Langmuirs Equation

Next, let the example of vanadium, which, in the as-reduced condition, may contain a variety of impurities (including aluminum, calcium, chromium, copper, iron, molybdenum, nickel, lead, titanium, and zinc) be considered. Vanadium melts at 1910 °C, and at this temperature it is considerably less volatile than many of the impurity metals present in it. The vapor pressure of pure vanadium at this temperature is 0.02 torr, whereas those of the impurity elements in their pure states are the following aluminum 22 torr calcium 1 atm, chromium 6 torr copper 23 torr iron 2 torr molybdenum 6 1CT6 torr nickel 1 torr lead 1 torr titanium 0.1 torr and zinc 1 atm. However, since most of these impurities form a dilute solution in vanadium, their actual partial pressures over vanadium are considerably lower than the values indicated. Taking this into account, the vaporization rate, mA, of an element A (the evaporating species) can be approximated by the following free evaporation equation (Langmuir equation) ... 

The original derivation of the Langmuir equation (Langmuir, 1916) is a kinetic one. The adsorbent surface is pictured as an array of N equivalent and independent sites for localized adsorption (one molecule per site). The fraction of sites occupied by N molecules is 8 = N /Ns. 

Based on the same assumptions as for the ideal-lattice-gas equation, Langmuir in 1918 derived Eq. (14.106) by noting that at equilibrium the rate of adsorption arrd the rate of desorption of gas molecules must be the same. For monolayer adsorption, the number of sites may be divided into the fraction occupied0 and the fraction vacant 1 — 0. By definition. 

Langmuir equation (Langmuir adsorption isotherm) An equation (Langmuir, 1918), originally formulated for... 

Chen, S.G. and Yang, R.T. (1994). Theoretical basis for the potential theory adsorption isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov equations. Langmuir, 10, 4244-9. 

Stoeckli, F., L6pez-Ram6n, M.V., and Moreno-Castilla, C. (2001). The adsorption of phenolic compounds from aqueous solutions by activated carbons described by the Dubinin—Astakhov equation. Langmuir, 17, 3301-6. 

In addition to the Gibbs equation, three other equations have been suggested that relate concentration of the surface-active agent at the interface, surface or interfacial tension, and equilibrium concentration of the surfactant in a liquid phase. The Langmuir equation (Langmuir, 1917)... 

The fundamental equations, Langmuir, Volmer, Fowler-Guggenheim and Hill de Boer, will form a basis for the study of heterogeneous adsorbents as we shall discuss briefly in Chapter 3 and in further detail in Chapter 6. 

Grzelak EM, Errington JR (2010) Nanoscale limit to the applicability of Wenzel s equation. Langmuir 26(16) 13297-13304... 

The relationship between the amoimt of a substance adsorbed per unit mass of adsorbent at constant temperature and its concentration in the equihbrium solution is called the adsorption isotherm. The more common models used to investigate the adsorption isotherm are Langmuir and Freundlich equations (Langmuir, 1918 Freimdlich, 1939). The experimental results of this study were fitted with these two models. The equihbrium adsorption capacity of the adsorbent was calculated by the following Eq. (12.2). The equihbrium adsorption of Cr (VI) ion solutions by hydrogel Aj was measured (50 mL of 50 mg/L-400 mg/L Solutions) after equilibrium time. 

Nosonovsky M (2007) On the range of applicability of the Wenzel and Cassie equations. Langmuir 23 9919-9920... 

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