mostly with statistical mechanics rather than quantum mechanics, but the two appear together in SP 74. Chapter 9 contains one of Pauling s most celebrated papers, SP 73, in which he explains the experimentally measured zero-point entropy of ice as due to water-molecule orientation disorder in the tetrahedrally H-bonded ice structure with asymmetric hydrogen bonds (in which the bonding proton is not at the center of the bond). This concept has proven fully valid, and the disorder phenomenon is now known to affect greatly the physical properties of ice via the... 

Entropy of activation (A5 ), which is the difference in entropy between the starting compounds and the transition state, becomes important when two reacting molecules must approach each other in a specific orientation in order for the reaction to take place. For example, the reaction between a simple noncyclic alkyl chloride and hydroxide ion to give an alkene (17-12) takes place only if, in the transition state, the reactants are oriented as shown. 

Not only must the OH be near the hydrogen, but the hydrogen mu.st be oriented anti to the chlorine atom. When the two reacting molecules collide, if the OH should be near the chlorine atom or near R or R, no reaction can take place. In order for a reaction to occur, the molecules must surrender the freedom they normally have to assume many possible arrangements in space and adopt only that one that leads to reaction. Thus, a considerable loss in entropy is involved, that is, A5, is negative. 

As we shall see in Chapter 17, with some molecules elimination is also possible if the hydrogen is oriented syn, instead of anti, to the chlorine atom. Of course, this orientation also requires a considerable loss of entropy. 

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