Molecules proportions

When the density n of XY molecules is sufficiently high, the reaction on the first line maintains a supply of (XY) molecules proportional to n. Each (XY) has a certain probability per unit time to decompose according to the second line. Hence the overall reaction proceeds at a rate proportional to n rather than n2. See the Exercise below. 

A measure of the polarity of a bond (or a molecule), proportional to the product of the charge separation times the bond length, (p. 10)... 

Studies on structure and dynamics of liquids have recently been extended to solvate structure of ions in non-aqueous solutions, and to the structure of complexes with relatively complicated ligands. We can also handle special problems like hydrophobic solvation is. Diffraction studies have been performed on new solvents as e.g. trifluoroethanol [23] and tetramethyl urea [26], and on solvent mixtures [27-30]. More recently the preferential solvation of ions has been subjected by an XD investigation in MgCh-water-methMol ternary systems [31], and the solvation structure around the cations proved to undergo the change of solvent molecules proportionally to the relative concentration of the two solvents. 

Electrostriction is an effect that requires the existence of inhomogeneities in the intensity of the electric field. The inhomogeneous field creates a force on the molecules proportional to the gradient of the intensity of the electric field. A typical value for electrostriction susceptibility is x ... 

The left-hand side of the equation is the third power of the root-mean-square radius of gyration of the molecule (proportional to the volume or size of the molecule). The right-hand side contains the product of the molecular weight M of the polymer and the intrinsic viscosity [> ]. is a proportionality constant. Braoit et at. (4) found that he could obtain a univer curve for aU polymers when he plotted the product of the intrinsic viscosity and the molecular weight against the dution volume. 

When adsorption is such that 0< 1, the rate of adsorption is, neglecting interaction between adsorbed molecules, proportional to p(l - 6), where (1 - 0) is the fractional available space for adsorption and p is the gas pressure. The model is similar to that describing the kinetics of a bimolecular reaction. The rate of desorption is proportional to 0 or the concentration of the adsorbed species. At equilibrium... 

The drcumference of the drde in Figure A2-16 represents the same pore diameter as in Figure A2-15. Similarly, there are three water molecules — proportionally sized to the pore diameter. With just a few very smalF water molecules, there is a low probability that any one molecule will be located near the wall. So at low levels of water in the air fed to the adsorber bed, one would exped litde adsorption of water. 

Light scattering processes are related to the polarizability of the molecule which interacts with the EM field. A simple classical EM field description of Raman spectroscopy can be used to explain many of the important features of Raman band intensities. The incident EM field Eioc(ci>inc). induces a dipole moment pind in the molecule proportional to the molecular polarizability amoiecuie ... 

Expressed in moles this leads to the factor n. The attraction is limited to not too large particle-to-particle separation. We assume that two particles feel attracted if they are in the same volume element AV. The probability that two particular particles are found within A V simultaneously (A V/ V). Assuming this to be true for all possible pairs leads to an overall number of attracted molecules proportional to (n/ V). The resulting Eq. (4.1) is the van der Waals equation of state for gases and... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Boyle s law At constant temperature the volume of a given mass of gas is inversely proportional to the pressure. Although exact at low pressures, the law is not accurately obeyed at high pressures because of the finite size of molecules and the existence of intermolecular forces. See van der Waals equation. 

Raoult s law When a solute is dissolved in a solvent, the vapour pressure of the latter is lowered proportionally to the mole fraction of solute present. Since the lowering of vapour pressure causes an elevation of the boiling point and a depression of the freezing point, Raoult s law also applies and leads to the conclusion that the elevation of boiling point or depression of freezing point is proportional to the weight of the solute and inversely proportional to its molecular weight. Raoult s law is strictly only applicable to ideal solutions since it assumes that there is no chemical interaction between the solute and solvent molecules. 

Customarily, it is assumed that e is unity and that ]l = p,cos 9, where 0 is the angle of inclination of the dipoles to the normal. Harkins and Fischer [86] point out the empirical nature of this interpretation and prefer to consider only that AV is proportional to the surface concentration F and that the proportionality constant is some quantity characteristic of the film. This was properly cautious as there are many indications that the surface of water is structured and that the structure is altered by the film (see Ref. 37). Accompanying any such structural rearrangement of the substrate at the surface should be a change in its contribution to the surface potential so that AV should not be assigned too literally to the film molecules. 

The chemical reactivity of a self-similar surface should vary with its fractional dimension. Consider a reactive molecule that is approaching a surface to make a hit. Taking Fig. VII-6d as an illustration, it is evident that such a molecule can see only a fraction of the surface. The rate of dissolving of quartz in HF, for example, is proportional to where Dr, the reactive... 

Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

The exchange-repulsion energy is approximately proportional to the overlap of the charge densities of the interacting molecules [71, 72 and 73]... 

As described at the end of section Al.6.1. in nonlinear spectroscopy a polarization is created in the material which depends in a nonlinear way on the strength of the electric field. As we shall now see, the microscopic description of this nonlinear polarization involves multiple interactions of the material with the electric field. The multiple interactions in principle contain infomiation on both the ground electronic state and excited electronic state dynamics, and for a molecule in the presence of solvent, infomiation on the molecule-solvent interactions. Excellent general introductions to nonlinear spectroscopy may be found in [35, 36 and 37]. Raman spectroscopy, described at the end of the previous section, is also a nonlinear spectroscopy, in the sense that it involves more than one interaction of light with the material, but it is a pathological example since the second interaction is tlirough spontaneous emission and therefore not proportional to a driving field... 

The canonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same temperature T. This corresponds to putting the systems in a thennostatic bath or, since the number of systems is essentially infinite, simply separating them by diathennic walls and letting them equilibrate. In such an ensemble, the probability of finding the system in a particular quantum state / is proportional to where UfN, V) is tire energy of the /th quantum state and /c, as before, is the Boltzmaim... 

While a laser beam can be used for traditional absorption spectroscopy by measuring / and 7q, the strength of laser spectroscopy lies in more specialized experiments which often do not lend themselves to such measurements. Other techniques are connnonly used to detect the absorption of light from the laser beam. A coimnon one is to observe fluorescence excited by the laser. The total fluorescence produced is nonnally proportional to the amount of light absorbed. It can be used as a measurement of concentration to detect species present in extremely small amounts. Or a measurement of the fluorescence intensity as the laser frequency is scaimed can give an absorption spectrum. This may allow much higher resolution than is easily obtained with a traditional absorption spectrometer. In other experiments the fluorescence may be dispersed and its spectrum detennined with a traditional spectrometer. In suitable cases this could be the emission from a single electronic-vibrational-rotational level of a molecule and the experimenter can study how the spectrum varies with level. 

For molecules exposed to the intensity of sunlight at the earth s surface this would suggest that the molecule might be excited once in the age of the universe. However, the probability is proportional to the square of the light intensity. For a molecule exposed to a pulsed laser focused to a small spot, the probability of being excited by one pulse may be easily observable by fluorescence excitation or multiphoton ionization teclnhques. 

---