The approach motion of molecules

We discuss the motion of two molecules as they approach one another. Our ultimate goal is to understand chemical reactions. Toward this goal we need to know how close the two molecules will approach and how much energy is available to their relative motion. A chemical reaction requires the reactants to approach one another and the ideas of this section are applied to reactivity in Chapter 3. The same tools will allow us to examine the entire course of the collision of structureless particles, the theme of Chapter 4. By Chapter 5 we will frilly recognize the internal structure of the colliding molecules. 

This section has two parts. We first discuss the approach motion using the conservation of energy as our tool. Our purpose is to argue that in every collision 

In this section we center attention on the relative distance R of the colliding A - - B molecules. The motion of the coordinate R looks like that of the motion of a single particle with mass p, = mAmB/(niA + wb), usually called the reduced mass. This is discussed in textbooks of classical mechanics and in Section 2.2.7 below. 

To connect the impact parameter with more familiar notions, start with the angular momentum vector L defined as L = R x /rv. Here x denotes the vector product We show below that for a potential that depends only on the distance il, L is conserved. Lisa vector and so both its direction and its magnitude are constant The direction of L is perpendicular to both R and v. The constancy of the direction of L confines the motion to a plane. The magnitude L of the angular momentum is 

We begin by turning off the force between the two molecules. Newton s first law tells us that in the absence of a force the velocity v will continue to be along the same direction. At any time t we can therefore use the Pythagorean theorem to write the distance as a function of time  

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Obviously, the magic angle spinning techniques described briefly above rely on coherent averaging out of resonance line-broadening interactions. An alternative approach is to cause solids to assume the incoherent motion of molecules in the liquid phase. Essentially, two such methods exist. The first is the so-called ultra-fine particle NMR (UFPNMR) method which was proposed originally by Yesinowski [21 ] and developed further by Kimura [22,23]. The second is the sonically induced narrowing of the NMR spectra of solids (SINNMR) that was demonstrated recently by Homer et al. [24,25]. 

So far we have discussed the approach motion of the reactants. But there are usually other requirements, besides the closeness of the approach, for a reaction to take place. In particular, there may be steric requirements, some configurations of the colliding molecules may be more conducive to reaction. A direct experimental verification of the steric requirement can be obtained using oriented reactant molecules. The experiment allows the determination of the reactive asymmetry or the relative reactivity for the two configurations, say... 

In the reactant channel leading up to the transition region, motion along represents the FI atom approaching the molecule, while motion along / is the vibrational motion of the atom. The initial wavepacket is chosen to represent the desired initial conditions. In Figure 2, the FI2 molecule is initially in the ground... 

The inner layer (closest to the electrode), known as the inner Helmholtz plane (IHP), contains solvent molecules and specifically adsorbed ions (which are not hilly solvated). It is defined by the locus of points for the specifically adsorbed ions. The next layer, the outer Helmholtz plane (OHP), reflects the imaginary plane passing through the center of solvated ions at then closest approach to the surface. The solvated ions are nonspecifically adsorbed and are attracted to the surface by long-range coulombic forces. Both Helmholtz layers represent the compact layer. Such a compact layer of charges is strongly held by the electrode and can survive even when the electrode is pulled out of the solution. The Helmholtz model does not take into account the thermal motion of ions, which loosens them from the compact layer. 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

As in scattering theory in general, one can treat the role of V in either a time independent or a time dependent point of view. The latter is simpler if the perturbation V is either explicitly time dependent or can be approximated as such, say by replacing the approach motion during the collision by a classical path. Algebraic methods have been particularly useful in that context,2 where an important aspect is the description of a realistic level structure for H0. Figure 8.3 is a very recent application to electron-molecule scattering. 

The steric effects may be more pronounced in heterogeneous catalysts than in homogeneous reactions in solution. The rigid, solid surface restricts the approach of the reactants to the active centers and interaction between the reactants. The steric requirements are quite stringent when a two-point adsorption is necessary and when, in consequence, the internal motion of the adsorbed molecules is limited. In this way, the stereoselectivity of some heterogeneous catalytic reactions, for example, the hydrogenation of alkenes on metals (5) or the dehydration of alcohols on alumina and thoria (9), have been explained. 

It is possible that microbubble shell may be shattered during the interaction with an ultrasound pulse. Indeed, drastic variation of microbubble size, up to several-fold in less than a microsecond, has been reported [33], with linear speeds of the wall motion of microbubble approaching hundreds of meters per second in certain conditions. At these rates, it is easy to shatter the materials that would otherwise flow under slow deformation conditions. In some cases (e.g., lipid monolayer shells, which are held together solely by the hydrophobic interaction of the adjacent molecules), after such shattering the re-formation of the shell maybe possible in other cases - e.g., with a solid crosslinked polymer or a denatured protein shells - the detached iceberg-like pieces of the microbubble shell coat would probably not re-form and anneal, and the acoustic response of microbubbles to the subsequent ultrasound pulses would be different [34]. 

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