Adsorption by nanoporous materials has a number of applications, including gas separation, purification and storage. The suitability of a particular material for any given application can be assessed by determining its adsorption properties, such as absolute uptakes with respect to pressure and temperature, and adsorption enthalpy with respect to surface coverage. High pressure measurements are required for gas storage.
The accuracy of absolute uptake depends on the experimental data quality and the conversions of excess adsorption to absolute adsorption. Also, it is necessary to choose a suitable isotherm model to analytically represent and interpolate data for determining the isosteric enthalpy of adsorption, ΔH_{iso}_{,} by van 't Hoff plots.
This article demonstrates the evaluation of the Langmuir, Freundlich, Sips (LangmuirFreundlich) and Toth isotherm equations using an IMIHTP high pressure sorption analyzer. These equations can be used to describe the adsorption of methane by a commercial activated carbon for pressures of up to approximately 6.5 MPa between 307 K and 329 K. The calculation of ΔH_{iso} in terms of surface coverage or methane uptake is also performed using the data obtained from the Sips and Toth expressions.
Isotherm Equations
The localized adsorption of a monolayer of atoms or molecules on a homogeneous surface without any adsorbateadsorbate interactions is given by the Langmuir equation as follows:

(1) 
where n_{a} is the amount adsorbed and n_{m} is the monolayer capacity, θ = n_{a} / n_{m} is the surface coverage, P is the pressure, and b is the adsorption coefficient, which is exponentially related to the positive value of the energy of adsorption.
The Freundlich equation is given as follows:

(2) 
where K and n are empirical parameters. The Freundlich equation is empirical in nature. However, the multisite Langmuir equation can also be used to derive the Freundlich equation, by assuming an exponential adsorption site energy distribution.
The LangmuirFreundlich isotherm is given as:

(3) 
where k is the same as b in the Langmuir equation, and n is a measure of the heterogeneity of the adsorbent. Bradley initially proposed this equation as an empirical relation. However, Sips later derived this equation by assuming Langmuir isotherm behaviour and a nearGaussian distribution of adsorption site energies.
The Tóth equation uses another exponent, t, to represent the heterogeneity, and is given by:

(4) 
where b and t are specific to a particular adsorbateadsorbent system.
Experimental Procedure
The measurements were carried out using a fully automated IMIHTP manometric analyzer, as shown in the Figure 1. The gas sorption properties of materials of up to 20 MPa pressure can be determined using the instrument. 1.755 g of Filtrasorb F400 was used to determine the methane adsorption isotherms with ±0.1 K stability at temperatures of 307 K, 312 K, 316 K, 320 K, 325 K and 329 K. The calibrated dosing manifold and the sample cell have their respective volumes of 6.32 cm^{3} and 16.47 cm^{3}, determined by pycnometry, in which helium is used as the calibration fluid.
Figure 1. IMIHTP high pressure sorption analysis instrument
The manifold temperature was maintained constant at 308 K while stability was maintained at ±0.1 K. A 10.0 MPa range transducer, with an accuracy of ±0.04% of full scale, was used to measure the pressure. Methane compressibility was determined using a linear interpolation between tabulated compressibility values produced using the NIST REFPROP database. The NIST database uses the Setzmann and Wagner equation for methane.
Results and Discussion
Figure 2 shows the absolute uptake isotherms calculated from excess uptake isotherms recorded using the IMIHTP. These were fitted using equations (1) to (4), and the isoteric enthalpies of adsorption were calculated from van 't Hoff plots constructed using the best fit parameters. Figure 3 shows the comparative data fits for absolute adsorption measured at 316 K. The Sips equation was the best fit for all excess adsorption isotherms. The Sips and Tóth equations provided significantly better fits than the Langmuir and Freundlich equations for both excess and absolute adsorption. The Sips and Tóth equations are suitable for providing absolute adsorption data over the measured temperature range, as shown in the Figure 4. The residuals in the lower plot indicate that the Tóth equation is the better fit over the Sips equation.
The Sips and Tóth equation fits for the excess adsorption data over the measured temperature range is shown in the Figure 5.
The residuals in the lower plot of the Figure 5 show that the Sips equation is a better overall fit than the Tóth equation. However, the magnitude of the residuals at low surface coverage looks to be approximately equal.
Figure 6 shows ΔH_{iso} as a function of coverage, calculated using the parameters from the Tóth and Sips model and isotherm data recorded at different temperatures. The ΔH_{iso} calculated in this study vary beyond the approximate 23 to 25 kJmol^{1} range reported elsewhere for methane adsorbed onto activated carbons, although ΔH_{iso} is known to change as a function of both temperature and surface coverage. It is evident from the different trends of Figure 6 that a clear correlation between the calculated enthalpies and the isotherm temperatures or specific surface coverage was not achieved. This demonstrates that different values of ΔH_{iso} can be calculated using the same experimental data depending on the model chosen to interpolate the data.
Figure 2. Absolute methane adsorption by 1.755 g activated carbon (Filtrasorb F400) at temperatures between 307 K and 329 K.
Figure 3. Langmuir (blue), Freundlich (red), Sips (grey) and Tóth (green) fits, top, and residuals, bottom, for absolute uptake data at 316 K.
Figure 4. Sips (blue) and Tóth (red) fits, top, and residuals, bottom, for absolute uptake isotherms at temperatures between 307 K and 329 K
Figure 5. Sips (blue) and Tóth (red) fits, top, and residuals, bottom, for excess uptake isotherms at temperatures between 307 K and 329 K.
Figure 6. The isosteric enthalpies of adsorption for methane on F400 carbon, calculated from various isotherms fitted using the Sips and Tóth equations.
Conclusion
The Toth equation was found to be the best fit for all absolute adsorption isotherm data. However, the excess adsorption data is best modeled using the Sips equation. Also, calculation of a wide range of ΔH_{iso} values using the same experimental data but different analysis methods indicates that ΔH_{iso} values calculated from one method could result in a large potential error. The significant effect of the isotherm model and the selected data temperature range on the calculated values of ΔH_{iso} indicates the importance of clearly stating the method while reporting adsorption isosteric enthalpies.
The article also demonstrated the successful determination of the high pressure adsorption of methane using an activated carbon in the IMI HTP analyzer.
This information has been sourced, reviewed and adapted from materials provided by Hiden Isochema.
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