Equilibrium fundamental fact

We will narrow our task and attempt to ehoose a funetion S sueh that its partial derivatives obey Equation (2.338). Of eourse, this does not allow us to find all possible figures of equilibrium, but for our purposes it is not important, because we are only interested to show that a rotating homogeneous spheroid, describing Earth, can be under certain conditions a figure of equilibrium. This fundamental fact was established by MacLauren in the 18th century. In this case we have... 

It can be seen from equation (34.18) that for an ideal liquid system, the vapor will always be relatively richer than the liquid in the more volatile constituent, i.e., the one with the higher vapor pressure. For example, if, as in Fig. 21, the component 1 is the more volatile, p will be greater than p, and hence Ni/i 2 will exceed Ni/ns, and the vapor will contain relatively more of the component 1 than does the liquid with which it is in equilibrium. This fact is fundamental to the separation of liquids by fractional distillation. The limitations arising in connection with nonideal systems will be considered in 35b, 35c. 

To reiterate, systems at stable equilibrium must be described in general in terms of two fixed state variables. If two different equilibrium states having the same values of two state variables exist, then either both are metastable, or one is stable and one is metastable, and the metastable states in fact should be described in terms of three or more constraint variables. Usually, however, the third constraint is an activation energy barrier and is not thought of as a third variable (though in principle it is). For each choice of the two state variables that the two states have in common, there exists a function (another state variable) that is minimized or maximized at stable equilibrium therefore, by comparing values of this variable one can tell which of the two states is more stable. Note finally that although we have many potentials, the existence of the entropy parameter is the fundamental fact that allows us to define them all. It appears in one way or another in all thermodynamic potentials. 

Ionic polymers contain two types of ions, namely bound ions, which are part of the structure, and the counterions, which are free. In a medium in which the ionic polymer is insoluble, the counterions are exchanged for similar ions from the surrounding medium and an equilibrium is established, the kinetics of the process being dependent on factors such as the physical form of the insoluble polyion, its porosity, and surface area. The fundamental fact in this exchange is that the counterions have free movement into... 

The most insightful use of the thermometer was made by Joseph Black (1728-1799), a professor of medicine and chemistry at Glasgow. Black drew a clear distinction between temperature or degree of hotness, and the quantity of heat. His experiments using the newly developed thermometers established the fundamental fact that, in thermal equilibrium, the temperatures of all the... 

Chemical systems do not tend to equilibrium by minimizing their internal energy. Another driving force is provided by spontaneous randomization of the position of molecules in space and of the distribution of energies among available energy levels. This fundamental fact is embodied in the macroscopic property called entropy. 

The properties of a system at equilibrium do not depend on how the system arrived at equilibrium. Therefore, Eq. (5.1-5) is valid for any system at equilibrium, not only for a system that arrived at equilibrium under conditions of constant T and P. We call it the fundamental fact of phase equilibrium In a multiphase system at equilibrium the chemical potential of any substance has the same value in all phases in which it occurs. 

Section 5.1 The Fundamental Fact of Phase Equilibrium 5.3 For water at equilibrium at 23.756 torr and 298.15 K,... 

The vapor pressure that we have discussed thus far is measured with no other substances present. We are often interested in the vapor pressure of a liquid that is open to the atmosphere. The other gases in the atmosphere exert an additional pressure on the liquid that modifies its vapor pressure. Small amounts of the other gases dissolve in the liquid, but we neglect these impurities in the liquid. Denote the vapor pressure corresponding to a total pressure of P by P. From the fundamental fact of phase equilibrium for a one-component system,... 

The fundamental fact of phase equilibrium is that at equilibrium... 

We now show that a component of an ideal solution obeys Raoult s law if the solution is at equilibrium with an ideal gas mixture. From the fundamental fact of phase equilibrium the chemical potential of component / has the same value in the solution and in the vapor ... 

Physical chemists always want to write a single equation that applies to as many different cases as possible. We would like to write equations similar to Eq. (6.1-8) for the chemical potential of every component of every solution. Consider a dilute solution in which the solvent and the solute are volatile. We equilibrate the solution with a vapor phase, which we assume to be an ideal gas mixture. Using Henry s law, Eq. (6.2-1), for the partial vapor pressure of substance number i (a solute) and using the fundamental fact of phase equilibrium ... 

Consider a solid solute that is soluble in a liquid solvent but insoluble in the solid solvent. Assume that the pure solid solvent (component number 1) is at equilibrium with a liquid solution containing the dilute solute. From the fundamental fact of phase equilibrium. 

Consider a volatile solvent (component 1) and a nonvolatile solute (component 2) in a solution that is at equilibrium with the gaseous solvent at a constant pressure. We assume that the gas phase is an ideal gas and that the solvent acts as though it were ideal. Our development closely parallels the derivation of the freezing point depression formula earlier in this section. The fundamental fact of phase equilibrium gives... 

Hydrochloric acid, HCl, is one of a half-dozen strong acids, which means that its acid ionization constant is too large to measure accurately. We must find a way to handle the activity of unionized species such as HCl in spite of their unmeasurably small concentrations. Since aqueous HCl has an appreciable vapor pressure we assume that aqueous unionized HCl in an aqueous solution of HCl is at equilibrium with gaseous HCl. From the fundamental fact of phase equilibrium... 

We call M/.chem the chemical part of the chemical potential. It is assumed to be independent of the electric potential and depends only on temperature, pressure, and the composition of the system. The chemical potential including the electric potential term is the true chemical potential that obeys the fundamental fact of phase equilibrium. Some electrochemists use the term electrochemical potential for the chemical potential in Eq. (8.1-7) and refer to the chemical part of the chemical potential as the chemical potential. We will use the term chemical potential for the tme chemical potential and the term chemical part of the chemical potential for /r,diein-... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

So far we have not touched on the fact that the important topic of solvation energy is not yet taken into account. The extent to which solvation influences gas-phase energy values can be considerable. As an example, gas-phase data for fundamental enolisation reactions are included in Table 1. Related aqueous solution phase data can be derived from equilibrium constants 31). The gas-phase heats of enolisation for acetone and propionaldehyde are 19.5 and 13 keal/mol, respectively. The corresponding free energies of enolisation in solution are 9.9 and 5.4 kcal/mol. (Whether the difference between gas and solution derives from enthalpy or entropy effects is irrelevant at this stage.) Despite this, our experience with gas-phase enthalpies calculated by the methods described in this chapter leads us to believe that even the current approach is most valuable for evaluation of reactivity. 

Most catalytic cycles are characterized by the fact that, prior to the rate-determining step [18], intermediates are coupled by equilibria in the catalytic cycle. For that reason Michaelis-Menten kinetics, which originally were published in the field of enzyme catalysis at the start of the last century, are of fundamental importance for homogeneous catalysis. As shown in the reaction sequence of Scheme 10.1, the active catalyst first reacts with the substrate in a pre-equilibrium to give the catalyst-substrate complex [20]. In the rate-determining step, this complex finally reacts to form the product, releasing the catalyst... 

Solubility and kinetics methods for distinguishing adsorption from surface precipitation suffer from the fundamental weakness of being macroscopic approaches that do not involve a direct examination of the solid phase. Information about the composition of an aqueous solution phase is not sufficient to permit a clear inference of a sorption mechanism because the aqueous solution phase does not determine uniquely the nature of its contiguous solid phases, even at equilibrium (49). Perhaps more important is the fact that adsorption and surface precipitation are essentially molecular concepts on which strictly macroscopic approaches can provide no unambiguous data (12, 21). Molecular concepts can be studied only by molecular methods. 

As noted above, often the kinetic equations are written as a function of i0 rather than k°. One of the advantages of using i0 is that the faradaic current can be described as a function of the difference between the potential applied to the electrode, E, and the equilibrium potential, Eeq, rather than with respect to the formal electrode potential, E01, (which, as previously mentioned, is a particular case of equilibrium potential [COx(0,f) = CRed(0,t)], and at times may be unknown). In fact, dividing the fundamental expression of i by that of i0 one obtains ... 

Equilibrium stable isotope fractionation is a quantum-mechanical phenomenon, driven mainly by differences in the vibrational energies of molecules and crystals containing atoms of differing masses (Urey 1947). In fact, a list of vibrational frequencies for two isotopic forms of each substance of interest—along with a few fundamental constants—is sufficient to calculate an equilibrium isotope fractionation with reasonable accuracy. A succinct derivation of Urey s formulation follows. This theory has been reviewed many times in the geochemical... 

The notion of fluid strains and stresses and how they relate to the velocity field is one of the fundamental underpinnings of the fluid equations of motion—the Navier-Stokes equations. While there is some overlap with solid mechanics, the fact that fluids deform continuously under even the smallest stress also leads to some fundamental differences. Unlike solid mechanics, where strain (displacement per unit length) is a fundamental concept, strain itself makes little practical sense in fluid mechanics. This is because fluids can strain indefinitely under the smallest of stresses—they do not come to a finite-strain equilibrium under the influence of a particular stress. However, there can be an equilibrium relationship between stress and strain rate. Therefore, in fluid mechanics, it is appropriate to use the concept of strain rate rather than strain. It is the relationship between stress and strain rate that serves as the backbone principle in viscous fluid mechanics. 

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