Strong Acid Research Articles

Tensile-Strained Cu Penetration Electrode Boosts Asymmetric C-C Coupling for Ampere-Level CO2-to-C2+ Reduction in Acid.

The synthesis of multicarbon (C2+) products remains a substantial challenge in sustainable CO2 electroreduction owing to the need for sufficient current density and faradaic efficiency alongside carbon efficiency. Herein, we demonstrate ampere-level high-efficiency CO2 electroreduction to C2+ products in both neutral and strongly acidic (pH = 1) electrolytes using a hierarchical Cu hollow-fiber penetration electrode (HPE). High concentration of K+ could concurrently suppress hydrogen evolution reaction and facilitate C-C coupling, thereby promoting C2+ production in strong acid. By optimizing the K+ and H+ concentration and CO2 flow rate, a faradaic efficiency of 84.5% and a partial current density as high as 3.1 A cm-2 for C2+ products, alongside a single-pass carbon efficiency of 81.5% and stable electrolysis for 240 h were demonstrated in a strong acidic solution of H2SO4 and KCl (pH = 1). Experimental measurements and density functional theory simulations suggested that tensile-strained Cu HPE enhances the asymmetric C-C coupling to steer the selectivity and activity of C2+ products.

Tensile‐Strained Cu Penetration Electrode Boosts Asymmetric C–C Coupling for Ampere‐Level CO2‐to‐C2+ Reduction in Acid

The synthesis of multicarbon (C2+) products remains a substantial challenge in sustainable CO2 electroreduction owing to the need for sufficient current density and faradaic efficiency alongside carbon efficiency. Herein, we demonstrate ampere‐level high‐efficiency CO2 electroreduction to C2+ products in both neutral and strongly acidic (pH = 1) electrolytes using a hierarchical Cu hollow‐fiber penetration electrode (HPE). High concentration of K+ could concurrently suppress hydrogen evolution reaction and facilitate C–C coupling, thereby promoting C2+ production in strong acid. By optimizing the K+ and H+ concentration and CO2 flow rate, a faradaic efficiency of 84.5% and a partial current density as high as 3.1 A cm–2 for C2+ products, alongside a single‐pass carbon efficiency of 81.5% and stable electrolysis for 240 h were demonstrated in a strong acidic solution of H2SO4 and KCl (pH = 1). Experimental measurements and density functional theory simulations suggested that tensile‐strained Cu HPE enhances the asymmetric C–C coupling to steer the selectivity and activity of C2+ products.

CO2 reduction in strong acid

CO2 reduction in strong acid

Study of a Copper Oxide Leaching in Alkaline Monosodium Glutamate Solution

Oxide copper minerals are commonly extracted via acidic leaching, using acids such as H2SO4, HCl, or HNO3. These strong acids are the most widely used because of their high dissolution kinetics. However, their main concern is the high acid consumption because copper oxide deposits contain large amounts of acid-consuming gangue. This paper proposes using an alternative aqueous alkaline monosodium glutamate (MSG) system to leach copper oxide minerals. Tenorite (CuO) was used as the copper oxide mineral under study. The influence of process variables (such as temperature and glutamate concentration) and kinetics of this system on copper leaching from tenorite were studied. The results showed that temperature has a significant effect on copper dissolution rates. Increased temperature from 15 °C to 60 °C enhanced the copper extraction from 9.1% to 97.7% after 2 h. Leaching kinetics were analyzed using the shrinking core model (SCM) under various conditions, indicating that the leaching rate presented a mixed control. This method, however, fails to describe leaching for broad particle sizes due to its requirement for single-sized solid grains. This study demonstrated that a large particle size distribution in tenorite supported a successful extension of the SCM for leaching it from mixed glutamate solutions. The activation energy for the 15–60 °C temperature range was calculated to be 102.6 kJ/mol for the chemical control.

Ischemic stroke: focus on hyperuricemia

Uric acid is a weak organic acid with a molecular weight of 168.112 g/mol. Most uric acid at normal blood pH circulates as urates, negatively charged weak salts derived from uric acid. Being the end product of the process of purine catabolism, uric acid is excreted from the human body in urine until kidney function is impaired. Hyperuricemia may occur due to decreased excretion, increased production, or a combination of both mechanisms. Over the past decades, several studies in the adult population have attempted to establish the correlation between the risk of stroke and serum uric acid concentrations, and how these levels influence the patient's neurological outcome after stroke. Our review is devoted to the study of the mechanisms of normal purine metabolism, disorders of purine metabolism and pathogenetic mechanisms of the development of ischemic stroke in hyperuricemia.

Comprehensive nontargeted analysis of fluorosurfactant byproducts and reaction products in wastewater from semiconductor manufacturing

Semiconductor manufacturing employs per- and polyfluoroalkyl substances (PFAS) as fluoroasurfactants to enhance the quality of photolithography lines. Our research, employing a fragment-based approach to investigate nontarget PFAS, overcomes conventional homologous series limitations. In a mixture of PFAS standards used as a quality control sample, 92% (36 out of 39 compounds spanning 11 compound classes) were detectable through the fragment-based nontarget procedure. This indicates the effectiveness of this approach in identifying the hydrophobic and hydrophilic characteristics of various fluorosurfactants. Eighty-three PFAS were detected in wastewater and effluent samples from semiconductor industry, including 29 newly discovered compounds, categorized into three groups. First, besides detecting perfluorobutane sulfonamido ethanol (FBSE), various fluoroalkyl chain structures of FBSE derivatives were initially identified in wastewater. These include perfluorobutyl ether sulfonamido ethanol, unsaturated FBSE, and hydrogen-substituted FBSE. These derivatives were also detected in trace amounts in commercial authentic standards of FBSE. To quantify their presence, we analyzed the FBSE derivatives from the authentic standard, and the relative proportions of these derivatives contribute to approximately 0.5%. This suggests that the FBSE derivative series detected in wastewater may arise from byproducts of chemical formulations used in the manufacturing of semiconductors. Second, transformation products from FBSE during oxidation, including the first identified intermediate transformation compounds, perfluorobutane sulfonamido acetaldehyde and its hydrate, were discovered. Third, diverse reaction products are generated from the intricate processes of semiconductor manufacturing, which utilize strong acids, bases, and solvents under UV light or heated conditions. These processes include the formation of PFAS-related compounds through hydration, sulfonation, oxidation, and nitrification. This study revealed 25 isomeric PFAS, encompassing headgroup isomers and functional tail group isomers. These findings underscore the importance of comprehending diverse reactions and the overall emission compositions of PFAS in semiconductor wastewater, highlighting its complexity and presenting challenges for subsequent wastewater treatment.

The First Main Group Element Lewis Acid Thionyl chloride Adduct and its Chemistry.

Although the adduct of aluminum trichloride with thionyl chloride has been reported, no thionyl chloride adduct of a main group element Lewis acid or organometallic compound has been structurally characterized. In this communication we present the synthesis and reactivity of the structurally ascertained adduct of thionyl chloride with tris(pentafluoroethyl)gallane as a representative of a main group element Lewis acid. Gallium and indium compounds with electron withdrawing groups, e.g. the pentafluoroethyl ligand, display versatile properties. While gallates and indates, [MR4]-, behave as weakly coordinating anions, neutral gallanes and indanes, MR3, are strong Lewis acids. Salts with the tetrakis(pentafluoroethyl)gallate and -indate, [M(C2F5)4]- (M=Ga, In), have recently been studied in detail. In contrast to this, work on the syntheses of the free Lewis superacids M(C2F5)3 (M= Ga, In) is scarce and underdeveloped. The hydrates [M(C2F5)3(OH2)2] proved to be suitable starting materials, particularly due to their thermal stability. Herein we report on synthesis and characterization of reactive adducts, [M(C2F5)3D], with the weak donor molecules (D) SOCl2 and Me3SiF. The effective Lewis acidities of Ga(C2F5)3 and In(C2F5)3 were experimentally deter-mined by the (modified) Gutmann-Beckett method and their catalytic potential is showcased.

The First Main Group Element Lewis Acid Thionyl chloride Adduct and its Chemistry

Although the adduct of aluminum trichloride with thionyl chloride has been reported, no thionyl chloride adduct of a main group element Lewis acid or organometallic compound has been structurally characterized. In this communication we present the synthesis and reactivity of the structurally ascertained adduct of thionyl chloride with tris(pentafluoroethyl)gallane as a representative of a main group element Lewis acid. Gallium and indium compounds with electron withdrawing groups, e.g. the pentafluoroethyl ligand, display versatile properties. While gallates and indates, [MR4]–, behave as weakly coordinating anions, neutral gallanes and indanes, MR3, are strong Lewis acids. Salts with the tetrakis(pentafluoroethyl)gallate and ‐indate, [M(C2F5)4]– (M=Ga, In), have recently been studied in detail. In contrast to this, work on the syntheses of the free Lewis superacids M(C2F5)3 (M= Ga, In) is scarce and underdeveloped. The hydrates [M(C2F5)3(OH2)2] proved to be suitable starting materials, particularly due to their thermal stability. Herein we report on synthesis and characterization of reactive adducts, [M(C2F5)3D], with the weak donor molecules (D) SOCl2 and Me3SiF. The effective Lewis acidities of Ga(C2F5)3 and In(C2F5)3 were experimentally deter­mined by the (modified) Gutmann‐Beckett method and their catalytic potential is showcased.

Effects of Different Concentrations of Weak Acid Fracturing Fluid on the Microstructure of Coal

Effects of Different Concentrations of Weak Acid Fracturing Fluid on the Microstructure of Coal

Enterobacter sp. HIT-SHJ4 isolated from wetland with carbon, nitrogen and sulfur co-metabolism and its implication for bioremediation

Both autotrophic and heterotrophic denitrification are known as important bioprocesses of microbe-mediated nitrogen cycle in natural ecosystems. Actually, mixotrophic denitrification co-driven by organic matter and reduced sulfur substances are also common, especially in hypoxic environments such as estuarine sediments. However, carbon, nitrogen and sulfur co-metabolism during mixotrophic denitrification in natural water ecosystems has rarely been reported in detail. Therefore, this study investigated the co-metabolism of carbon, nitrogen and sulfur using samples collected from four distinct natural water ecosystems. Results demonstrated that samples from various sources all exhibited the ability for co-metabolism of carbon, nitrogen and sulfur. Microbial community analysis showed that Pseudomonas and Paracoccus were dominant bacteria ranging from 65.6% to 75.5% in mixotrophic environment. Enterobacter sp. HIT-SHJ4, a mixotrophic denitrifying strain which owned the capacity for co-metabolism of carbon, nitrogen and sulfur, was isolated and reported here for the first time. The strain preferred methanol as its carbon source and demonstrated remarkable efficiency for removing sulfide and nitrate with below 100 mg/L sulfide. Under weak acid conditions (pH 6.5–7.0), it exhibited enhanced capability in converting sulfide to elemental sulfur. Its bioactivity was evident within a temperature from 25 °C to 40 °C and C/N ratios from 0.75 to 3. This study confirmed the widespread presence of microbial-mediated synergistic carbon, nitrogen and sulfur metabolism in natural aquatic ecosystems. HIT-SHJ4 emerges as a novel strain, shedding light on carbon, nitrogen and sulfur co-metabolism in natural water bodies. Furthermore, it also serves as a promising candidate microorganism for in-situ ecological remediation, particularly in dealing with contamination posed by nitrate, sulfide, and organic matter.

Straw Biochar and Graphite Oxide Enhanced External Pressure Ultrafiltration for Leaded Wastewater Treatment.

In order to solve the problem of the low treatment efficiency of wastewater containing heavy metals in mining areas, straw biochar and graphene oxide enhanced external pressure ultrafiltration (SGU) was used to treat wastewater containing high concentrations of Pb2+. The operation parameters such as pH and temperature were optimized, and the removal efficiency of CODCr, NH3-N, turbidity and Pb2+ via SGU, straw biochar ultrafiltration (SU), ultrafiltration (UF), and conventional treatment (CT) were systematically investigated. The results showed that the pH and temperature of polluted water were 4.8-5.2 and 21-30 °C, respectively, the average removal rates of CODCr, NH3-N, turbidity and Pb2+ by SGU reached 91%, 97%, 98% and 95%, respectively, and the removal effect was better than that of other processes. In addition, under the backwash conditions of clean water, weak acid, and weak alkali, the membrane flux recovered 65%, 88%, and 89% of the new membrane, respectively. This study provides scientific and theoretical support for the advanced treatment of polluted water in mining areas.

Bioinspired Diiron Complex with Proton Shuttling and Redox-Active Ligand for Electrocatalytic Hydrogen Evolution.

A μ-oxo diiron complex, featuring the pyridine-2,6-dicarboxamide-based thiazoline-derived redox-active ligand, H2L (H2L = N2,N6-bis(4,5-dihydrothiazol-2-yl)pyridine-2,6-dicarboxamide), was synthesized and thoroughly characterized. [FeIII-(μ-O)-FeIII] showed electrocatalytic hydrogen evolution reaction activity in the presence of different organic acids of varying pKa values in dimethylformamide. Through electrochemical analysis, we found that [FeIII-(μ-O)-FeIII] is a precatalyst that undergoes concerted two-electron reduction to generate an active catalyst. Fourier transform infrared spectrum of reduced species and density functional theory (DFT) investigation indicate that the active catalyst contains a bridged hydroxo unit which serves as a local proton source for the Fe(III) hydride intermediate to release H2. We propose that in this active catalyst, the thiazolinium moiety acts as a proton-transferring group. Additionally, under sufficiently strong acidic conditions, bridged oxygen gets protonated before two-electron reduction. In the presence of exogenous acids of varying strengths, it displays electro-assisted catalytic response at a distinct applied potential. Stepwise electron-transfer and protonation reactions on the metal center and the ligand were studied through DFT to understand the thermodynamically favorable pathways. An ECEC or EECC mechanism is proposed depending on the acid strength and applied potential.

Synergistic enzymatic mechanism of lepidolite leaching enhanced by a mixture of Bacillus mucilaginosus and Bacillus circulans

Numerous studies have demonstrated that the co-leaching of ores by different silicate bacteria significantly improves the performance of bioleaching systems. Nevertheless, the mechanism of different silicate bacteria synergistically or complementarily enhanced the leaching process of lithium-containing silicate remains unclear. This study discussed the leaching impact of the combined presence of two metabolically distinct silicate bacteria on lepidolite, with the aim of comprehending the synergistic effect resulting from the presence of Bacillus mucilaginosus and Bacillus circulans in the leaching process. The results indicated that the polysaccharides and proteins secreted by bacteria-containing functional groups such as -OH and -COOH, which played an important role in the complex decomposition of ores. Organic acids played the role of acid etching and complexation. Bacillus mucilaginosus and Bacillus circulans exhibited low individual leaching efficiency, primarily due to their weak organic acid secretion. Moreover, the prolific polysaccharide production by Bacillus mucilaginosus led to bacterial aggregation, diminishing contact capability with minerals. Bacillus circulans decomposed the excessive polysaccharides produced by Bacillus mucilaginosus through enzymatic hydrolysis in the co-bioleaching process, providing later nutrient supply for both strains. The symbiosis of the two strains enhanced the synthesis and metabolic capabilities of both strains, resulting in increased organic acid secretion. In addition, protein and humic acid production by Bacillus mucilaginosus intensified, collectively enhancing the leaching efficiency. These findings suggested that the primary metabolic products secreted by different bacterial strains in the leaching process differ. The improvement in bioleaching efficiency during co-leaching was attributed to their effective synergistic metabolism. This work contributes to the construction of an efficient engineering microbial community to improve the efficiency of silicate mineral leaching, and reveals the feasibility of microbial co-culture to improve bioleaching.

Independent variables of pH: Ten Knights of the Hydrogen Ion Kingdom-Part I. A prospective observational study.

CO2, HCO3, SID, and total weak acids have been defined as pH's independent variables. However, according to Gamble, HCO3 should be equal to the difference between the sum of cations and the sum of anions besides HCO3. Therefore, if this mathematical expression is substituted for HCO3 in the Henderson-Hasselbalch equation, all independent variables of pH can be demonstrated. Our aim is to test this theory in this study. This prospective observational study was conducted between 2019 and 2020. All admitted patients to the intensive care unit who were >18 years old were included. Demographic data, blood gas parameters, albumin, magnesium, and inorganic phosphorus levels, and outcomes were recorded twice (at admission and at the 24th hour). The multivariate linear regression model was used to determine pH's independent variables. In the multivariate linear regression model, pH was significantly increased by each unit increase in Na, K, Ca, and Mg (mmol L-1). In contrast, pH was significantly decreased by each unit increase in CO2, Cl, lactate, albumin (g dL-1), inorganic phosphorus (mg dL-1), and the strong ion gap. Ten independent variables can accurately predict the changes in pH. For this reason, all ten independent variables should be separately evaluated when interpreting the acid-base status. With this understanding, all algorithms regarding acid-base evaluation may become unnecessary.

Smart nanozymes coupled with dynamic magnet field and laser exposures for cancer therapy

Developing nanozymes for cancer therapy has attracted great attention from researchers. However, enzymes-loaded magnetic particles triggered by both a low-frequency vibrating magnetic field (VMF) and laser for inhibiting tumor growth have never been reported. Herein, we developed a magnetic nanozyme with 3D flower-like nanostructures for cancer therapy. Specifically, the flower-like nanozymes exposed to a VMF could efficiently damage the mitochondrial membrane and cell structure, and inhibit tumor growth through magneto-mechanical force. In parallel, magnetic nanozymes in a weak acid environment containing glucose could generate abundant hydrogen peroxide through glucose oxidase-catalyzed oxidation of glucose, and further significantly promote the Fenton reaction. Interestingly, both glucose oxidase- and Fenton-based catalytic reactions were significantly promoted by the VMF exposure. Flower-like magnetic nanospheres upon a near-infrared laser irradiation could also damage cancer cells and tumor tissues through photothermal effect. The cell-killing efficiency of magnetic nanozymes triggered by the VMF or laser significantly increased in comparison with that of nanozymes without exposures. Mouse tumors grown after injection with magnetic nanozymes was inhibited in a significant way or the tumors disappeared after exposure to a VMF and laser due to the synergistic effect of four major stimuli, viz., magneto-mechanical force, photothermal conversion, improved Fenton reaction, and intratumoral glucose consumption-based starvation effect. This is a great platform that may be suitable for treating many solid tumors.

Hydrogen bond regulation to promote the direct production of diosgenin from Dioscorea zingiberensis rhizomes

Herein, a renewable system consisted of tartaric acid and isopropanol was developed to produce diosgenin efficiently from Dioscorea zingiberensis rhizomes. The yield of diosgenin reached 4.15% comparable to the highest value reported in previous researches. The combination of experiment and DFT calculation revealed that the acid-alcohol system could regulate the hydrogen bond interaction of each component in the rhizomes to disassemble the connection network and promote the release of saponin. Meanwhile, the intramolecular hydrogen bond of tartaric acid and the steric hindrance effect of isopropanol could prevent the esterification reaction and maintain the strong acidity of the system for the high yield of diosgenin. The scale-up experiment with a 100-fold feeding capacity was also conducted successfully. Furthermore, the technological process integrated with purification of diosgenin, recovery of organic acid and solvent and upgrading of residual solid to porous carbon materials has been established systematically to verify the prospective of industrialization.

Recent Trends in Tailoring External Acidity in Zeolites for Catalysis.

Zeolites are crystalline aluminosilicates with well-defined microporous structures that have found several applications in catalysis. In recent years, great effort has been devoted to defining strategies aimed at tuning structural and acidity properties to improve the catalytic performance of zeolites. Depending on the zeolitic structure, the acid sites located inside the crystals catalyze reactions by exploiting the internal channel shape-selectivity. In contrast, strong acid sites located on the external surface do not offer the possibility to control the size of molecules involved in the reactions. This aspect generally leads to a loss of selectivity toward desired products and to the uncontrolled production of coke. Passivating surface acidity is a promising way to overcome deactivation issues and to enhance the catalytic performance of zeolites. This Mini-Review aims to provide, for the first time, a complete overview of the techniques employed in recent years to neutralize strong external acid sites. Both chemical and liquid vapor deposition of silicates have been widely employed to passivate the external surface acidity of zeolites. In recent years, the epitaxial growth of layers of aluminum-free zeolite, e.g., silicalite-1, over the surface of the acidic zeolite has been proposed as a new approach to neutralize strong external acid sites controlling diffusional phenomena. NH3-TPD, FT-IR, SEM-EDX, and other techniques have been used to provide information about the level of control of the external strong acidity of passivated zeolites. In this Mini-Review, both passivation treatments and characterization techniques are compared and advantages and disadvantages deeply discussed to elucidate the effect of passivation procedures on physical features and especially the catalytic behavior.

A theoretical investigation of the competition between σ- and π-holes on the stability of the complexes resulting from the interaction between the COCl2 and OCY molecules (Y = S, Se, and Te)

Carbonyl dichloride (COCl2), also known as phosgene, is a highly toxic gas, whose exposure or acute inhalation causes severe respiratory complications, such as pulmonary edema, pulmonary emphysema, and even death. In this study, we theoretically investigated the complexes formed by the interaction between the COCl2 (phosgene) and OCY molecules (Y = S, Se, and Te). Complex geometry optimization yielded three distinct types of conformations. Conformation-I complexes were stabilized by two concurring chalcogen-bonding interactions (i.e. O⋯Y and Cl⋯Y) by forming a ring structure. However, only a single linear interaction (O⋯C) or (O⋯Cl) occurred in conformation-II and III complexes, forming complexes with the nature of tetrel and halogen bonds, respectively. According to the results of the molecular electrostatic potential (MEP), interaction energy (ΔET), natural bond orbital (NBO), and so on, the σ-hole of the Y atom in the OCY molecule exhibited higher acidic strength and formed more stable complexes than those resulting from the σ- and π-holes of the COCl2 molecule.

Hydrogen Evolving Mono‐Nuclear Manganese Complexes with Carefully Positioned Pyridine Base Ligands

AbstractEnormous efforts have been made by the scientific community in the development of bio‐mimetic/bio‐inspired hydrogen‐evolving complexes as catalysts for an alternative hydrogen energy carrier and a renewable energy sources. In this regard, new bio‐inspired mono‐nuclear Mn(I) carbonyl model complexes fac‐[(Mn(CO)3(κ2‐SN2C7H5)(κ1‐PPh2Py))] 1 and fac‐[(Mn(CO)3(κ2‐S2NC7H4)(κ1‐PPh2Py))] 2 with 2‐mercaptobenzimidazole (N,N) and 2‐mercaptobenzothiazole (S,N) ligands and a N‐based phosphine ligand (PPh2Py) have been synthesized, spectroscopically characterized and evaluated as electro‐catalysts for hydrogen generation. In contrast to related mono‐nuclear (N,N/S,N)−Mn−PPh3 complexes,, the newly introduced pyridine nitrogen atom of the PPh2Py ligand placed in an axial position in the second coordination sphere of the Mn(I) complexes, provides a first site for protonation and enables a fast intra‐molecular proton transfer to the central metal atom with a weaker acid. Theoretical DFT calculations enable a detailed picture of catalytic pathways, assign redox transitions and structural changes. Both the molecular complexes were tested as electro‐catalysts for hydrogen generation in CH3CN. They effectively catalyzed the electrochemical proton reduction with the weaker acetic acid as the proton source and displayed a turnover frequency (TOF/s−1; O.P., η/V) of 610; 1.02 and 615; 1.12, respectively. The computer‐aided design of dedicated proton transfer pathways from second coordination sphere ligands is thus able to allow a hydrogen evolution reactivity with a weaker acid (acetic acid).

Selective and Controllable Cracking of Polyethylene Waste by Beta Zeolites with Different Mesoporosity and Crystallinity.

Waste plastics bring about increasingly serious environmental challenges, which can be partly addressed by their interconversion into valuable compounds. It is hypothesized that the porosity and acidity of a zeolite-based catalyst will affect the selectivity and effectiveness, enabling a controllable and selective conversion of polyethylene (PE) into gas-diesel or lubricating base oil. A series of embryonic, partial- and well-crystalline zeolites beta with adjustable porosity and acidity are prepared from mesoporous SBA-15. The catalysts and catalytic systems are studied with nuclear magnetic resonance (NMR), X-ray diffraction (XRD), and adsorption kinetics and catalytic reactions. The adjustable porosity and acidity of zeolite-beta-based catalysts achieve a controllable selectivity toward gas-diesel or lubricating base oil for PE cracking. With a catalyst with mesopores and appropriate acid sites, a fast escape and reduced production of cracking of intermediates are observed, leading to a significant fraction (88.7%) of lubricating base oil. With more micropores, a high acid density, and strong acid strength, PE is multiply cracked into low carbon number hydrocarbons. The strong acid center of the zeolite is confirmed to facilitate significantly the activation of hydrogen (H2),and, an in situ ammonia poisoning strategy can significantly inhibit hydrogen transfer and effectively regulate the product distribution.