#
Palladium, Iridium, and Rhodium Supported Catalysts: Predictive H_{2} Chemisorption by Statistical Cuboctahedron Clusters Model

^{*}

## Abstract

**:**

_{M}), currently assuming a stoichiometry of one hydrogen atom H adsorbed per surface metal atom M. This assumption leads to a large error when estimating D, d, and S

_{M}, and a rigorous method is needed to tackle this problem. A model describing the statistics of the metal surface atom and site distribution on perfect cuboctahedron clusters, already developed for Pt, is applied to Pd, Ir, and Rh, using the density functional theory (DFT) calculation of the literature to determine the most favorable adsorption sites for each metal. The model predicts the H/M values for each metal, in the range 0–1.08 for Pd, 0–2.77 for Ir, and 0–2.31 for Rh, depending on the particle size, clearly showing that the hypothesis of H/M = 1 is not always confirmed. A set of equations is then given for precisely calculating D, d, and S

_{M}for each metal directly from the H chemisorption results determined experimentally, without any assumption about the H/M stoichiometry. This methodology provides a powerful tool for accurate determination of metal dispersion, metal particle size, and metallic specific surface area from chemisorption experiments.

_{2}chemisorption; adsorption sites; stoichiometric factors

## 1. Introduction

_{x}in automotive applications [8], and hydrogen production by steam reforming [9]. Iridium is generally used as a catalyst for propulsion applications [10] or ring opening reactions [11]. In catalysis, the activity of catalysts is currently expressed in the literature by the turnover frequency (TOF), exhibiting the activity per active site. In catalysis by metals, the mean metal particle size and the dispersion are required to be known precisely, to determine the TOF.

_{S}) according to the following reaction (R1):

_{2}chemisorption measurements may be estimated, using the following equation (Equation (2)):

_{S}) and the nature of atomic planes exposed on the surface, the particle size (d(nm)) and the metallic specific surface area (S

_{M}) of noble fcc metals catalysts can be obtained [14]. The common assumption is that the values of H/M

_{S}= 1 for Pt, Pd, Ir, and Rh metals [15,16]. However, some data also report H/M

_{S}stoichiometry factor exceeding unity for Pt, Pd, Rh, and Ir supported catalysts. For instance, data compiled by Bartholomew show chemisorption stoichiometric factor (H/M

_{S}) values of 1.0–1.2 for Pt, Pd, Rh, and Ir catalysts [15] Kip et al. performed careful characterization of supported platinum, rhodium, and iridium catalysts by hydrogen chemisorption and EXAFS data analysis. They reported H/M ratios exceeding unity for Pt (H/Pt = 1.14) and Rh (H/Rh = 1.98), and even higher than 2 for Ir (H/Ir = 2.68) over highly dispersed metal catalysts supported on Al

_{2}O

_{3}and SiO

_{2}[17]. McVicker et al. reported a H/Ir ratio close to 2 for small particle sizes (<0.6 nm) over highly dispersed Ir catalysts on Al

_{2}O

_{3}[18]. Krishnamurthy et al. have shown that 0.48 wt% Ir/Al

_{2}O

_{3}catalyst adsorbed up to 2.72 hydrogen atoms per iridium atom [19].

_{S}ratios higher than unity, such as (i) spillover of H atoms from the metal to the support [20], (ii) hydride formation [21,22], (iii) the support ionicity (with zeolite) [23] or (iv) multiple adsorption on corners and edges for small metal particles [17,24].

_{2}chemisorption literature data for the Pt catalysts [24,26,27]. For this purpose, a model describing the statistics of the surface atoms and sites (top, bridge, hollow) on perfect cuboctahedron clusters was developed. This model allowed us to assess values of D(%), d and S

_{Pt}, assuming the most favorable adsorption sites based on DFT calculation from the literature [28]. Thus, it successfully predicted, precisely, the H/Pt

_{S}stoichiometry, which ranges from 1 to 2 for the smallest cluster (d

_{Pt}= 0.7 nm), and the experimental values of D, d, and S

_{Pt}determined from H

_{2}chemisorption data. A set of simple equations was provided for the accurate determination of these parameters from chemisorption experiments on Pt. This approach, based on the combination of identification and quantification of adsorption sites for a given cluster shape, is expected to be valid for other fcc metals, such as Pd, Rh, and Ir.

_{S}using a simple methodology (statistical model) by the same philosophy as that developed in our previous work [25]. The proposed statistical model will be confronted with the H/M ratios and particle size values obtained from literature data.

## 2. Model Calculation

#### 2.1. Dispersion, Size, Metallic Specific Surface Area, and Adsorption Surface Sites of the Cuboctahedron Crystallite

_{T}, N

_{S}, N

_{B}, and N

_{Ci}representing the total number of atoms, surface atoms, bulk atoms, and atoms of i coordination number, respectively), dispersion (D), size (d), metallic specific surface area (S

_{M}), and adsorption sites (top, bridge, and hollow sites) for metal cuboctahedron cluster (Figure 1). Based on our previous work, Table 1 summarizes the enumeration and the equations giving statistics of atoms, dispersion, size, metallic specific surface area, and the number of each adsorption site for a given value of m (defined as the number of atoms lying on equivalent edge, corners atoms included, of the chosen crystallite) for Pd, Ir, and Rh metal cuboctahedron clusters, respectively [25].

#### 2.2. Surface Hydrogen Adsorption Sites on Metal Cuboctahedron Crystallite (H/M) and H Chemisorption Stoichiometric Factor (H/M_{S})

#### 2.2.1. Case of Pd

#### 2.2.2. Case of Ir

_{4}cluster. These additional adsorption sites are top (corresponding to the ${N}_{1}^{\left(5\right)}$ adsorption site for cuboctahedron clusters) and bridge position at Ir–Ir bonds (corresponding to ${N}_{2}^{\left(5,5\right)}$ adsorption sites for cuboctahedron clusters) [34]. Starting from m = 3, an additional bridge site ${N}_{2}^{\left(7,8\right)}$ appears and has to be considered as another adsorption site.

#### 2.2.3. Case of Rh

_{4}and octahedron Rh

_{6}) indicated that bridge sites are the most stable [35], corresponding to ${N}_{2}^{\left(5,5\right)}$, for a small cuboctahedron cluster (m = 2). When the cluster size increases, ${N}_{2}^{\left(5,7\right)}$ (starting from m = 3) and ${N}_{2edge}^{\left(7,7\right)}$ (starting from m = 4) equivalent adsorption sites are created, due to the additional appearance of edge atoms. As shown for Pd clusters, the ${N}_{4}^{\left(8,8,8,8\right)}$ sites for (100) faces can lead to the creation of additional 4-fold sites (${N}_{4}^{\left(5,5,5,5\right)}+{N}_{4}^{\left(5,7,7,8\right)}+{N}_{4}^{\left(7,7,8,8\right)}$) as the cluster size decreases. Finally, the number of H atoms that can be adsorbed on the Rh cuboctahedron surface (for a given m, denoted ${N}_{H,Rh}$) can be calculated as follows (Equation (5)):

#### 2.2.4. Determination of the Stoichiometric Factor and Correlation between Experimental and Model Calculations

_{H,M}value, as well as the N

_{S}number for each m value, it is possible to calculate the theoretical chemisorption stoichiometric factors with the following equation (Equation (7)):

_{S}theoretical chemisorption stoichiometric factors versus the theoretical H/M ratio are depicted in Figure 2d. The adsorption of one hydrogen atom per surface M atom (M

_{S}) is reasonably constant (near unity) for H/Pd < 0.54, H/Ir < 0.28, and H/Rh < 0.36, which corresponds to the large particle size domain. However, when H/Pd ≥ 0.44, H/Ir ≥ 0.28, and H/Rh ≥ 0.36 (small particle size domain), the H/M

_{S}ratio increases with the H/M ratio to reach a maximum value of 1.17, 3.00, and 2.50 for Pd, Ir, and Rh, respectively. This particular behavior directly originates from the different sites considered for hydrogen adsorption (Equations (3)–(5)), as well as their relative proportion (Table 1).

#### 2.3. Determination of the Dispersion, Particle Size, and Metallic Specific Surface Area from H/M Ratios

_{T}, N

_{S}, N

_{H}, D (%), d (nm), and S

_{M}$\left({\mathrm{m}}^{2}{\mathrm{g}}_{\mathrm{M}}^{-1}\right)$) for any value of m allows drawing correlations with the value of H/M (M corresponding to the chosen metal), the latter being accessible from a chemisorption experiment (Figure 3a–c). It can be seen that the evolution of dispersion, particle size, as well as metallic surface area, are clearly differing from one metal to another. The physical reason for these differences lies in the different adsorption sites between Pd, Rh, and Ir. For a convenient determination of D (%), d (nm), and S

_{M}$\left({m}^{2}{g}_{M}^{-1}\right)$, a general fifth order polynomial trend line (with the R

^{2}value equal to 1) is provided. The expression of dispersion, reciprocal particle size, and metallic surface area (see Table 1) are given below (Equations (8)–(10)), and are plotted as a function of H/M on Figure 3:

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$d$ | particle size (particle diameter) |

${d}_{M}$ | metallic diameter |

$D\mathrm{or}{D}_{M}$ | dispersion |

fcc | face centered cubic |

hcp | hexagonal close packing |

H | hydrogen |

$\frac{H}{M}$ | number of adsorbed hydrogen per total number of metal atoms |

$i$ | coordination number |

Ir | iridium |

$m$ | number of atoms lying on equivalent edge, corners atoms included |

$M$ | metal |

${M}_{S}$ | atom on metal surface |

${N}_{B}$ | total number of bulk atoms |

${N}_{Ci}$ | total number of atoms of i coordination number |

${N}_{H,M}$ | number of hydrogen atoms adsorbed on the metal surface |

${N}_{S}$ | total number of surface atoms |

${N}_{T}$ | total number of atoms |

${N}_{1}^{\left(i\right)}$ | top adsorption site (for example ${N}_{1}^{\left(5\right)}$ represents the top adsorption site over a surface atom of 5 coordination number) |

${N}_{2}^{\left(i,i\right)}$ | bridge adsorption site (for example ${N}_{2}^{\left(5,5\right)}$ represents the bridge adsorption site between two surface atoms of 5 coordination number) |

${N}_{3}^{\left(i,i,i\right)}$ | hollow (3-fold) adsorption site (for example ${N}_{5}^{\left(5,5,5\right)}$ represents the hollow (3-fold) adsorption site between three surface atoms of 5 coordination number) |

${N}_{4}^{\left(i,i,i,i\right)}$ | hollow (4-fold) adsorption site (for example ${N}_{4}^{\left(5,5,5,5\right)}$ represents the hollow (4-fold) adsorption site between four surface atoms of 5 coordination number) |

Pd | palladium |

Pt | platinum |

Rh | rhodium |

${S}_{ci}$ | accessible surface area of the surface atom of type ${N}_{Ci}$ |

${S}_{M}$ | metallic specific surface area |

${\rho}_{M}$ | density of the metal |

$\left(\frac{1}{d}\right)$ | reciprocal particle size of the considered metal |

$2\alpha or\frac{H}{{M}_{S}}$ | chemisorption stoichiometric factor of hydrogen atoms over the metal surface |

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**Figure 1.**Representation of the perfect cuboctahedron (with m = 4) and its adsorption sites over triangular and square faces. The numbers 5 (grey), 7 (red), 8 (blue), and 9 (green) represent the coordination number of the atoms located in the corners, edges, faces (100), and faces (111), respectively. Top sites: white circle with a T; bridge sites: yellow circle with a B; and hollow sites: purple circle with a H (for more details, see ref. [25]).

**Figure 2.**Evolution of the H/M ratio versus the particle size: M = Pd (

**a**), M = Ir (

**b**) and M = Rh (

**c**). Evolution of H/M

_{S}ratio versus H/M ratio (

**d**). Full square, triangle, and circle: literature data for Pd, Ir, and Rh, respectively (see Table 3); and open square, triangle and circle: result of the statistical model calculation of this work for Pd, Ir, and Rh, respectively.

**Figure 3.**Evolution of the theoretical dispersion versus H/M theoretical ratio (M = Pd, Ir, or Rh) (

**a**). Evolution of the theoretical reciprocal particle size versus H/M theoretical ratio (

**b**). Evolution of the theoretical metallic specific surface area versus H/M theoretical ratio (

**c**). Open square, triangle, and circle: result of the statistical model calculation of this work for Pd, Ir, and Rh, respectively. The black, blue, and red curves are the fitting result (R

^{2}= 1.000) with a 5th order polynomial trend line (see Equations (8)–(10)) for Pd, Rh, and Ir, respectively.

**Table 1.**Statistics of atoms, dispersion, size, metallic specific surface area, and adsorption site numbering for metal cuboctahedron cluster. ${d}_{M}$ and ${\rho}_{M}$ represent the metallic diameter (${d}_{Pd}=0.274\mathrm{nm}$, ${d}_{Rh}=0.270\mathrm{nm}$ and ${d}_{Ir}=0.272\mathrm{nm}$ ), and the density of the metal (${\rho}_{Pd}=12.020{\mathrm{g}\mathrm{cm}}^{-3}$, ${\rho}_{Rh}=12.410{\mathrm{g}\mathrm{cm}}^{-3}$ and ${\rho}_{Ir}=22.562{\mathrm{g}\mathrm{cm}}^{-3}$). ${S}_{C5},{S}_{C7},{S}_{C8},\mathrm{and}{S}_{C9}$ represent the surface area of the surface atom of type N

_{C5}, N

_{C7}, N

_{C8}, and N

_{C9}, respectively (for more details, see ref. [25]).

Type | m | ||||
---|---|---|---|---|---|

2 | 3 | 4 | $\ge $5 | ||

Atoms | ${N}_{T}$ | 13 | 55 | 147 | $\frac{10}{3}\times {m}^{3}-5\times {m}^{2}+\frac{11}{3}\times m-1$ |

${N}_{S}$ | 12 | 42 | 92 | $10\times {m}^{2}-20\times m+12$ | |

${N}_{B}$ | 1 | 13 | 55 | $\frac{10}{3}\times {m}^{3}-15\times {m}^{2}+\frac{71}{3}\times m-13$ | |

${N}_{C5}$ | 12 | 12 | 12 | 12 | |

${N}_{C7}$ | 0 | 24 | 48 | $24\times \left(m-2\right)$ | |

${N}_{C8}$ | 0 | 6 | 24 | $6\times {\left(m-2\right)}^{2}$ | |

${N}_{C9}$ | 0 | 0 | 8 | $4\times \left(m-2\right)\times \left(m-3\right)$ | |

D (%) | Pd, Ir and Rh | 92.3 | 76.4 | 62.6 | ${N}_{S}/{N}_{T}\times 100$ |

d (nm) | Pd | 0.7 | 1.2 | 1.6 | $1.105\times {\left({N}_{T}\right)}^{\frac{1}{3}}\times {d}_{M}$ |

Ir | 0.7 | 1.1 | 1.6 | ||

Rh | 0.7 | 1.1 | 1.6 | ||

S_{M}(m ^{2} g^{−1}) | Pd | 1352.2 | 937.3 | 705.7 | $\frac{\left({S}_{C5}+{S}_{C7}+{S}_{C8}+{S}_{C9}\right)\times {10}^{-18}}{\frac{4}{3}\pi \times {\left(\frac{{d}_{M}}{2}\times {10}^{-7}\right)}^{3}\times {N}_{T}\times {\rho}_{M}}$ |

Ir | 725.7 | 503.0 | 378.8 | ||

Rh | 1329.1 | 921.3 | 693.7 | ||

Top sites | ${N}_{1}^{\left(5\right)}$ | 12 | 12 | 12 | 12 |

${N}_{1}^{\left(7\right)}$ | 0 | 24 | 48 | $24\times \left(m-2\right)$ | |

${N}_{1}^{\left(8\right)}$ | 0 | 6 | 24 | $6\times {\left(m-2\right)}^{2}$ | |

${N}_{1}^{\left(9\right)}$ | 0 | 0 | 8 | $4\times \left(m-2\right)\times \left(m-3\right)$ | |

Bridge sites | ${N}_{2}^{\left(5,5\right)}$ | 24 | 0 | 0 | 0 |

${N}_{2}^{\left(5,7\right)}$ | 0 | 48 | 48 | 48 | |

${N}_{2edge}^{\left(7,7\right)}$ | 0 | 0 | 24 | $24\times \left(m-3\right)$ | |

${N}_{2face}^{\left(7,7\right)}$ | 0 | 24 | 24 | 24 | |

${N}_{2}^{\left(7,8\right)}$ | 0 | 24 | 48 | $24\times \left(m-2\right)$ | |

${N}_{2}^{\left(8,8\right)}$ | 0 | 0 | 24 | $12\times \left(m-2\right)\times \left(m-3\right)$ | |

${N}_{2}^{\left(7,9\right)}$ | 0 | 0 | 48 | $48\times \left(m-3\right)$ | |

${N}_{2}^{\left(9,9\right)}$ | 0 | 0 | 0 | $12\times \left(m-3\right)\times \left(m-4\right)$ | |

Hollow sites | ${N}_{3hcp}^{\left(5,5,5\right)}$ | 8 | 0 | 0 | 0 |

${N}_{3hcp}^{\left(5,7,7\right)}$ | 0 | 24 | 24 | 24 | |

${N}_{3hcp}^{\left(7,7,7\right)}$ | 0 | 8 | 0 | 0 | |

${N}_{3fcc}^{\left(7,7,9\right)}$ | 0 | 0 | 24 | 24 | |

${N}_{hcp}^{\left(7,7,9\right)}$ | 0 | 0 | 24 | $24\times \left(m-3\right)$ | |

${N}_{3fcc}^{\left(7,9,9\right)}$ | 0 | 0 | 0 | $24\times \left(m-4\right)$ | |

${N}_{3fcc}^{\left(9,9,9\right)}$ | 0 | 0 | 0 | $4\times \left(m-4\right)\times \left(m-5\right)$ | |

${N}_{3hcp}^{\left(9,9,9\right)}$ | 0 | 0 | 0 | $4\times \left(m-3\right)\times \left(m-4\right)$ | |

${N}_{4}^{\left(5,5,5,5\right)}$ | 6 | 0 | 0 | 0 | |

${N}_{4}^{\left(5,7,7,8\right)}$ | 0 | 24 | 24 | 24 | |

${N}_{4}^{\left(7,7,8,8\right)}$ | 0 | 0 | 24 | $24\times \left(m-3\right)$ | |

${N}_{4}^{\left(8,8,8,8\right)}$ | 0 | 0 | 6 | $6\times {\left(m-3\right)}^{2}$ |

**Table 2.**Most favored hydrogen adsorption sites for Pd, Ir, and Rh flat surfaces and clusters determined from DFT/ab initio calculations.

Metal | Surface or Shape | H Adsorption Favored Sites | Ref |
---|---|---|---|

Pd | (100) | Hollow 4-fold | [30] |

(111) | Hollow 3-fold fcc | [31] | |

Cuboctahedron (Pd_{13}) | Hollow 4-fold and 3-fold hcp | [32] | |

Ir | (100) | Bridge | [33] |

(111) | Top | [31] | |

Truncated octahedron (Ir_{38}) | Bridge (edge) | [33] | |

Tetrahedron (Ir_{4}) | Top (corner) and Bridge (at Ir–Ir bonds) | [34] | |

Rh | (100) | Hollow 4-fold | [30] |

(111) | Hollow 3-fold fcc | [31] | |

Tetrahedron (Rh_{4}) | Bridge (edge) | [35] | |

Octahedron (Rh_{6}) | Bridge (edge) | [35] |

**Table 3.**Literature results of H

_{2}chemisorption measurements and average particle sizes (determined by TEM) for Pd, Ir, and Rh catalysts.

M/Support | H/M | d (nm) | Ref |
---|---|---|---|

Pd/SiO_{2} | 0.40 | 2.5 | |

Pd/SiO_{2} | 0.13 | 6.5 | |

Pd/Al_{2}O_{3} | 0.41 | 2.5 | [37] |

Pd/Al_{2}O_{3} | 0.06 | 13 | |

Pd/Al_{2}O_{3} | 0.54 | 2.8 | |

Pd/Al_{2}O_{3} | 0.52 | 1.4 | |

Pd/Al_{2}O_{3} | 0.52 | 5.1 | |

Pd/Al_{2}O_{3} | 0.14 | 7.7 | [38] |

Pd/Al_{2}O_{3} | 0.26 | 6 | |

Pd/Al_{2}O_{3} | 0.23 | 7.2 | |

Pd/Al_{2}O_{3} | 0.91 | 0.9 | [39] |

Pd/Al_{2}O_{3} | 0.26 | 5 | |

Pd/Al_{2}O_{3} | 0.44 | 2.7 | |

Pd/Al_{2}O_{3} | 0.37 | 3.2 | [40] |

Pd/Al_{2}O_{3} | 0.38 | 4.2 | |

Pd/Al_{2}O_{3} | 0.71 | 1.4 | |

Pd/Al_{2}O_{3} | 0.71 | 1.2 | |

Ir/Al_{2}O_{3} | 1.96 | <0.6 | |

Ir/Al_{2}O_{3} | 1.57 | <0.6 | |

Ir/Al_{2}O_{3} | 0.98 | 0.81 | [18] |

Ir/Al_{2}O_{3} | 0.51 | 2.9 | |

Ir/Al_{2}O_{3} | 0.13 | 12.7 | |

Rh/Al_{2}O_{3} | 0.92 | 0.9 | [39] |

Rh/Al_{2}O_{3} | 0.22 | 4.8 | |

Rh/Al_{2}O_{3} | 0.80 | 1.7 | [41] |

Rh/Al_{2}O_{3} | 0.45 | 2.4 | |

Rh/Al_{2}O_{3} | 0.082 | 15 | |

Rh/SBA-15 | 0.49 | 1.9 | |

Rh/SBA-15 | 0.49 | 1.9 | |

Rh/SBA-15 | 0.48 | 2.4 | |

Rh/SBA-15 | 0.23 | 3.6 | [42] |

Rh/SBA-15 | 0.13 | 5.1 | |

Rh/SBA-15 | 0.16 | 6.7 | |

Rh/SBA-15 | 0.11 | 11.3 |

**Table 4.**Values of the constants ${a}_{Y},{b}_{Y},{c}_{Y},{d}_{Y},\mathrm{and}{e}_{Y}$ for Equation (11). (M: metal; range of validity of equation 11: 0–1.08 for H/Pd, 0–2.31 for H/Rh, and 0–2.77 for H/Ir).

Equation | ${\mathit{Y}}_{\mathit{M}}={\mathit{a}}_{\mathit{Y}}\times {\left(\frac{\mathit{H}}{\mathit{M}}\right)}^{5}+{\mathit{b}}_{\mathit{Y}}\times {\left(\frac{\mathit{H}}{\mathit{M}}\right)}^{4}+{\mathit{c}}_{\mathit{Y}}\times {\left(\frac{\mathit{H}}{\mathit{M}}\right)}^{3}+{\mathit{d}}_{\mathit{Y}}\times {\left(\frac{\mathit{H}}{\mathit{M}}\right)}^{2}+{\mathit{e}}_{\mathit{Y}}\times \left(\frac{\mathit{H}}{\mathit{M}}\right)$ | |||||
---|---|---|---|---|---|---|

${\mathit{Y}}_{\mathit{M}}$ | $\mathit{M}$ | ${\mathit{a}}_{\mathit{Y}}$ | ${\mathit{b}}_{\mathit{Y}}$ | ${\mathit{c}}_{\mathit{Y}}$ | ${\mathit{d}}_{\mathit{Y}}$ | ${\mathit{e}}_{\mathit{Y}}$ |

${D}_{M}(\%)$ | Pd | −5.055 | 71.208 | −117.720 | 38.434 | 98.775 |

Ir | −2.116 | 13.163 | −20.633 | −23.073 | 100.361 | |

Rh | −8.599 | 46.065 | −73.064 | 2.015 | 101.969 | |

${\left(\frac{1}{d}\right)}_{M}\left({\mathrm{nm}}^{-1}\right)$ | Pd | 1.912 | −2.665 | 0.875 | 0.288 | 0.737 |

Ir | 0.000 | 0.038 | −0.171 | 0.099 | 0.743 | |

Rh | −0.063 | 0.390 | −0.771 | 0.414 | 0.753 | |

${S}_{M}\left({\mathrm{m}}^{2}{\mathrm{g}}_{\mathrm{M}}^{-1}\right)$ | Pd | 1053.493 | −1139.725 | −119.922 | 463.061 | 903.061 |

Ir | −12.169 | 81.710 | −176.563 | 39.215 | 487.871 | |

Rh | −106.053 | 576.518 | −1005.933 | 413.062 | 900.314 |

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**MDPI and ACS Style**

Drault, F.; Comminges, C.; Can, F.; Pirault-Roy, L.; Epron, F.; Le Valant, A.
Palladium, Iridium, and Rhodium Supported Catalysts: Predictive H_{2} Chemisorption by Statistical Cuboctahedron Clusters Model. *Materials* **2018**, *11*, 819.
https://doi.org/10.3390/ma11050819

**AMA Style**

Drault F, Comminges C, Can F, Pirault-Roy L, Epron F, Le Valant A.
Palladium, Iridium, and Rhodium Supported Catalysts: Predictive H_{2} Chemisorption by Statistical Cuboctahedron Clusters Model. *Materials*. 2018; 11(5):819.
https://doi.org/10.3390/ma11050819

**Chicago/Turabian Style**

Drault, Fabien, Clément Comminges, Fabien Can, Laurence Pirault-Roy, Florence Epron, and Anthony Le Valant.
2018. "Palladium, Iridium, and Rhodium Supported Catalysts: Predictive H_{2} Chemisorption by Statistical Cuboctahedron Clusters Model" *Materials* 11, no. 5: 819.
https://doi.org/10.3390/ma11050819