Potential energy substitution

Figure B3.4.1. The potential surfaee for the eollinear D + H2 DH + H reaetion (this potential is the same as for H + H2 — H2 + H, but to make the produets and reaetants identifieation elearer the isotopieally substituted reaetion is used). The D + H2 reaetant arrangement and the DH + H produet arrangement are denoted. The eoordinates are r, the H2 distanee, and R, the distanee between the D and the H2 eentre of mass. Distanees are measured in angstroms the potential eontours shown are 4.7 eV-4.55 eV,.. ., -3.8 eV. (The potential energy is zero when the partieles are far from eaeh other. Only the first few eontours are shown.) For referenee, the zero-point energy for H2 is -4.47 eV, i.e. 0.27 eV above the H2 potential minimum (-4.74 eV) the room-temperature thennal kinetie energy is approximately 0.03 eV. The graph uses the aeeiirate Liu-Seigbalm-Triihlar-Horowitz (LSTH) potential surfaee [195].

The use of isotopic substitution to detennine stmctures relies on the assumption that different isotopomers have the same stmcture. This is not nearly as reliable for Van der Waals complexes as for chemically bound molecules. In particular, substituting D for H in a hydride complex can often change the amplitudes of bending vibrations substantially under such circumstances, the idea that the complex has a single stmcture is no longer appropriate and it is necessary to think instead of motion on the complete potential energy surface a well defined equilibrium stmcture may still exist, but knowledge of it does not constitute an adequate description of the complex. 

I the sum of the kinetic and potential energy of an electron in the orbital lUg in the electro-atic field of the two bare nuclei. This integral can in turn be expanded by substituting the... 

Section 4 9 The potential energy diagrams for separate elementary steps can be merged into a diagram for the overall process The diagram for the reac tion of a secondary or tertiary alcohol with a hydrogen halide is charac terized by two intermediates and three transition states The reaction is classified as a ummolecular nucleophilic substitution, abbreviated as SnI... 

If the Lewis base ( Y ) had acted as a nucleophile and bonded to carbon the prod uct would have been a nonaromatic cyclohexadiene derivative Addition and substitution products arise by alternative reaction paths of a cyclohexadienyl cation Substitution occurs preferentially because there is a substantial driving force favoring rearomatization Figure 12 1 is a potential energy diagram describing the general mechanism of electrophilic aromatic substitution For electrophilic aromatic substitution reactions to... 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

The dissociation energy is unaffected by isotopic substitution because the potential energy curve, and therefore the force constant, is not affected by the number of neutrons in the nucleus. However, the vibrational energy levels are changed by the mass dependence of 03 (proportional to where /r is the reduced mass) resulting in Dq being isotope-... 

If further AU = AE when the kinetic and potential energies in Equation 2.36 do not change. Equation 2.35 can be rewritten, substituting U for E, changing to the specific notation and putting the equation in differential form. 

A. Potential energy of substituted acetaldehydes as a iunction of the OCCR angle, relative to 4k energy of the syn (ZOCCR = 0°) rotamer isolated molecules. [Reproduced with permission of Ejsevier Science Publishing.]... 

The ionization mechanism for nucleophilic substitution proceeds by rate-determining heterolytic dissociation of the reactant to a tricoordinate carbocation (also sometimes referred to as a carbonium ion or carbenium ion f and the leaving group. This dissociation is followed by rapid combination of the highly electrophilic carbocation with a Lewis base (nucleophile) present in the medium. A two-dimensional potential energy diagram representing this process for a neutral reactant and anionic nucleophile is shown in Fig. 

The concept of ion pairs in nucleophilic substitution is now generally accepted. Presumably, the barriers separating the intimate, solvent-separated, and dissociated ion pairs are quite small. The potential energy diagram in Fig. 5.4 depicts the three ion-pair species as being roughly equivalent in energy and separated by small barriers. 

Fig. 5.5. Potential energy diagrams for substitution mechanisms. A is the S l mechanism. B is the Sjy2 mechanism with intermediate ion-pair or pentacooidi-nate species. C is the classical S).(2 mechanism. [Reproduced from T. W. Bentley and P. v. R. Schleyer, Adv.

Fig. 10.6. Various potential energy profiles for eleetrophilie aromatie substitution.

Ashton solved this problem approximately by recognizing that the differential equation, Equation (5.32), is but one result of the equilibrium requirement of making the total potential energy of the mechanical system stationary relative to the independent variable w [5-9]. An alternative method is to express the total potential energy in terms of the deflections and their derivatives. Specifically, Ashton approximated the deflection by the Fourier expansion in Equation (5.29) and substituted it in the expression for the total potential energy, V ... 

Figure 12.1 is a potential energy diagram describing the general mechanism of electrophilic aromatic substitution. For electrophilic aromatic substitution reactions to... 

FIGURE 12.1 Potential energy diagram for electrophilic aromatic substitution. 

Consider a reactant molecule in which one atom is replaced by its isotope, for example, protium (H) by deuterium (D) or tritium (T), C by C, etc. The only change that has been made is in the mass of the nucleus, so that to a very good approximation the electronic structures of the two molecules are the same. This means that reaction will take place on the same potential energy surface for both molecules. Nevertheless, isotopic substitution can result in a rate change as a consequence of quantum effects. A rate change resulting from an isotopic substitution is called a kinetic isotope effect. Such effects can provide valuable insights into reaction mechanism. 

Fig. 7. Potential-energy diagrams for nucleophilic heteroaromatic substitutions. A, solid line very probable and common B, solid line probable but less frequent A and B, dotted lines scarcely probable and/or infrequent.

Complete and Incomplete Ionic Dissociation. Brownian Motion in Liquids. The Mechanism of Electrical Conduction. Electrolytic Conduction. The Structure of Ice and Water. The Mutual Potential Energy of Dipoles. Substitutional and Interstitial Solutions. Diffusion in Liquids. 

On the other hand, if Fig. 24b is compared with Fig. 23a, it will be seen that here each solute particle occupies a position that in the pure solvent would be occupied by a solvent particle. Such a solution, which can be formed by one-for-one substitution, is called a substitutional solution."1 This kind of solution will not be formed if the forces of attraction between adjacent solute and solvent particles are weak, while the forces of attraction between adjacent solvent particles are strong. For, if we look at Fig. 24b, we see that each solute particle prevents three solvent particles from coming together under their mutual attraction—that is to say, it prevents them from falling to a state of much lower potential energy. We can be certain that, when neon or argon is dissolved in water, the solute particles will not tend to take up such positions, which are suitable only for a solute particle which attracts an adjacent solvent particle with a force at least as great as the force of attraction between two adjacent solvent particles. 

The first term on the right is the potential energy of the solute molecule at the center of the cavity and the second term is the average of the extra potential energy of the molecule due to its motion in the field w(r). By substituting the values of w(o) and d In g/dT following from the L-J-D theory, one obtains... 

Fig. 12-7. Potential energy contour diagram showing the course of an aromatic substitution X+ + ArH - ArX + H+ (after Zollinger, 1956 a).

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

In highly exothermic reactions such as this, that proceed over deep wells on the potential energy surface, sorting pathways by product state distributions is unlikely to be successful because there are too many opportunities for intramolecular vibrational redistribution to reshuffle energy among the fragments. A similar conclusion is likely as the total number of atoms increases. Therefore, isotopic substitution is a well-suited method for exploration of different pathways in such systems. 

Fig. 7.6 Torsion angle distribution for ortbo-substituted phenol ethers (bars) and the derived potential energy (closed line).

The possible orientations of L with respect to B for the case 1 = 3 are illustrated in Figure 5.6. Classically, all values between 0 and Jt are allowed for the angle 6. When equations (5.81) and (5.88) are substituted into (5.83), we find that the potential energy Fis also quantized... 

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