Langmuir’s law

The solubility coefficient indicates the sorption capacity of a polymer with respect to a particular sorbate. The simplest solubility coefficient is defined by Henry s law of solubility, which is valid at low concentration values for most substances. For other combinations, such as CO2 in PET at high pressure, Henry s and Langmuir s laws must be combined. These are also discussed in Chapter 14. 

Sorption does not necessarily follow Henry s law. For a glassy polymer an assumption is made that there are small cavities in the polymer and the sorption at the cavities follows Langmuir s law. Then, the concentration in the membrane is given as the sum of Henry s law adsorption and Langmuir s law adsorption... 

Yet another explanation of the reverse thermal effect was offered by Suh et al. (2001) for a carbon/epoxy composite. They also consider that absorbed water consists of free and bound components, Mp and Mp, respectively. Similarly to previous considerations (Vieth and Sladek 1965 Vieth et al. 1976), the authors relate the bound water by means of Henry s law and the free water by means of Langmuir s law. They assume that the sorption of bound water is a reversible process and consider the uptake of free water to be irreversible - thereby not being squeezed out upon changes in hygrothermal conditions. It thus follows that... 

Make a plot of Eq. XVII-69 as 6 versus P, and, for comparison, one of a Langmuir adsorption isotherm of same limiting or Henry s law slope. Comment on the comparison. 

In glassy polymers tire interactions of tire penetrant molecules witli tire polymer matrix differ from one sorjDtion site to anotlier. A limiting description of tire interaction distribution is known under tire name of tire dual-soriDtion model [, 60]. In tliis model, tire concentration of tire penetrant molecules consists of two parts. One obeys Henry s law and tire otlier a Langmuir isotlienn ... 

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

With the adsorbate concentration low enough, the Langmuir isotherm transforms into the linear equation and becomes the simplest isotherm of adsorption, as described by Henry s law. 

Correspondence of experimental data to the Langmuir model does not mean that the stated assumptions are valid for the particular system being studied, because departure from the assumptions can have a canceling effect. An advantage of this model is that it can approach Henry s law at low concentrations. 

To derive a rate of extraction in terms of easily measurable quantities, we express the concentration of the reagent adsorbed at the liquid-liquid interface as function of its bulk organic concentration. This can be done by utilizing the Langmuir s adsorption law that is,... 

This region of the isotherm is the Henry s law region the uptake is directly proportional to the pressure. At high pressures, KP 1, so that Langmuir s equation becomes IV =... 

Another concept is that the electronic work function changes linearly with the amount adsorbed or that the dipole moment is independent of the concentration. The (1 — 0) concept states that the rate of adsorption for a constant arrival rate is proportional to the fraction of the surface which has not yet been covered. The last two concepts permitted Langmuir to derive his famous adsorption isotherm, which has been verified by experiment in many systems (see the discussion in section IV). Langmuir s experimental work for Cs on W convinced him that the (1 — 0) law was not applicable in this system. This work also led to the concept that the energies involved in surface migration were much smaller than the energies involved in evaporation. 

Earlier work on the application of the concept of dual mode sorption and diffusion to glassy polymer-gas systems has been reviewed in detail 6) and important aspects of more recent work have been dealt with in more recent reviews 7 10). Eq. (5) was first applied by Michaels et al U). Sorption in the polymer matrix and in the specific sorption sites was represented by linear (Henry s law) and Langmuir isotherms respectively so that Sj in Eq. (5) is given by... 

The concept of unrelaxed volume in glassy polymers is used to interpret sorption and transport data for pure and mixed penetrants A review of recent sorption and permeation data for mixed penetrants indicates that competition for sorption sites associated with unrelaxed gaps between chain segments is a general feature of gas/glassy polymer systems This observation provides convincing support for the use of the Langmuir isotherm to describe deviations from simple Henry s law sorption behavior. 

PET) in Figure 6 (21) in terms of a progressive increase in the fraction of the local concentration present in the higher mobility Henry s law environment as the local Langmuir capacity saturates at increasingly higher pressures. 

Although not necessary in terms of phenomenological applications, it is interesting to consider possible molecular meanings of the coefficients, Dq and D If two penetrants exist in a polymer in the two respective modes designated by "D" and "H" to indicate the "dissolved (Henry s law) and the "hole" (Langmuir) environments, then the molecules can execute diffusive movements within their respective modes or they may execute intermode jumps ... 

The first term in Eq (7) describes transport related to the Henry s law environment, while the second term is related to the Langmuir environment The tendency for the permeability to asymptotically approach the limiting value of kDDD at high pressures derives from the fact that after saturation of the upstream Langmuir capacity at high pressures, additional pressure increases result in additional flux contributions only from the term related to Henry s law which continues to increase as upstream pressure increases ... 

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