Gibbs adsorption equation, calculation

Using the Gibbs adsorption equation, calculate the surface excess concentra-... 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

Calculate the surface energies of each of these liquids and plot a graph of y for the CTAB solutions as a function of logio(conc.). Use your results and the Gibbs adsorption equation (see later) to estimate the minimum surface area per CTAB molecule adsorbed at the air-water interface. 

Surface Adsorption. From Fig.l and Fig.2 we can calculate the total surface adsorption (17 ) of the RDH-surfac-tant mixture by applying the Gibbs adsorption equation(7). In the case of a mixed aqueous solution with a constant ionic strength, the equation is written as... 

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

By use of Gibbs adsorption equation, the spreading force can also be expressed as a function of the rate of evaporation of the atoms. From the figures for the spreading force, the dipole moments m can be calculated. In this way the following values were found by Langmuir ... 

At constant temperature, the interfacial tension y of a water oil system containing a single surfactant solute can be calculated at thermodynamic equilibrium starting with the Gibbs adsorption equation... 

In this case (in the absence of a surface charge) the interface is neutral. The dependence of the interfacial tension y on the uni-univalent electrolyte concentration cE can be calculated using the Gibbs adsorption equation ... 

The change in the interfacial tension due to the electrolyte ions can be calculated using the Gibbs adsorption equation (Eqs. (12)—(14)). The surface adsorptions are given by ... 

For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface [23]). Therefore, one should first identify the mechanism of formation of the double layer. In this case,... 

Once the additional interactions are known, the distribution of the concentration of ions near the interface can be obtained from eqs la and lb (for the appropriate boundary conditions), and consequently, the surface tension of electrolytes can be easily calculated, via the Gibbs adsorption equation. 

Enthalpy Change. The enthalpy change measured by the heats of immersion (smooth curve) and calculated with Equation 7 (plotted points) is compared in Figure 6. The agreement is satisfactory, since the integration of the Gibbs adsorption equation depends so strongly upon the extrapolation of the adsorption isotherm to x = 0. 

Equation (2.34) is often referred to as the Gibbs adsorption equation where the interdependence of r and p is given by the adsorption isotherm. TTie Gibbs adsorption equation is a surface equation of state which indicates that, for any equilibrium pressure and temperature, the spreading pressure II is dependent on the surface excess concentration r. The value of spreading pressure, for any surface excess concentration, may be calculated from the adsorption isotherm drawn with the coordinates n/p and p, by integration between the initial state (n = 0, p = 0) and an equilibrium state represented by one point on the isotherm. 

Figure 9.5. Gibbs adsorption equation. The surface excess F can be obtained (cf. equation 19) from a plot of surface tension 7 versus log activity (concentration) of adsorbate. The area occupied per molecule or ion adsorbed can be calculated.

Proteins are found either not to desorb or to desorb only with great difficulty from quiescent interfaces. Langmuir and Schaefer (1939) calculated, on the basis of the Gibbs adsorption equation, that compression of a monolayer of protein of molecular weight 35,000 by 15 mN m-1 should increase its solubility by a factor of 1095. This results from the large area occupied by the molecule at the interface and the resultant large pressure increment of solubility. The failure of protein monolayers to desorb readily on compression was thus taken as an indication of irreversibility. 

In these equations 6 is the adsorbate surface coverage calculated from surface pressure data by means of the Gibbs adsorption equation, x, Xg are the mole fractions of the adsorbate and solvent respectively in the bulk solution, a is the activity of the adsorbate in the bulk solution, II(= 7 — 7) is the experimental surface pressure of the adsorbed film, 7 is the surface tension of the test solution, 7° is the value of 7 of the pure solvent, R is the gas constant and T is the temperature. 

The above discussion clearly illustrates that, in order to calculate T2 from the y-log C curve it is important to consider the nature of the surfactant and the composition of the medium. For nonionic surfactants the Gibbs adsorption equation [Eq. (5.9)] can be used directly but for an ionic surfactant, in the absence of electrolytes, the right-hand side of Equation (5.9) should be divided by 2 to account for surfactant dissociation. This factor disappears in the presence of a high concentration of an indifferent electrolyte. 

We also compare the above film characteristics with those of NaDS absorbed from a simple, salt-free, aqueous solution. For this purpose we use the composite tt-A curve of Pethica (17) calculated from the 2 form of the Gibbs adsorption equation—viz.,... 

When positive adsorption takes place at the solution surface, it lowers the surface tension of the pure solvent, y0, and the surface tension of the solution, y, can be determined experimentally. If the solution is dilute, then yean also be calculated from the Gibbs adsorption equation (Equation 224). The spreading pressure (or surface pressure), n, is defined as the decrease in the surface tension with the presence of an adsorbed monolayer... 

An important application of the Gibbs adsorption equation is to the calculation of the relative adsorption from measurements of the variation of surface tension with concentration ... 

As mentioned in the discussion around Eqs. (21)-(24), we need values of the five parameters listed in Eqs. (53) and of the surfactant concentration, Csa, to be able to calculate values of various properties of microemulsions (O/W + O or W/O-fW). Of these five parameters the area per surfactant molecule, (t.,., can be determined accurately from the Gibbs adsorption equation, Eq. (1), and there is only a small uncertainty regarding the values of / and c. 

The Gibbs adsorption equation allows for calculation of area per molecule from very simple measurements of surface (or interfacial) tension versus surfactant concentration in the solution. This calculation, in turn, enables one to study the relative area/molecule of a surfactant. Tighter molecular packing in the adsorbed film lowers the interfacial tension. 

The preceding discussion of the Gibbs adsorption equation was referenced to a fluid-fluid interface in which the surface excess, T, is calculated based on a measured quantity, a, the interfacial tension. For a sohd-fluid interface, the interfacial tension cannot be measured directly, but the surface excess concentration of the adsorbed species can be, so that the equation is equally useful. In the latter case. Equation (9.16) provides a method for determining the surface tension of the interface based on experimentally accessible data. 

Show that for an ionic surfactant that completely dissociates in water, a factor of 2 must be included in the Gibbs adsorption equation (Equation 1.26). Then calculate the smface concentration Fj just below the critical micelle concentration (i.e., where the discontinuity in slope occurs) for the three surfactants shown in the figure (Getter and Hoffman, 1988). DMAO is dimethylamine oxide DMPO is the corresponding phosphine oxide. 

The interface in extraction systems is usually studied by measuring the interfacial tension, viscosity and potential. To study the adsorption kinetics, one usually plots the isotherms of the interfacial tension and uses the Gibbs adsorption equation to calculate the surface concentration of the extractant [15-22]. In practice, the concentration of the extractant is selected so as to saturate the monolayer at the interface. In such systems a rise in extractant concentration does not affect the extraction rate if the limiting stage is the surface reaction or a reaction in the adjoining layers [17-20,23]. 

This criterion is very important when predicting multicomponent adsorption systans by the ideal adsorption theory (lAST) because of the integration limits in the Gibbs adsorption equation. Meanwhile, (mmol/g) was calculated from mass balance equation (6.3)... 

Surface Activity Parameters The surface activity parameters of sinfac-tants are mainly obtained via surface tension measurement [36, 37, 40, 42, 44, 46]. From the surface tension, isotherms of the saturation surface excess, can be calculated using the Gibbs adsorption equation, which is used as a measure of maximum extent of adsorption of surfactant at the air/solution or solution/solution interface. 

Another interesting aspect to discuss is the calculation of the apparent area of the species adsorbed at the interface and its relationship with the molecular weight of said species. As is well known, Gibbs adsorption equation allows us calculate the adsorption T (the concentration of the species that is adsorbed at the surface) from measurements of tension as a function of the concentration of the surfactant in the solution (Hiemmenz, 1998). 

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