Mirror-image

In Fig. 13.7d, the grand composite curve for the reactor and that for the rest of the process are superimposed. To obtain maximum overlap, one of the curves must be taken as a mirror image. It can be seen in Fig. 13.7d that the reactor is appropriately placed relative to the rest of the process. Had the reactor not been appropriately placed, it would have been extremely... 

A molecule is chiral if it cannot be superimposed on its mirror image (or if it does not possess an alternating axis of symmetry) and would exhibit optical activity, i.e. lead to the rotation of the plane of polarization of polarized light. Lactic acid, which has the structure (2 mirror images) shown exhibits molecular chirality. In this the central carbon atom is said to be chiral but strictly it is the environment which is chiral. 

In certain crystals, e.g. in quartz, there is chirality in the crystal structure. Molecular chirality is possible in compounds which have no chiral carbon atoms and yet possess non-superimposable mirror image structures. Restricted rotation about the C=C = C bonds in an allene abC = C = Cba causes chirality and the existence of two optically active forms (i)... 

Some representative plots of entropies of adsorption are shown in Fig. XVII-23, in general, T AS2 is comparable to Ah2, so that the entropy contribution to the free energy of adsorption is important. Notice in Figs. XVII-23 i and b how nearly the entropy plot is a mirror image of the enthalpy plot. As a consequence, the maxima and minima in the separate plots tend to cancel to give a smoothly varying free energy plot, that is, adsorption isotherm. 

Unlike most words in a glossary of tenns associated with the theoretical description of molecules, the word synnnetry has a meaning in every-day life. Many objects look exactly like their mirror image, and we say that they are syimnetrical or, more precisely, that they have reflection synnnetry. In addition to having reflection synnnetry, a pencil (for example) is such that if we rotate it tln-ough any angle about its long axis it... 

The zeroth-order solution to the above equations is tire Gotiy-Chapman theory dating from the early part of the 20th cenPiry [20], In this solution, the ionic aPnosphere is ignored, as is the mirror image potential for the ion. Equation A2.4.90 can therefore be ignored and equation A2.4.89 reduces to... 

Next, we denote the line between the centres of the two particles at the point of closest approach by the unit vector k. In figure A3.1.7 it can also be seen that the vectors -g and g are each other s mirror images in the direction of kin the plane of the trajectory of particles ... 

If the tctravalcnt carbon atom has three difFcrcnt substituents, the molecule is chiral and it is not possible to superimpose it onto its mirror image. Our feet are also chiral objects the right foot is a mirror image of the left one and does not ftt into the left shoe. 

An example of a chiral compound is lactic acid. Two different forms of lactic acid that are mirror images of each other can be defined (Figure 2-69). These two different molecules are called enantiomers. They can be separated, isolated, and characterized experimentally. They are different chemical entities, and some of their properties arc different (c.g., their optical rotation),... 

A matrix is symmetric if it is a square matrix with elements such that the elements above and below the diagonal are mirror images = A . 

There is no need to form the eorresponding states with negative Ml or negative Ms values beeause they are simply "mirror images" of those listed above. For example, the state with Ml= -1 and Ms = -1 is p.iPpoP, whieh ean be obtained from the Ml = 1, Ms = 1 state piapoOt by replaeing a by P and replaeing pi by p L... 

For each M , Ml combination for which one can write down only one product function (i.e., in the non-equivalent angular momentum situation, for each case where only one product function sits at a given box row and column point), that product function itself is one of the desired states. For the p2 example, the piapoot and piap.ia (as well as their four other Ml and Ms "mirror images") are members of the 3p level (since they have Ms = 1) and IpiapiPI and its Ml mirror image are members of the level (since they have Ml... 

As shown in Figure 7 1 the two mirror images of bromochlorofluoromethane cannot be superimposed on each other Because the two mirror images of bromochlorofluoro methane are not supenmposable BrClFCH is chiral... 

The surest test for chirality is a careful examination of mirror image forms for superimposabihty Working with models provides the best practice m dealing with mol ecules as three dimensional objects and is strongly recommended... 

FIGURE 7 1 A molecule with four different groups attached to a single carbon is chiral Its two mirror image forms are not superimposable... 

Certain structural features can sometimes help us determine by inspection whether a mol ecule IS chiral or achiral For example a molecule that has a plane of symmetry or a cen ter of symmetry is superimposable on its mirror image and is achiral... 

A plane of symmetry bisects a molecule so that one half of the molecule is the mirror image of the other half The achiral molecule chlorodifluoromethane for exam pie has the plane of symmetry shown m Figure 7 3... 

FIGURE 7 4 (a) Struc tural formulas A and B are drawn as mirror images (b) The two mirror images are superimposable by rotating form B 180 about an axis passing through the center of the molecule The cen ter of the molecule is a center of symmetry... 

As outlined m Table 7 1 (+) 2 butanol has the S configuration Its mirror image IS (—) 2 butanol which has the R configuration... 

We can view this reaction as the replacement of one or the other of the two methylene protons at C 2 of butane These protons are prochiral atoms and as the red and blue protons m the Newman projection indicate occupy mirror image environments... 

Stereoisomer I is not a mirror image of III or IV so it is not an enantiomer of either one Stereoisomers that are not related as an object and its mirror image are called diastereomers diastereomers are stereoisomers that are not enantiomers Thus stereoisomer I is a diastereomer of III and a diastereomer of IV Similarly II is a diaste reomer of III and IV... 

Because diastereomers are not mirror images of each other they can have quite different physical and chemical properties For example the (2R 3R) stereoisomer of 3 ammo 2 butanol is a liquid but the (2R 3S) diastereomer is a crystalline solid... 

Fischer projection formulas can help us identify meso forms Of the three stereoisomeric 2 3 butanediols notice that only in the meso stereoisomer does a dashed line through the center of the Fischer projection divide the molecule into two mirror image halves... 

Structures A and A are nonsuperimposable mirror images of each other Thus although as 1 2 dichloro cyclohexane is chiral it is optically inactive when chair-chair interconversion occurs Such interconver Sion IS rapid at room temperature and converts opti cally active A to a racemic mixture of A and A Because A and A are enantiomers interconvertible by a conformational change they are sometimes re ferred to as conformational enantiomers... 

Steroids are another class of natural products with multiple chirality centers One such compound is cholic acid which can be obtained from bile Its structural formula IS given m Figure 7 12 Cholic acid has 11 chirality centers and so a total (including cholic acid) of 2" or 2048 stereoisomers have this constitution Of these 2048 stereoiso mers how many are diastereomers of cholic acid s Remember Diastereomers are stereoisomers that are not enantiomers and any object can have only one mirror image Therefore of the 2048 stereoisomers one is cholic acid one is its enantiomer and the other 2046 are diastereomers of cholic acid Only a small fraction of these compounds are known and (+) cholic acid is the only one ever isolated from natural sources... 

Occasionally an optically inactive sample of tartaric acid was obtained Pasteur noticed that the sodium ammonium salt of optically inactive tartaric acid was a mixture of two mirror image crystal forms With microscope and tweezers Pasteur carefully sep arated the two He found that one kind of crystal (m aqueous solution) was dextrorota tory whereas the mirror image crystals rotated the plane of polarized light an equal amount but were levorotatory... 

The CIS and trans isomers of 4 methylcyclohexanol are stereoisomers but they are not related as an ob ject and its mirror image they are diastereomers... 

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