Quantum Effects in the Liquid Phase

Introduction 374. 2. High Temperature Expansions 380. 3. Ground State 382. 

Cell Model 385. 5. Effect of Quantum Statistics 389. 6. Isotope Effect and 

This part of the book is devoted to isotopic mixtures and especially to the study of the deviations of such solutions from the laws of perfect solutions. We begin with a preliminary summary of quantum effects in one-component condensed systems (liquids or solids). These considerations will be developed only as far as required for our main pmpose the problem of solutions. 

We shall consider molecules interacting according to a (6-12) law. We have seen in Ch. II, 5, that such systems statisfy a theorem of corresponding states of the form 

The classical case corresponds to the vanishing de Broglie wave length. Formula (18.1.2) permits to visualize quite easily the factors which determine quantum effects. For heavy particles A is small and we may n lect the quantum effects. Also an increase in the interaction parameter e lowers A. If we consider for example the rare gases, both factors work in the same direction because with increasing number of electrons there is an increase in the polarisability of the molecule and consequently of e. For this reason the value of A drops rapidly if we go from He to A (cf. Table 18.1.1). Table 18.1.1 gives the values of the quantum mechanical parameter a for some atoms and molecules. 

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The previous discussion pointed out some well-known limitations of the different theoretical approaches currently used to analyse polarization effects and estimate the dipole moment in the liquid phase. We will focus the present analysis of polarization effects in HB liquids on results obtained by using the sequential statistical mechanics/quantum mechanics approach. 

Until now we had been talking of gas reactions. Many substances undergo photochemical reactions in liquid state. Again the reaction in initiated by Stark-Einstein law by direct light absorption on the part of reactants. However, it maybe anticipated that quantum efficiency of these reactions will be less than for the same reaction in the gas phase. The reason for this is that in the liquid state an active molecule may readily be deactivated by frequent collisions with other molecules. Furthermore, because of the very short mean free path in the liquid phase free radicals or atoms when formed photochemically will tend to recombine before they have a chance to get very far from each other. The net effect of these processes will be to keep the quantum yield relatively low. In fact, only those reactions may be expected to proceed to any extent for which the primary products of the photochemical act are relatively stable particles. Otherwise the active intermediates will tend to recombine with the solvent and thereby keep the yield low. 

In practice, we approximate the exact transmission coefficient by a mean-field-type of approximation that is we replace the ratio of averages by the ratio for an average or effective potential. For gas-phase reactions with small reaction-path curvature, this effective potential would just be the vibrationally adiabatic ground-state potential. In the liquid phase and enzymes we generalize this with the canonical mean-shape approximation. In any event, though, the transmission coefficient should not be thought of as a perturbation. The method used here may be thought of as an approximate full-dimensional quantum treatment of the reaction rate. 

In the liquid phase, the relation between the chemical potential and the equilibrium density of i can be modified in two ways. First, there is the coupling work of i against the entire solvent, which we denote by W i l). Second, the solvent molecules can perturb the internal degrees of freedom of the solvated molecule. The latter is essentially a quantum-mechanical problem. We shall always assume that the effect of the solvent on the... 

Here v and m denote the volume and mass of the molecule or atom, respectively. The r.h.s of Equation 32 denotes the ground-state energy of a quantum mechanical particle enclosed in a potential well (particle in a box problem [Martin and Leonard, 1970]). This condition is not satisfied for liquid helium and liquid hydrogen, while liquid neon is a borderline case. For the theoretical description of their thermophysical properties, application of the Maxwell-Boltzmann statistics sometimes does not suffice. Another assumption states that the internal degrees of freedom of the molecules or atoms are the same in the gas phase and in the liquid phase. In other words, it is assumed that the molecules can rotate and vibrate freely in the liquid phase, too. Molecular rotation may be hindered in the case of long-chain hydrocarbons or silicone fluids with side groups but also for small, nonspherical molecules such as N2,02, CS2, and others, rotation around two axes is restricted due to steric hindrance. Polar molecules exhibit restricted rotation due to the effect of dipolar orientation. 

Classically it is impossible for a substance to remain in the liquid phase as the temperature approaches absolute zero since the internal energy tends to zero. Quantum mechanically a condensed phase at absolute zero has a zero-point energy (see Quantum Mechanics by P.C.W. Davies in this series), but, generally, this is too small to have an important effect on the behaviour. The one known exception to the classical rule is that of helium, which remains liquid down to the lowest temperatures unless... 

The effects of transfer of atoms by tunneling may play an essential role in a number of phenomena involving the transfer of atoms and atomic groups in the condensed phase. One may expect that these effects may exist not only in the proton transfer reactions considered above but also in such processes as the diffusion of hydrogen atoms and other light ions (e.g., Li+) in liquids, tunnel inversion and isomerization in some molecules, quantum diffusion of defects and light atoms in the electrode at cathodic incorporation of the ions, ion transfer across the liquid/solid interface, and low-temperature chemical reactions. 

One of the most important motivations for the study of gaseous systems, as repeatedly hinted at, is the hope of obtaining a better connection with theory and theoretical modeling. The structure of solvated adducts and charge-transfer pairs in solution cannot be deduced directly from experimental data. In the gas phase, rota-tionally resolved spectroscopy provides information on the structure. The method also allows a much better vibrational resolution than liquid-phase spectroscopy, allowing in principle the elucidation of subtle effects such as the role of torsional motion. All of these advantages are enhanced in supersonic jets, where only a small number of quantum states are initially populated. 

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