# A Wide-Ranging NMR Probe for Multiscale Structures
## 2 Basic Principles and Models
### Proposed Models
#### 2.2.2 Mesoporous Silica  
In 1993, a model for determining the parameters of the porous structure of mesoporous silica gels was proposed by [PERSON], [PERSON] and [PERSON] [8]. Their objective was to explain unexpectedly high \({}^{129}\)Xe chemical shift values obtained by analyzing various mesoporous silicas (again, in the absence of strong adsorption sites and paramagnetic species). Indeed, from a series of experiments, they obtained values similar to those obtained by [PERSON] on zeolites, i.e., around 100 ppm, while working in roughly the same pressure range. Since both nitrogen adsorption and mercury porosimetry did not reveal the presence of micropores, the authors concluded that the model proposed for zeolites was not applicable for this type of material, having average pore sizes in the range 20-400 A.  
They proposed that such scale is wide enough that xenon in the center of the pore behaves as free gas. The actual chemical shift is the result of a dynamic equilibrium between that population and the population adsorbed on the pore surface. Assuming the validity of the ideal gas law inside the pores and assuming that xenon gas is adsorbed according to [PERSON]'s law, the xenon chemical shift follows the relationship:  
\[\delta=\frac{\delta_{\rm S}}{1+\frac{V_{\rm g}}{K\overline{K}\overline{K} \overline{K}}} \tag{6}\]  
where \(R\) is the gas constant, \(T\) is the temperature, \(K\) is [PERSON]'s constant and \(V_{\rm g}/S\) is the volume-to-surface ratio of the pores occupied by xenon atoms. This equation shows that the xenon chemical shift depends on the parameters of the porous structure and allows their determination, provided that the [PERSON]'s constant is known. Equation (6) can also be expressed in terms of the mean pore size, \(D\), with:  
\[D=\frac{\eta V_{\rm g}}{S} \tag{7}\]  
where \(\eta\) is a parameter depending on pore geometry: for cylindrical pores, \(\eta=4\), for randomly packed spheres of equal sizes, \(\eta=2.8\), and, for slit-like pores, \(\eta=2\). These values assume a model with non-intersecting pores, and, formally, lower values must be used in the case of intersecting pores.  
By combining Equations (6) and (7), one obtains:  
\[\delta=\frac{\delta_{\rm S}}{1+\frac{D}{K\eta RT}} \tag{8}\]  
It must be remembered that the [PERSON]'s constant, \(K\), is in fact temperature dependent. Thus, Equations (6) and (8) are valid only in isothermal conditions. Nonetheless, the energetic parameters of xenon adsorption can be determined by analyzing the temperature dependence of \({}^{129}\)Xe chemical shifts. The [PERSON]'s constant can be expressed as:  
\[K=\frac{K_{0}}{\sqrt{T}}\mathrm{e}^{\frac{\Delta_{\mathrm{ads}}H}{K^{H}}} \tag{9}\]  
where \(K_{0}\) is a temperature-independent pre-exponent and \(\Delta_{\mathrm{ads}}H\) is the heat of adsorption of the pores under study. Note that, in some papers, the similarity of notation between \(K\) and \(K_{0}\) created a certain confusion. By combining Equations (6) and (9), the following correct relationship is obtained:  
\[\delta=\frac{\delta_{\mathrm{S}}}{1+\frac{V_{\mathrm{g}}}{K_{0}5K\sqrt{T}} \mathrm{e}^{-\frac{\Delta_{\mathrm{ads}}H}{K^{H}}}} \tag{10}\]  
This equation can be fit to variable temperature data to obtain the enthalpy of xenon adsorption.