## 3 Results and Discussion
### Adsorbents Characteristics
#### 3.1.2 Structure of Nano-Sized Materials by XRD and Small-Angle X-ray Scattering
This indicates a greater proportion of larger pores (up to 25 nm) for the sample P1 and a greater proportion of smaller structures for the sample P2. From the small-angle scattering data, the size distribution of the scattering objects (pores) Dv(R) can be determined if they have approximately the same shape. It should be emphasised that changing the shape of scattering objects has a smaller impact on the shape of the small-angle scattering curve than changing their size. This size can be determined by the length of the dimension characteristic of a given shape. For spherical objects, the characteristic dimension may be their radius or diameter. If R is the value of the characteristic dimension of the scattering particle, and D(R) is the function of the particle size distribution, then D(R)dR will represent the probability of occurrence of objects with dimensions in the range R to R + dR. Figure 8 shows the scattering objects size distribution function, where Dv(R)dR is proportional to the volume of pores ranging in size from R to R + dR. The Dv(R) curves presented in Figure 8 determine the volume-weighted particle (or pore) size distribution Dv(R) from the scattering curve of an ensemble of spherical objects (or pores) having a homogeneous inner electron density distribution, and not showing interparticle interaction effects. The Dv(R) function refers to the volume-weighted particle size distribution ('volume distribution'). It may be seen as a histogram of the radii R of the particles (or pores) that are present in the sample. The height of the Dv(R) function is proportional to the volume (not to the number) of particles that can be found within a given size interval. In the case of carbon samples, Dv(R) curves of carbons (Figure 8A) have a single, well-pronounced peak, which enables one to determine the most probable particle size (by volume) and stable solutions from the size distribution analysis. In the case of polymer samples (Figure 8B), the given sphericity criteria may not be accurately defined and deviate from Gaussian or Log-normal functions. In this case, the well-pronounced peak is disturbed, as evidenced by fluctuations in the curve in the area above 50 A. Assuming that the tested materials are a porous system, it should be taken into account that all pores present are determined, including closed pore systems that are normally not accessible to the analysing molecules, such as nitrogen molecules. This results in possible differences in their size distributions as well as in the determined S\({}_{\text{SAXS}}\) interface. The interphase surface S\({}_{\text{SAXS}}\) refers to the boundary between two materials or phases in a sample. This surface can be characterized by analysing the scattering intensity of X-rays as they pass through the sample. [PERSON]'s law describes  
Figure 8: Comparison of volume size distribution function (Dv(R)) of the scattering heterogeneities in investigated materials: the mesoporous carbons C1–C4 (**A**), and the mesoporous polymer (**B**) materials. of a scattering curve towards high scattering angles. Scattering curves of particle systems and porous materials with smooth surfaces decay proportionally to q\({}^{-4}\) at large angles: I(q) \(\approx\) K\({}_{4}\)/q\({}^{4}\) for non-smeared data (K\({}_{4}\) is the Porod constant). For slit approximation, the decay is proportional to q\({}^{-3}\) due to the slit-smearing effects (I(q) \(\approx\) K\({}_{3}\)/q\({}^{3}\) for smeared data where K\({}_{3}\) is the smeared Porod constant). Moreover, the surface-to-volume ratio is based on the Porod constant K and the scattering invariant Q as S/V = (K\({}_{3}\)/Q).4. The invariant Q was estimated using linear extrapolation to q = 0 as a function I:q\({}^{n}\) vs. q where n = 1 for smeared data. The Q values were estimated automatically during SAXS data treatment (Porod analysis) by EasySAXS (PANalytical) software.