Equation Harkins-Jura

According to equation (7.7a), the Harkins-Jura equation, a plot of In P/Pq ) versus l/W should give a straight line with a slope equal to — fi and an intercept equal to A. The surface area is then calculated as... 

The term K in equation (7.9) is the Harkins-Jura (HJ) constant and is assumed to be independent of the adsorbent and dependent only on the adsorbate. 

Although it may give satisfactory values for Asp, the Harkins-Jura equation leaves something to be desired at the molecular level. For example, the linear 7r versus o equation of state —the starting point of the derivation of the Harkins-Jura isotherm —represents the relatively incompressible state of the surface phase (i.e., 6 = 0.7 in Fig. 9.6b). (This equation is obtained in analogy with the approximately linear ir versus a equation for insoluble mono-layers discussed in Chapter 7.) However, in most instances of physical adsorption, no satura-... 

Figure 2. Dependencies of the adsorption potential on the pore width calculated according to the equations (3) (SF curve), (4) (curve KJSe) and (5) (curve KJSc). The KJSc curve was calculated via equation (5) using the Harkins-Jura-type expression for the statistical film thickness t, which gives a good representation of the experimental t-curve only in the range of relative pressures from 0.1 to 0.95. Therefore, this curve deviates from points at high values of A, which correspond to low values of p/po. Data for the ZLZ curve are from Zhu et al. [29].

Figure 7. Comparison of PSDs obtained by tbe HK method with the A(w) relationships given by equations (4) (circles) and (5) (lines). While in equation (4) the experimental t-curve was used, equation (5) was obtained by using the Harkins-Jura fit for the t-curve, which is much less accurate for relative pressures smaller than 0.1.

Equations (33) and (34) form the theoretical basis for the absolute Harkins-Jura (HJ) method [76,94] to estimate the solid surface area. However, in the earlier calorimetric experiments applying the Harkins-Jura principle, the term QjJJ, was always neglected. Neglecting it may lead to certain discrepancies between the surface areas determined by the Harkins-Jura and BET methods in the case of water adsorbed on oxides. 

Nitrogen physisorption measurements were performed on a Micromeritics Tristar 3000 apparatus at -196 °C. Prior to analysis the samples were dried in a helium flow for 14 horns at 120 °C. Surface areas (St), and micropore (Vmicro) and mesopore (Vmeso) volumes were determined using the t-method [13] with the Harkins-Jura thickness equation. There is no standard method for the determination of blocked mesopore volume (Vmeso,bi)- For this we used the pore size distribution from the desorption branch of the isotherm calculated using BJH theory [14]. The total amoimt of Vmeso,bi was determined considering that the volume in pores with a diameter of 2 - 5 run is (partially) blocked. 

Measured by atomic absorption spectrometry. Reference untreated NaY. c).d) Low pressure argon adsorption. t-Plot method of Lippens-De Boer, Harkins-Jura equation. 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

Rearranging the Harkins-Jura equation (2.3-39) into the form of adsorbed amount versus the reduced pressure, we have ... 

Thus, the Harkins-Jura isotherm equation can be written as... 

For the Harkins-Jura equation to describe the Type II isotherm, it must have an inflexion point occurring at the reduced pressure between 0 and 1, that is the restriction on the parameter B is ... 

Figure 2.3-5 shows typical plots of the Harkins-Jura equation. 

We see that many isotherm equations (linear, Volmer, Hill-deBoer, Harkins-Jura) can be derived from the generic Gibbs equation (2.3-13). Other equations of state relating the spreading pressure to the surface concentration can also be used, and thence isotherm equations can be obtained. The following table (Table 2.3-1) lists some of the fundamental isotherm equations from a number of equations of state (Ross and Olivier, 1964 Adamson, 1984). 

Although the BET theory is used almost regularly as a convenient tool to evaluate the surface area of a solid, other isotherms such as the Harkins-Jura equation, obtained in Chapter 2 can also be used to determine the surface area. Analogous to a monolayer film on liquids, Harkins and Jura (1943) obtained the following equation ... 

To separate contributions due to micropore filling on one hand and the formation of mono- and multilayers on the other hand which are superimposed at relative pressures below 0.2- 0.3, the f-plot or the aj-plot approach can be applied. Both methods use empirical reference isotherms to be compared with the isotherms taken for the sample under investigation. In the f-plot method, the statistical layer thickness t of a nonmicroporous material is related to the relative pressure plp. One of the most frequently used relationship for the layer thickness is the empirical Harkins-Jura equation [67] derived for metal oxides ° ... 

Brown and EveretF proposed the Harkins Jura type equation... 

The surface areas of all the samples were measured using the B.E.T. method with nitrogen adsorption at 77 K and a Micromeritics ASAP 2000 for the determination of the pore size distribution for the most interesting ones. Mesopore size distributions were calculated using the Barrett, Joyner and Halenda (BJH) method, assuming a cylindrical pore model (IS). In the analysis of micropore volume and area, the t-plot method is used in conjunction with the Harkins-Jura thickness equation (16). 

Complete details about the method of construction of 3-D porous networks through Monte Carlo simulation can be found elsewhere [10] similarly, the precise algorithm employed to replicate sorption processes in porous networks has been reported somewhere else [11]. Here, we will only mention several key aspects regarding these porous network and sorption simulations. First, the critical conditions required for cavities (hollow spheres) and necks (hollow cylinders open at both ends) to be fully occupied by either condensate or vapor have been calculated by means of the Broekhoff-de Boer (BdB) equation [12], while the thickness of the adsorbed film has been approximated via the Harkins-Jura equation [13]. Some other important assumptions that are made in this work are (i) the pore volume is exclusively due to sites (ii) bonds are considered as volumeless windows that communicate neighboring sites (ili) bonds can merge into a site without suffering of any geometrical interference with adjacent throats. 

The Gibbs isotherm assumes that adsorbed layers behave like liquid films, aud that the adsorbed molecules are free to move over the surface. This isotherm cau be derived then using classical thermodynamics using Gibbs free energy equations. This results in an isotherm of the form given in Equation 8.10. This is known as the Harkins-Jura (HJ) equation[90]. For details of derivation please refer to (3). 

Equation 17.23 has the form of an adsorption isotherm since it relates the amount adsorbed to the corresponding pressure. This is known as the Gibbs Adsorption Isotherm. For it to be useful, an expression is required for T. Assuming an analogy between adsorbed and liquid films, Harkins and Jura(15) have proposed that ... 

Equation 17.26, derived by Harkins and Jura(15) may be plotted as In (P/To) against I / V2 to give a straight line. The slope is proportional to A2. The constant of proportionality may be found by using the same adsorbate on a solid of known surface area. Since the equation was derived for mobile layers and makes no provision for capillary condensation, it is most likely to fit data in the intermediate range of relative pressures. 

Mention should be made of an important relation first brought out by Jura and Harkins (1944). It is simply that the adsorption isotherm is closely represented by the equation... 

Some doubt has been indicated as to the proper choice of a value for the cross-sectional area of the nitrogen molecule involved in the calculation of the surface area from the B.E.T. plot. The data were replotted according to the following equation rerived from an analysis of Harkins and Jura (119). 

The second method, that described by Harkins and Jura (14), applies the semiempirical equation... 

It is due to Harkins and Jura and was derived on the basis of an empirical two-dimensional equation of state. Here, A and B are constants. 

Whilst the use of the Kelvin equation can be questioned in the case of smaller mesopores, this is not the case in the present case where, on the contrary, the pores are situated in the upper mesopore range. However, use of the BJH method implies the use of a t-curve. On commercial adsorption equipment, the software proposes the use of several equations to fit the t-curve. In the present case, the Harkins and Jura equation or Halsey equation is proposed. Unfortunately neither of these fit the original t-curve data of de Boer very well. 

More extensive and accurate data and additional calculations are necessary to obtain s , e , and from isotherm data over what is required to get the differential energy and entropy from the isosteric equation. The first complete calculation of ss, e and , as well as the differential quantities, has recently been made by Hill, Emmett, and Joyner (95). This paper shows in detail how the methods of this section can be applied in practice. Using heats of immersion, Harkins and Jura (96) made earlier equivalent calculations, but the relationship of their calculated quantities to the thermodynamic functions of the adsorbed molecules was not pointed out until recently by Jura and Hill (92). 

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