When studying gas StA, a certain volume of either a storage container or sorbent pores is given, as well as pressure and temperature of the gas. The product of the gas volume V and the pressure P in a closed vessel at a constant temperature T

( P V ) T = const (1)

is the equation of state for an ideal gas.

In a broad sense, PV is energy (work), where P is force and V is displacement [2]. In the case of a real gas, PV is a natural generalized variable whose deviation from constancy indicates the presence of additional variables (proximity to critical points, molecular dissociation, etc.). In the case of gas storage, an additional variable may be the forces of gas attraction (physical sorption) in narrow pores (overlapping effect [4]). This effect can be fixed by the inversely proportional dependence of the capacitance on the pore size. For a porous material, any textural property (pore volume V_p, surface area S_BET, and pore size) can serve as the displacement V in Equation (1). The force in the gas phase is pressure P. To go to the energy of the solid part of the sorbent, imagine a closed container, for example, a children’s rubber balloon, on the walls of which a strain gauge is glued. As the gas pressure increases, the elongation and wall stress will increase until the balloon bursts. Let us change in Equation (1) the pressure of the gas P that destroys the cylinder by a certain tensile strength of its material P_m (the subscript _m-mechanical). So, the equation of state will be

( P m V ) = const (2)

and P_mV can be called the energy of the sorbent. This is an example of a process (P_m) - structural (V) generalized variable. Previously [SI-1] for a granule of a porous material, the mechanical strength of the pore walls was accepted as a result (as replica) of gas (steam) pressure that occurs inside the pores of the material during its preparation or operation; the gas pressure leading to the destruction of the wall corresponds to the crushing strength of the wall P_m. In practice, such destruction of the walls of the pores occurs when the high pressure in the catalyst bed is abruptly released or when the wet granules are dried too intensively.

P_mV was expected to be constant over a range of process variables in the production of molded materials. P_m was a type of mechanical strength, depending on the loading scheme (device). The choice of the appropriate pore volume (total V_p, meso-V_me, micro-V_mi, etc.) was based on the statistical nature of the crushing of the catalyst tablets under uniaxial compression, each tablet is crushed differently. One of the statistical theories of strength, the theory of the weak link-says that the destruction begins from the weakest point in the product and then develops catastrophically. This point can be in the wall of a pore of any size, so every kind of pore volume V must be tested. Indeed, for many materials and loading schemes (tables in SI-1) existеnсе of P_mV ≈ const in the dependences of P_mV onx (x is a technological variable) has been established. In the current paper, we have added S_BET surface area and pore size, either represented by D_p or by generalized V_p/S_BET, as displacements.

Taking into account that V_p/S_BET is the average thickness τ (nm) of the probing gas layer (usually N₂) and having the average pore diameter D_p, one more meaning of the dimensionless pore shape factor F_d can be found. From the “equation in separated generalized variables” for the average pore diameter [1] we obtain the pore shape factor

F d = D p / ( V p / S B E T ) (3)

Thus, the shape factor F_d has one more physical meaning: it is the average number of probing gas layers in the pore cross-section. In other words, the average number of probing gas layers is determined by the average pore shape.

As mentioned above, if the D_p value is not given in the work under study, then V_p/S_BET can be used instead and (use kinetic diameters σ of the nitrogen probing gas-0.364 nm and stored gases: hydrogen-0.289 nm, methane-0.380 nm, and carbon dioxide-0.330 nm) the average number of layers of these gases in pore cross-section, which (number) is simultaneously a dimensionless pore shape factor, can be estimate

F d = τ / σ (4)

For 1 < F_d< 2, one of the pore shape models can be represented by two layers of molecules with some distance between neighboring molecules in the bottom row (in contrast to the BET assumption); the molecules of the upper layer are partially immersed between molecules of the lower layer. If they are completely immersed, we get one layer (linear packing) and F_d = 1; if they do not sink at all, we have two parallel layers and F_d = 2 (square packing). When they are partially submerged, we have a triangle packing of molecules inside a smooth-walled split pore (note the packing of molecules, not the shape of the pore [12]). The case F_d< 1 means either one layer of molecules with some distance between them, or gas molecules trapped between layers of a layered adsorbent.

Another imaginary model of pores can be a “stalactite cave”, pores with rough walls, in which, at some small distance between the roughness peaks of opposite walls, overlapping forces arise to attract gas molecules. Such a model can fit any F_d. Another molecular packing model that seems to be acceptable for MOF and COF is the Chinese cherry with pinpoint adsorption active places.

Excluding the gram in dimension of adsorbent capacity G, g/g, gives: 1) G/S_BET, g/m², surface activity, which indicates whether we need to activate the surface, for example by depositing some amount of metal to provide an H₂ spillover [15], 2) G/V_p, g/cm³, a kind of volumetric capacity, 3) conditional G/V_mi, g/cm³, imaginary filling of micropores, provided that the total amount of gas fills the pores of a certain volume (V_mi, V_me, etc.). Comparing these capacities, we can detect the location of micropore, in form of either cracks or pits on the surface of larger micropores or mesopores.

1) K. Xia et al. [16] activated by CO₂ mesoporous carbon SMK-3 at a temperature of 1223 K for 2, 4, 6, and 8 h to store hydrogen. Activation time was the only variable. With its increase, the adsorption of H₂ G, wt.% (1 bar/77 K), the total surface area S_BET, the total volume V_p, the volumes of micropores V_{<2 nm}, V_{1-2 nm}, V_{<1 nm} and their diameter DHK increase.

The surface activity G/S_BET decreases, so the increase in G is associated with all micropore volumes (directly proportional). The best correlation (R² = 0.9995) shows G vs. V_{<1 nm}. G versus diameter D_HK gives a direct proportionality, which contradicts the idea of attraction. So, we tested the generalized size τ = V_p/S_BET (Figure 1).

It can be seen from Figure 1 that a decrease in the generalized pore size from 1.1 to 0.95 nm leads to a strong increase in H₂ adsorption (according to the attraction effect [4]), but then (from 0.95 to 0.76 nm) adsorption slows down. It is obvious that repulsive forces arise, and V_p/S_BET = 0.95 nm is some critical value for this material and gas.

Imagine the density of a gas under the condition that the amount of gas G fits completely into the pores of a given size (Table 1). Pores > 2 nm were called

G: capacity, V: volume of pores, subscript: pore size.

mesoporous, and their volume V_me was obtained as V_me = V_p − V_{< 2 nm}. Note that the serial STD of all discussed volumes V_{<1 nm}, V_{1÷2 nm}, etc. as well G, are in the range of 21% ÷ 38%.

First of all, let’s compare STD of gravimetric G (30.92%) and volumetric G/V_p(7.83%) capacities. Excluding gram of carbon from the dimension of the uptake G allows to reduce STD by 4 times; this means that the time of carbon activation significantly affects the density of carbon. The average G/V data illustrate the decrease in gas density with increasing pore size. Impressive is STD (only 0.6%) G/V_{<1 nm}, which can be explained by the attraction of gas by the walls; the highest STD shows G/V_me where there is no gas attraction.

2) Texier-Mandoki et al. [17] presented uptakes H₂ by 7 carbon species at 77 K and pressures 1, 3. 7 and 10 bars; given are S_BET and V_p obtained by low temperature adsorption of N₂; D_p is missing. We calculate the average layer thicknesses N₂ τ = V_p/S_BET, nm, take them as estimates of pore sizes and then calculate the pore shapes (H₂ molecules packings) F_d = τ/σ_H (σ_H = 0.289 nm is the kinetic diameter of H₂ molecules). Figure 2 shows dependencies H₂ uptakes G, wt.%, on the pore size (left) and on the H₂ molecules packing τ/σ_H(right).

It can be seen from Figure 2 (left) that in the narrowest pores (0.46 nm) the walls have a repulsive effect on gas molecules and uptake is low; an increase in pressure suppresses repulsive forces and increases absorption. It can also be seen that, starting from a pore size of 0.53 nm, G, wt.%, is inversely proportional to the size.

Figure 2 (right) shows the effect of the number of gas layers (packing of gas molecules in layers) on G. Noteworthy are two packings of H₂ molecules: 1.63 and 1.85, which ensure high adsorption. Packing between them, F_dh= 1.7, is a special case, it gives markedly lower uptake (the sample is Norit R0.8 carbon). This result can be explained by the presence of two PSD peaks (see [18], Fig. 2) at pore diameters d₁ < 0.49 nm and d₂ > 0.49 nm, which gives an average value of d = 0.49 nm; both real diameters impair adsorption.

Returning to the work of Fiero [8], mentioned in the introductory part, we

believe that the decrease in the absorption of H₂ at a very large (excessive) surface area and volume of micropores is explained by the formation of micropores of such small sizes that repulsive forces appear.

3) Sun et al. [19] noted that ultra-micropore sorbent (<0.7 nm) should be avoided for natural gas storage. Obviously, for methane the forces of attraction act in the range of pore diameters range of 0.7 [19] ÷ 1.3 [1] nm.

1) Examples of P_mV constancy for various volumes in conventional catalysts and sorbents are given in author’s report [SI-1]. The report is given in SI form because the publisher did not refer to it, and the only evidence in English for its existence is the review by Shchukin et al. [20].

2) Valekar et al. [21] obtained by wet granulation the spheres from MOFs: MIL-100(Fe), MIL-101(Cr), UiO-66(Zr), and UiO-66(Zr)_NH₂ with ρ-Al₂O₃ as a binder. Total surface area S_BET (m²/g), total pore volume V_p (cm³/g) from low-temperature N₂ adsorption, and sphere crushing strength P_m(N) are given. We tested correlations of P_m with S_BET, V_p and V_p/S_BET; no correlations were found. In the same time the sorbent energy P_mV_p showed close relationships with S_BET (logarithmic function, R² = 0.819) and especially with size V_p/S_BET (Figure 3).

Figure 3 (left) reflects the effect of pore size reduction on increased attraction of gas molecules. Figure 3 (right) shows that the energy of the material is inversely proportional to and closely related to the packing of N₂ molecules (R² = 0.975). It is noteworthy that the maximum energy (point 1.19/6.56) corresponds to τ/σ ≈ 1, i.e., to the pore shape F_d which provides a single-layer packing of N₂ molecules.

3) Dhainaut et al. [22] tested tablets hardness P_m (N) of MOFs: UiO-66, UiO-66-NH2, UiO-67, and HKUST-1. As displacement V for P_mV, the given S_BET, V_mi, S_BET/V_mi, crystal density and bulk density of tablets can be tested. We also note two facts related to V_mi: 1) V_mi of powders and tablets, made from

them, are about equal (see also Table 3), 2) generalized micropore size (thicknesses of probing gas) V_mi/S_BET are almost equal for all powder and tablet samples. To find samples with close sorbent energy P_mV graphs like Figure 4 were plotted.

On Figure 4 the highest P_mS_BET has UiO-67. Similar plot with highest (but little lower) energy for the same sorbent was obtained for displacement V_mi (not shown). According to authors ( [22], Fig. S6) this MOF possess the highest volumetric capacity.

Mahmoud [23] in a review, on methane storage, gave total and deliverable uptakes, pressure and temperature for 15 MOFs samples, 12 porous organic polymers (POP) and 20 AC samples, their S_BET and V_p. D_p is missing, so we used V_p/S_BET. For MOF volumetric G, cm³/cm³, for POPs and AC, gravimetric G, g/g, capacities are presented.

a) MOF. G versus V_p, V_p(P/T), S_BET, S_BET(P/T) and V_p/S_BET.

Among the 15 experiments with MOF, the closest dependence shows deliverable G, cm³/cm³, on the total pore volume V_p (Figure 5, left).

On Figure 5 left, the only separated point (0.94/306) belongs to the sample MOF-519 with the highest pressure in the series of 250 Bar. After excluding this sample, the remaining samples form a straight regression line with R² = 0.8918. The gas energy V_p(P/T), as an independent variable (Figure 4, right), returns MOF-519 in the MOF family, but separates samples ST-2 (200 Bar/303 K, 200 Bar/289 K). After excluding ST-2, the rest of the samples form a straight line with R² = 0.8849. The absence of points between the MOF-519 and ST-2 samples on this graph suggests that G = 306 cm³/cm³ is the maximum possible capacity

for the studied MOFs; at very high pressure (dense gas), repulsive forces prevent the capacitance from growing.

When S_BET was taken as the displacement, the variable S_BET(P/T) combined all the samples into one line G-s (not shown), but with a noticeably larger scattering, R² = 0.7591. With the exception of MOF-519, which is furthest from the line, R² = 0.8324 was obtained, which is less than for G vs. V_p(P/T).

G, cm³/cm³, depending on another displacement, the size V_p/S_BET, is shown in Figure 6.

Figure 6 shows the absence of any relationship between the capacity of this series of MOF and the pore size. The separated point (0.15/175) is Cu-tbo-MOF 5 material (see below).

The packing of CH₄ molecules (number of layers) τ/σ.

In this MOF series, the τ/σ range is 0.39 ÷ 1.47. The smallest value τ/σ = 0.39, the only value < 1, belongs to the mentioned Cu-tbo-MOF 5 sample. This MOF is characterized by “the presence of very strong CH₄ adsorption sites near the organic linker with similar adsorption energetics as the open metal sites” [24]. This packing of knots and molecules resembles a Chinese cherry or a stalactite cave with repeatedly changing distances between floor and ceiling; both can serve as pore models. From the remaining data τ/σ (1.00 - 1.47) it can be concluded that these samples have a flat split shape, allowing 1 ÷ 1.5 layers of methane molecules.

b) POP.

The dependences of G, g/g, on S_BET and V_p are directly proportional with R² = 0.9465 and 0.79 correspondingly. The gravimetric capacity G, g/g, makes it possible to estimate the surface activity G/S_BET, g/m²; the most active surface has COF-5. Graphs of G vs. S_BET(P/T) and vs. V_p(P/T) look like in Figure 5 (left); “pushed out” sample from the series, is PPN-4. Excluding this sample, we get straight lines with R² = 0.9787 and R² = 0.9033 for G vs. S_BET(P/T) andG vs. V_p(P/T) respectively, i.e. values higher than for G depending on S_BET and V_p.

There was no evidence of an effect of V_p/S_BET pore size on G, g/g. Assuming that this is due to fluctuations in the density of POPs, the gravimetric uptake of methane G, g/g, was related to cm³ of the total pore volume V_p, cm³/g. G/V_p, g/cm³, kind of volumetric capacity. The plot of G/V_p versus V_p/S_BET was approximated by a power function (Figure 7).

Figure 7 shows a very sharp increase in volumetric capacity as a result of a decrease in pore size. Although the relationship between the variables does not seem very strong (R² = 0.7338), the exclusion of the not very important (due to high V_p/S_BET) COF-5 sample (point 0.77/0.144) raises R² to 0.8483.

c) Activated carbons.

The processing conditions were: constant P = 35 Bar and T = 296 ÷ 300 K (practically constant), so the gas force in Equation (1) was constant. The

strengths of the connections (R²) between G and displacements S_BET, V_p and V_p/S_BET were tested (Table 2).

Among the AC samples, MAXSORB – 3 has the highest S_BET = 3140 m²/g and lowest V_p= 0.179 cm³/g, which gives the smallest generalized pore size V_p/S_BET = 0.057 nm (0.15 methane layers). The latter may explain the highest G = 0.408, g/g, of this sample and suggest a stalactite pore model. Use the volumetric delivery capacity G/V_p, i.e., elimination of fluctuations in sorbent density, provides a very strong relationship (R² = 0.9658) between this capacity and pore size V_p/S_BET.

Fomkin et al. [25] studied the adsorption of H₂ G, wt.%, by 5 different carbon materials at 3 combinations (a, b, c) of pressure and temperature (P/T):(P/T)_a= 1.01 Bar/20.38 K = 0.050, (P/T)_b = 100 Bar/303 K = 0.330, and (P/T)_c = 200 Bar/393 K = 0.509. So, P/T was processing generalized variable. The S_BET and V_mi of the samples were measured by the adsorption of N₂ and benzene, respectively.

The S_BET and V_mi of the samples fluctuated markedly: the STDs were 37% and 68%, respectively. Therefore, as an independent textural characteristic of the AC samples, we took the surface topography (roughness) [1] V_mi/S_BET, cm³/m², the STD of which was lower (29%). The dependent variable was the energy in the form G(P/T) and for each (P/T) the behavior of the function G(P/T) =f(V_mi/S_BET) was studied; shapes of curves were similar. For compactness, in order to fit all the data on one graph, the average G(P/T) values for each P/T were calculated, individual G(P/T) were divided by average (normalized, subscript n) and normalized G(P/T)_na, G(P/T)_nb and G(P/T)_nc were plotted as functions of V_mi/S_BET (Figure 8).

It can be seen that the shapes of curves are similar. The closest are curves for P/T: 0.0500 and 0.509. This is one more example of similarity in PM [1].

Bambalaza et al. [26] compared the absorption H₂ by MOF UiO-66 both by powder and by the same material pressed into granules. Two series of adsorption

G, capacity, V_p, S_BET, total pore volume and surface area, R-squared values represent the scatter around the regression line.

carried out at temperatures: 77 K (dense gas) and 298 K (≈ STP gas); pressures varied from 0 to 100 Bar. On Fig. 3 (left) in [25] shows that a dense gas at high pressure experiences repulsive forces (absorption decreases), but we want also to know the effect of micropore sizes.

Textural parameters: S_BET, V_p, S_mi, V_mi and PSD are given, but the average values of D_p and D_mi are missing. Thus, we have used the generalized average pore size V_p/S_BET and the micropore size V_mi/S_mi, which are the thicknesses of the N₂ gas layers. Dividing the sizes by the kinetic diameters of N₂ and H₂ we obtain the average shape factors of pores and micropores (numbers of layers) F_dn, F_dh, i.e., packings of N₂ and H₂ molecules (subscripts n and h correspondingly) (Table 3).

It follows from the table that tableting the powder did not lead to a change in the micropore size V_mi/S_mi. This closeness is confirmed by the PSD peaks (Fig. 2 in [25]); in the region of micropores, the pore sizes are very close for powder and tablets (3 pairs: d₁ = 0.6 and 0.6 nm, d₂ = 0.70 and 0.76 nm, and d₃ = 1.35 and 1.35 nm, respectively). Assuming a constant V_mi/S_mi and using the indicated sizes, we can estimate the individual micropore shape factors f_d1n, f_d2n and f_d3n (Table 3).

The value of the total packings F_dn, F_dh in pellets are higher than in powder, while the packings in micropores f_dn, f_dh have not changed.

The correlations between gas energy V_p(P/T) and gas uptake by powder and pellets were checked using the ratios the pellet parameters to powder parameters (Pellet/Powder) (Table 4).

The pellet gas energy is 0.84 of the powder energy at both temperatures. This value is close (0.82) to the reduction in both total and excess uptake at 298 K. The lack of correlation at 77 K for both total uptake and excess led us to the idea of introducing a residual amount of gas, which is the difference between the total uptake and excess. As can be seen, the decrease in the amounts of residual gas correlates with the energy of the gas at both temperatures.

Gomez-Gualdrón et al. [7] tested H₂ cryo-adsorption using the MOF series: NU-1101, NU-1102, and NU-1103, cycling between 100 Bar/77 K and one of two delivery conditions: 5 Bar/77 K and 5 Bar/160 K. Most of the data were obtained both experimentally and by simulations. We took for analysis, as a rule, the experimental data; from those obtained by simulation, we took the smallest d and largest D pore sizes as well as heat of adsorption Q_a. Effect of gas energy V_p(P/T) on the total gravimetric capacity G is shown on Figure 9. Exactly the same dependences (not shown) were got for excesses at 5 Bar/77 K and at 5 Bar/160 K.

From the 9 graphs it follows that the largest both total and deliverable capacities correspond to the highest energy of the gas.

V_p, S_BET, total pore volume and surface area, V_mi, S_mi, micropore volume and surface area, F_d, f_d, molecules packing: F, total,f, inmicropores pore,n, nitrogen,h, hydrogen.

V_p, total pore volume, P, pressure, T, temperature.

F_d, dimensionless pore shape factor, V_p, S_BET, total both pore volume and surface area, Packing is number H₂ molecules across pore size.

The reported total surface area S_BET and volume V_p, as well as the diameters of the smallest d and largest D pores, make it possible to estimate the dimensionless shape factors (the number of layers N₂) F_d of these pores and the packing of hydrogen molecules in them (kinetic diameter σ = 0.289 nm).

Comparing d and D with V_p/S_BET in the Table 5, it can be concluded that the proportions of both the smallest and largest pores are very small. F_d-s says that Nu-1101, Nu-1102 have split (F_d ≈ 2) smallest and circular (F_d ≈ 4) largest pores, while Nu-1103 has the smallest pores, having ≈ 50:50% split/circular motifs, and the largest pores with ≈ spherical shape. The average thicknesses of the N₂ layer (0.40 ÷ 0.44 nm) and the numbers of H₂ layers (packings) are almost the same (1.37 ÷ 1.53).

The dependences of the heat of adsorption Q_a on both diameters d and D (all received by simulation) have the form of straight lines with equidistant points and equal gradients, inversely proportional and having strong correlations: R² = 0.954 for d and R² = 0.982 for D. It is possible that small pores are located inside the largest ones. The dependence of Q_a on the experimental V_p/S_BET looks different, namely, as a graph in Figure 9 rotated counterclockwise to 180˚, i.e., confirms that the smaller the pore size, the higher the heat of adsorption.