Phases condensed

For condensed phases (liquids and solids) the molar volume is much smaller than for gases and also varies much less with pressure. Consequently the effect of pressure on the chemical potential of a condensed phase is much smaller than for a gas and often negligible. This implies that while for gases more attention is given to the volumetric properties than to the variation of the standard chemical potential with temperature, the opposite is the case for condensed phases. 

Variation of the standard chemical potential with temperature 

The thermodynamic properties of single-component condensed phases are traditionally given in tabulated form in large data monographs. Separate tables are given for each solid phase as well as for the liquid and for the gas. In recent years analytical representations have been increasingly used to ease the implementation of the data in computations. These polynomial representations typically describe the thermodynamic properties above room temperature (or 200 K) only. 

Polynomial expressions are conveniently used to represent a condensed phase which is stable in the whole temperature range of interest and which does not undergo any structural, electronic or magnetic transformations. The Gibbs energy of a compound is in the CALPHAD approach represented relative to the elements in their defined standard state at 298.15 K as a power series in terms of temperature in the form of [16]  

Here Hm is the sum (in the stoichiometric ratio of the compound in question) of A298-15tf° of the elements in their defined standard state, a, b, c and dn are coefficients and n integers. This form of expression is useful for storing thermodynamic information in databases. A number of such expressions are often required for a given phase to cover the whole temperature range of interest. From eq. (2.41) all other thermodynamic functions can be derived, e.g. 

Solids and liquids are referred to collectively as condensed phases. This name emphasizes the high density of the liquid or solid as compared with the low density of gases. This difference in density is one of the most striking differences between gases and solids or liquids. The mass of air in a room of moderate size would not exceed two hundred pounds the mass of liquid required to fill the same room would be some hundreds of tons. Conversely, the volume per mole is very large for gases and very small for liquids and solids. At STP a gas occupies 22,400 cm /mole, while the majority of liquids and solids occupy between 10 and 100 cm /mole. Under these conditions the molar volume of a gas is 500 to 1000 times larger than that of a liquid or solid. 

In gases the volume occupied by the molecules is small compared with the total volume, and the effect of the intermolecular forces is very small. In the first approximation these effects are ignored and any gas is described by the ideal gas law, which is strictly 

Such plots should produce a straight line with the slope being equal to -AH IR and the intercept equal to In(klh) + AS /R. As the available temperature range often is 100°C, the error in AH will typically be 0.5-2 kJ/mol. The activation entropy is determined by extrapolating outside the data points to r= oo (i/r = 0), and is usually somewhat less well defined a typical error may be 5J/mol K. 

Experimentalists often analyze their data in terms of an Arrhrenius expression instead of the TST expression eq. (13.39) by plotting In( rate) against T  

The connection with the TST expression (13.39) may be established from the definition in eq. (13.40) of the activation energy. 

Here An is the change in the number of molecules from the reactant to the TS, i.e. An = 0 for a unimolecular reaction, -1 for a bimolecular reaction, etc. For a solution phase reaction An is approximately 0. 

Note that for a reaction taking place by multiple reaction paths (e.g. conformational TS s), the observed activation energy is obtained from the observed rate constant, which is a sum over individual rate constants. 

Solids, and liquids under conditions of temperature and pressure not close to the critical point, are much less compressible than gases. Typically the isothermal compressibility, KT, of a liquid or solid at room temperature and atmospheric pressure is no greater than 1 X 10 bar (see Fig. 7.2 on page 165), whereas an ideal gas under these conditions has Kt = 1/p = bar . Consequently, it is frequently valid to treat V for a liquid or solid as essentially constant during a pressure change at constant temperature. Because kt is small, the product KtP for a liquid or solid is usually much smaller than the product aT. Furthermore, kt for liquids and solids does not change rapidly with as it does for gases, and neither does a. 

Thermodynamics and Chemistry, second e6 Wor, version 3 20 by Howard De ADe. Latest version www.chGm.umd.edu/themobook 

The electrochemical formation of NH2 by reduction of NH3 at Pt electrodes at 213 K was observed between —2.2 and -2.5 V in the presence of Nal as electrolyte [4]. Thermodynamic data for the formation of NH2 from NH3 by reduction with solvated electrons in NH3 solution at 233 K and estimated rate constants are as follows [5]  

The formation of NHJ during chemisorption of NH3 on dehydratyed Y-AI2O3 occurs only to a small extent and is attributed to the surface reaction + NH3 NH2+OH most of NH3 acts as a donor to Al centers see [11] and the literature cited there. NH2 ions at low concentrations were identified mass-spectrometrically after exposing pure Cu and Fe samples to high-frequency sparks. The ions are formed together with other cations and anions from impurities in the metals [12]. 

Deuterated amides can be prepared by reacting the metals with ND3 instead of NH3. A sample of NaNHD was obtained by melting together approximately equal amounts of NaNH2 and NaND2 [13]. The reaction of Ca with a mixture of NH3 and ND3 yielded a sample of Ca(NHD)2 [14]. 

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The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-... 

There has been much activity in the study of monolayer phases via the new optical, microscopic, and diffraction techniques described in the previous section. These experimental methods have elucidated the unit cell structure, bond orientational order and tilt in monolayer phases. Many of the condensed phases have been classified as mesophases having long-range correlational order and short-range translational order. A useful analogy between monolayer mesophases and die smectic mesophases in bulk liquid crystals aids in their characterization (see [182]). 

The three general states of monolayers are illustrated in the pressure-area isotherm in Fig. IV-16. A low-pressure gas phase, G, condenses to a liquid phase termed the /i uid-expanded (LE or L ) phase by Adam [183] and Harkins [9]. One or more of several more dense, liquid-condensed phase (LC) exist at higher pressures and lower temperatures. A solid phase (S) exists at high pressures and densities. We briefly describe these phases and their characteristic features and transitions several useful articles provide a more detailed description [184-187]. 

At lower temperatures a gaseous film may compress indefinitely to a liquid-condensed phase without a discemable discontinuity in the v-a plot. 

Condensed phases of systems of category 1 may exhibit essentially ideal solution behavior, very nonideal behavior, or nearly complete immiscibility. An illustration of some of the complexities of behavior is given in Fig. IV-20, as described in the legend. 

The total interaction between two slabs of infinite extent and depth can be obtained by a summation over all atom-atom interactions if pairwise additivity of forces can be assumed. While definitely not exact for a condensed phase, this conventional approach is quite useful for many purposes [1,3]. This summation, expressed as an integral, has been done by de Boer [8] using the simple dispersion formula, Eq. VI-15, and following the nomenclature in Eq. VI-19 ... 

Traditionally one categorizes matter by phases such as gases, liquids and solids. Chemistry is usually concerned with matter m the gas and liquid phases, whereas physics is concerned with the solid phase. However, this distinction is not well defined often chemists are concerned with the solid state and reactions between solid-state phases, and physicists often study atoms and molecular systems in the gas phase. The tenn condensed phases usually encompasses both the liquid state and the solid state, but not the gas state. In this section, the emphasis will be placed on the solid state with a brief discussion of liquids. 

Of course, condensed phases also exliibit interesting physical properties such as electronic, magnetic, and mechanical phenomena that are not observed in the gas or liquid phase. Conductivity issues are generally not studied in isolated molecular species, but are actively examined in solids. Recent work in solids has focused on dramatic conductivity changes in superconducting solids. Superconducting solids have resistivities that are identically zero below some transition temperature [1, 9, 10]. These systems caimot be characterized by interactions over a few atomic species. Rather, the phenomenon involves a collective mode characterized by a phase representative of the entire solid. 

A1.3.3 DENSITY FUNCTIONAL APPROACHES TO QUANTUM DESCRIPTIONS OF CONDENSED PHASES... 

The existence of intennolecular interactions is apparent from elementary experimental observations. There must be attractive forces because otherwise condensed phases would not fomi, gases would not liquefy, and liquids would not solidify. There must be short-range repulsive interactions because otherwise solids and liquids could be compressed to much smaller volumes with ease. The kernel of these notions was fomuilated in the late eighteenth century, and Clausius made a clear statement along the lines of this paragraph as early as 1857 [1]. 

This section discusses how spectroscopy, molecular beam scattering, pressure virial coeflScients, measurements on transport phenomena and even condensed phase data can help detemiine a potential energy surface. 

The standard state of a substance in a condensed phase is the real liquid or solid at 1 atm and T. 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

In this brief review of dynamics in condensed phases, we have considered dense systems in various situations. First, we considered systems in equilibrium and gave an overview of how the space-time correlations, arising from the themial fluctuations of slowly varying physical variables like density, can be computed and experimentally probed. We also considered capillary waves in an inliomogeneous system with a planar interface for two cases an equilibrium system and a NESS system under a small temperature gradient. 

At the limit of extremely low particle densities, for example under the conditions prevalent in interstellar space, ion-molecule reactions become important (see chapter A3.51. At very high pressures gas-phase kinetics approach the limit of condensed phase kinetics where elementary reactions are less clearly defined due to the large number of particles involved (see chapter A3.6). 

The introductory remarks about unimolecular reactions apply equivalently to bunolecular reactions in condensed phase. An essential additional phenomenon is the effect the solvent has on the rate of approach of reactants and the lifetime of the collision complex. In a dense fluid the rate of approach evidently is detennined by the mutual difhision coefficient of reactants under the given physical conditions. Once reactants have met, they are temporarily trapped in a solvent cage until they either difhisively separate again or react. It is conmron to refer to the pair of reactants trapped in the solvent cage as an encounter complex. If the unimolecular reaction of this encounter complex is much faster than diffiisive separation i.e., if the effective reaction barrier is sufficiently small or negligible, tlie rate of the overall bimolecular reaction is difhision controlled. 

In the sections below a brief overview of static solvent influences is given in A3.6.2, while in A3.6.3 the focus is on the effect of transport phenomena on reaction rates, i.e. diflfiision control and the influence of friction on intramolecular motion. In A3.6.4 some special topics are addressed that involve the superposition of static and transport contributions as well as some aspects of dynamic solvent effects that seem relevant to understanding the solvent influence on reaction rate coefficients observed in homologous solvent series and compressed solution. More comprehensive accounts of dynamics of condensed-phase reactions can be found in chapter A3.8. chapter A3.13. chapter B3.3. chapter C3.1. chapter C3.2 and chapter C3.5. 

The treatment of equilibrium solvation effects in condensed-phase kmetics on the basis of TST has a long history and the literature on this topic is extensive. As the basic ideas can be found m most physical chemistry textbooks and excellent reviews and monographs on more advanced aspects are available (see, for example, the recent review article by Tnihlar et al [6] and references therein), the following presentation will be brief and far from providing a complete picture. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

For very fast reactions, the competition between geminate recombmation of a pair of initially fomied reactants and its escape from the connnon solvent cage is an important phenomenon in condensed-phase kinetics that has received considerable attention botli theoretically and experimentally. An extremely well studied example is the... 

As these examples have demonstrated, in particular for fast reactions, chemical kinetics can only be appropriately described if one takes into account dynamic effects, though in practice it may prove extremely difficult to separate and identify different phenomena. It seems that more experiments under systematically controlled variation of solvent enviromnent parameters are needed, in conjunction with numerical simulations that as closely as possible mimic the experimental conditions to improve our understanding of condensed-phase reaction kmetics. The theoretical tools that are available to do so are covered in more depth in other chapters of this encyclopedia and also in comprehensive reviews [6, 118. 119],... 

Harris A L, Berg M and Harris C B 1986 Studies of chemical reactivity in the condensed phase. I. The dynamics of iodine photodissociation and recombination on a picosecond time scale and comparison to theories for chemical reactions in solution J. Chem. Phys. 84 788... 

Wang W, Nelson K A, Xiao L and Coker D F 1994 Molecular dynamics simulation studies of solvent cage effects on photodissociation in condensed phases J. Chem. Phys. 101 9663-71... 

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