Subcritical isotherm

Fig. 10.5. Schematic diagram of the mean number of particles, (N), versus chemical potential, /u for a subcritical and a supercritical isotherm of a one-component fluid. The curve for the supercritical isotherm has been shifted up for clarity. Reprinted by permission from [6], 2000 IOP Publishing Ltd...

Figure 2.8 Representative supercritical (T = 31 OK) and subcritical (T = 280K) Van der Waals isotherms for C02, showing the liquid-gas (L + G) condensation plateau (P = 52 atm) for T = 280K, and outlining the 2-phase liquid-gas coexistence dome (dotted line) topped by the critical point (x) at Tc = 304K, Pc = 73 atm.

Physical adsorption of gases is, undoubtedly, the most widely used technique [4], Due to the considerable sensitivity of nitrogen adsorption isotherms to the pore texture in both microporous and mesoporous ranges and to its relative experimental simplicity, measurements of subcritical nitrogen adsorption at 77 K are the most used. However, this technique has some limitations, and other complementary techniques are needed for the characterization of microporous solids. 

As discussed in Sec. 3.5 with respect to cubic equations of state for pu species, a subcritical isotherm on a PV diagram exhibits a smooth transit) from the liquid to the vapor region, shown by the curve labeled T2 < Tc on F 3.10. We tacitly assumed in that discussion independent knowledge of the va pressure at this temperature. In fact, this value is implicit in the equation of sta We reproduce in Fig. 14.1 the subcritical isotherm of Fig. 3.10, without... 

Fig. 3-2. P/V/T phase diagram of a pure substance (pure solvent) showing domains in which it exists as solid, liquid, gas (vapour), and/or sc-fluid (CP = critical point TP = triple point p = mass density). The inserted isotherms T2 (T2 > Tc) and Tj, T3 Tc) illustrate the pressure-dependent density p of sc-fluids, which can be adjusted from that of a gas to that of a Hquid. The influence of pressure on density is greatest near the critical point, as shown by the greater slope of isotherm T2 compared to that of T3, which is further away from Tc- Isotherm Ti demonstrates the discontinuity in the density at subcritical conditions due to the phase change. This figure is taken from reference [220].

Figures 1 to 4 show the pore size distribution functions (obtained by the H-K and lAE) methods and the comparison between the experimental results and the recalculated isotherms for three of the five adsorbates. The highest mean deviation is 5.66% for nitrogen. Consequently, our characterization method appears to be an efficient modelling tool as it allows to simulate adsorption isotherms of five different adsorbates in wide temperature and pressure conditions (subcritical and supercritical isotherm temperatures) using a unique pore size distribution flmction of the adsorbent.

Eq.(2) together with Eq.(3) describe isotherms not only for supercritical, but also for subcritical if the isotherms show type-I feature. It was shown recently [22] that this model works well for the isotherms distributed densely around the critical temperature as shown... 

It is well known that the saturation pressure, Ps> the border of subcritical adsorption, beyond which another phenomenon, condensation, happens. It seems there is not a similar border for supercritical adsorption because no matter how high the pressure is, "adsorption isotherms could always be recorded. However, such border must exist considering the cause of adsorption. It is the interaction between gas and solid that causes the density difference between gas phase and adsorbed phase. The strength of the intermolecular (atomic) attraction force is limited, therefore, the density difference resulted must be limited, and the pressure difference corresponds to the density difference must be limited. So, there is no reason to think of supercritical adsorption (or high-pressure adsorption as synonym) as borderless. 

The simplest application of an equation of state for VLE calculations is to a pure species to find its saturation or equilibrium vapor pressure at given temperature T. As discussed in Sec. 3.5 with respect to cubic equations of state for pure species, a subcritical isothenu on a P V diagram exliibits a smooth transitionfrom liquid to vapor tliis is shown on Fig. 3.12 by the curve labeled Ti < T. Independent knowledge was there assumed of vapor pressures. In fact, tliis infomiation is implicit in an equationof state. Figure 14.7 illustrates a realistic subcritical isotherm owa PV diagram as generatedby an equation of state. One of the features of such an isothenu for temperahires not too close to fr is tliat the ininimum lies below the F = 0 axis. 

Reduced temperatures below 1.0 are subcritical and the gas becomes liquefied with increasing pressure. The density changes from the low value of the gas phase to the high value of the liquid phase. Then it remains almost constant with rising pressure because the liquid is almost incompressible. As reduced temperatures approach unity the isothermal compressibility of the gas rises rapidly. At values above unity in the supercritical area there is no further liquefaction and the gas density can be adjusted continuously with increasing pressure, which offers the option to adjust the dissolving power. 

Theoretical isotherms obtained using DFT method rival the accuracy of model isotherms constructed from molecular simulation, as shown in the comparison in Fig. 12 for N2 adsorption results in model graphitic slit pores at 77 K [22]. Tlie DFT model correctly predicts the capillary condensation pressure, the most prominent feature of the subcritical isotherm, relative to the exact computer simulation results shown in Fig. 11. DFT also provides a good description of the secondary structure of the mesopore isotherm (e.g., H = 42.9 A in Fig. 12), in which capillary condensation may be preceded by one or more wetting transitions. 

The adsorption of gas mixtures has been extensively studied. For example, Wendland et al. [64] applied the Bom—Green—Yvon approach using a coarse grained density to study the adsorption of subcritical Lennard-Jones fluids. In a subsequent paper, they tested their equations with simulated adsorption isotherms of several model mixtures [65]. They compared the adsorption of model gases with an equal molecular size but different adsorption potentials. They discussed the stmcture of the adsorbed phase, adsorption isotherms, and selectivity curves. Based on the vacancy solution theory [66], Nguyen and Do [67] developed a new technique for predicting the multicomponent adsorption equihbria of supercritical fluids in microporous carbons. They concluded that the degree of adsorption enhancement, due to the proximity of the pore... 

Because of the similarity between the lattice fluid calculations and the MC simulations for the continuous model, it seems instructive to study the phase behavior in the latter if the confined fluid is exposed to a shear strain. This may be done quantitatively by calculating p as a function of p and as -For sufficiently low p, one expects a gas-like phase to exist along a subcritical isotherm (see Fig. 4.13) defined as the set of points (T = const)... 

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