Virial isotherm

Our purpose in this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. Our model parameters are isobaric-isothermal virial coefficients which are about an order of magnitude smaller than the osmotic virial coefficients in the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically did not find this to be very important at the level of accuracy commonly attainable in experimental phase diagrams for these systems. 

The three isothermal virial coefficients can also be determined experimentally from P-V-T data and assuming that a real gas approaches ideality as (1/VJ —> 0 and as P —> 0 then A = RT. Actually, writing the virial equation of state as follows ... 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

Let us compare the theoretical isotherm of Equation 3-1 with the tc-T relationship represented as the standard expansion with the second a and third 3 virial coefficients as in... 

Nevertheless, surfactant sorption isotherms on natural surfaces (sediments and biota) are generally non-linear, even at very low concentrations. Their behaviour may be explained by a Freundlich isotherm, which is adequate for anionic [3,8,14,20,30], cationic [7] and non-ionic surfactants [2,4,15,17] sorbed onto solids with heterogeneous surfaces. Recently, the virial-electrostatic isotherm has been proposed to explain anionic surfactant sorption this is of special interest since it can be interpreted on a mechanistic basis [20]. The virial equation is similar to a linear isotherm with an exponential factor, i.e. with a correction for the deviation caused by the heterogeneity of the surface or the energy of sorption. 

Fig. 5.4.7. Sorption isotherms of C12-LAS on different sediments. The lines were calculated with the virial equation (from Westall et al. [20]).

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (J, 2 to obtain cross term second virial coefficients Bj2- At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy-pressure coefficient pressure yield the quantity 4>c = B - TdB/dT where B is the second virial coefficient. From values of obtain values of B without recourse to pVT measurements. 

If the experimental data of adsorption are fitted into the virial isotherm, the equilibrium constant P" is given by the equation... 

Numerous representations have been used to describe the isotherms in Figure 5.5. Some representations, such as the Van der Waals equation, are semi-empirical, with the form suggested by theoretical considerations, whereas others, like the virial equation, are simply empirical power series expansions. Whatever the description, a good measure of the deviation from ideality is given by the value of the compressibility factor, Z= PV /iRT), which equals 1 for an ideal gas. 

Vii ial equation of state in two dimensions, 931 Virial isotherm, 936 Visible radiation, 797 Volcanoes, in electrocatalysis, 1284 Volmcr, Max, 1048,1474 Volmer. Weber, electrodeposition. 1303. 1306 Volta, 1423, 1455 Volta potential difference, 822 Voltammetry. 1432 1434 cyclic, 1422 1423 diffusion control reactions, 1426 electron transfer reaction, 1424... 

In Section 6.8.5 we were able to derive one specific isotherm, the virial isotherm. However, can all the adsorption systems be described by this isotherm There is a difficulty. The isotherms, similarly to the equations of state in the gas phase, have restrictions that make them suitable for use only under certain conditions.57 For... 

Here B T) is the second virial coefficient, C(T) the third virial coefficient, and so forth. Formally, the virial coefficients can be defined as successive partial derivatives of Z with respect to inverse molar volume (density) under isothermal conditions for example, B(T) is given by... 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

These coefficients appear in the virial isotherm Equation 4. Even for osmotically ideal solutions the coefficients A do not vanish in Equation 4. One has in this case... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

Figure 22. Excess internal energy, Eex/N, and virial pressure, PP/p, calculated with the ODS integral equation versus the reduced densities p = pa3, along the isotherms T = 297.6, 350 and 420 K (from bottom to top), by using the two-body potential alone (dotted lines) and the two- plus three-body potentials (solid lines). The experimental data (open circles) are those of Michels et al. [115], Taken from Ref. [129].

Derive an equation for the work of mechanically reversible, isothermal compression of 1 mol of a gas from an initial pressure P, to a final pressure P2 when the equation of state is the virial expansion [Eq. (3.10)] truncated to... 

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