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At least 127 records · Page 7

Accurate Machine Learning for Predicting the Viscosities of Deep Eutectic Solvents

Deep eutectic solvents (DESs) are emerging as environmentally friendly designer solvents for mass transport and heat transfer processes in industrial applications; however, the lack of accurate tools to predict and thus control their viscosities under both a range of environmental factors and formulations hinders their general application. While DESs may serve as designer solvents, with nearly unlimited combinations, this unfortunately makes it experimentally infeasible to comprehensively measure the viscosities of all DESs of potential industrial interest. To assist in the design of DESs, we have developed several new machine learning (ML) models that accurately and rapidly predict the viscosities of a diverse group of DESs at different temperatures and molar ratios using, to date, one of the most comprehensive data sets containing the properties of over 670 DESs over a wide range of temperatures (278.15–385.25 K). Three ML models, including support vector regression (SVR), feed forward neural networks (FFNNs), and categorical boosting (CatBoost), were developed to predict DES viscosity as a function of temperature and molar ratio and contrasted with multilinear and two-factor polynomial regression baselines. Further, quantum chemistry-based, COSMO-RS-derived sigma profile (σ-profile) features were used as inputs for the ML models. The CatBoost model is excellent at externally predicting DES viscosity, as indicated by high R 2 (0.99) and low root-mean-square-error (RMSE) and average absolute relative deviations (AARD) (5.22%) values for the testing data sets, and 98% of the data points lie within the 15% of AARD deviations. Furthermore, SHapley additive explanation (SHAP) analysis was employed to interpret the ML results and rationalize the viscosity predictions. The result is an ML approach that accurately predicts viscosity and will aid in accelerating the design of appropriate DESs for industrial applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Electron transfer in nonpolar media

The Marcus theory of electron transfer views fluctuating orientations of permanent dipoles as the nuclear mode bringing the donor and acceptor into the tunneling resonance. Electronic polarization of the solvent is excluded as the fast mode adiabatically following the electronic density. This view, valid for solids, does not apply to molecular liquids where molecular translations (density fluctuations) modulate the induction interaction of the donor-acceptor complex with the solvent. This mechanism of promoting radiation-less electronic transitions is considered here in the framework of the perturbation liquid-state theory. The reorganization energy of electron transfer in nonpolar solvents is nonzero and reaches the values of 0.1–0.3 eV for typical molecular sizes and solvents used in applications. The reorganization energy scales quadratically with the molecular polarizability of the solvent and decays as the inverse fifth power with the size of the donor and acceptor. The combination of the entropic character of the density fluctuations, driven by re-packing of molecular cores, with the short range of induction solute–solvent interactions leads to the violation of the fluctuation–dissipation theorem for the variance of the donor-acceptor energy gap. An explicit, approximately hyperbolic, dependence of the reorganization energy on temperature is predicted. It leads to a non-Arrhenius kinetic law for the rate constant of electron transfer in nonpolar liquid solvents.

14 SOLAR ENERGY↗

Mobile loop dynamics in adenosyltransferase control binding and reactivity of coenzyme B 12

Cobalamin is a complex organometallic cofactor that is processed and targeted via a network of chaperones to its dependent enzymes. AdoCbl (5'-deoxyadenosylcobalamin) is synthesized from cob(II)alamin in a reductive adenosylation reaction catalyzed by adenosyltransferase (ATR), which also serves as an escort, delivering AdoCbl to methylmalonyl-CoA mutase (MCM). The mechanism by which ATR signals that its cofactor cargo is ready (AdoCbl) or not [cob(II)alamin] for transfer to MCM, is not known. In this study, we have obtained crystallographic snapshots that reveal ligand-induced ordering of the N terminus of Mycobacterium tuberculosis ATR, which organizes a dynamic cobalamin binding site and exerts exquisite control over coordination geometry, reactivity, and solvent accessibility. Cob(II)alamin binds with its dimethylbenzimidazole tail splayed into a side pocket and its corrin ring buried. The cosubstrate, ATP, enforces a four-coordinate cob(II)alamin geometry, facilitating the unfavorable reduction to cob(I)alamin. The binding mode for AdoCbl is notably different from that of cob(II)alamin, with the dimethylbenzimidazole tail tucked under the corrin ring, displacing the N terminus of ATR, which is disordered. In this solvent-exposed conformation, AdoCbl undergoes facile transfer to MCM. The importance of the tail in cofactor handover from ATR to MCM is revealed by the failure of 5'-deoxyadenosylcobinamide, lacking the tail, to transfer. In the absence of MCM, ATR induces a sacrificial cobalt–carbon bond homolysis reaction in an unusual reversal of the heterolytic chemistry that was deployed to make the same bond. Finally, the data support an important role for the dimethylbenzimidazole tail in moving the cobalamin cofactor between active sites.

59 BASIC BIOLOGICAL SCIENCES↗

Elucidating the Molecular Origins of the Transference Number in Battery Electrolytes Using Computer Simulations

The rate at which rechargeable batteries can be charged and discharged is governed by the selective transport of the working ions through the electrolyte. Conductivity, the parameter commonly used to characterize ion transport in electrolytes, reflects the mobility of both cations and anions. The transference number, a parameter introduced over a century ago, sheds light on the relative rates of cation and anion transport. This parameter is, not surprisingly, affected by cation–cation, anion–anion, and cation–anion correlations. In addition, it is affected by correlations between the ions and neutral solvent molecules. Computer simulations have the potential to provide insights into the nature of these correlations. We review the dominant theoretical approaches used to predict the transference number from simulations by using a model univalent lithium electrolyte. In electrolytes of low concentration, one can obtain a quantitative model by assuming that the solution is made up of discrete ion-containing clusters–neutral ion pairs, negatively and positively charged triplets, neutral quadruplets, and so on. These clusters can be identified in simulations using simple algorithms, provided their lifetimes are sufficiently long. In concentrated electrolytes, more clusters are short-lived and more rigorous approaches that account for all correlations are necessary to quantify transference. Elucidating the molecular origin of the transference number in this limit remains an unmet challenge.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Effects of charged interfaces on electrolyte decomposition at the lithium metal anode

Lithium–Sulfur batteries are promising candidates to substitute conventional Li-ion batteries due to their higher energy density and reduced cost. However, several challenges related to the reactivity of the lithium metal anode have prevented this technology from becoming broadly commercialized. Lithium's high reactive nature leads to the continuous decomposition of the electrolyte and the formation of the solid-electrolyte interphase (SEI) layer. Thus, a comprehensive understanding of how the SEI film is formed is crucial to help elucidate improvements for this battery technology. In this work, we use density functional theory (DFT) based computational methods to investigate the effect of charged interfaces on the electrolyte degradation at the lithium anode. Here, several electrolyte mixtures were considered including 1,2-dimethoxyethane (DME) and 1,3-dioxolane (DOL) as solvents, and lithium bis(fluorosulfonyl)imide (LiFSI) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) as salts. It is found that the extent of salt decomposition is higher when the interface is charged. In addition, two types of solvent reduction mechanisms are identified: C–O bond cleavage and radical attacks. Charge transfer and evolution analysis are characterized in detail. Finally, solvent decomposition reactions are found to become more thermodynamically favorable and their activation barriers to be diminished under the effect of constant potentials and electric fields.

25 ENERGY STORAGE↗

Mass transfer intensification through increased surface wetting and liquid turbulence using 3D printing structured packing for CO2 capture

The absorber column is one of the most expensive pieces of equipment to construct in the solvent-based post-combustion carbon capture unit. In order to decrease the absorber size and reduce capital costs, novel polymer packings were proposed by enhancing surface wettability and local turbulence within liquid solvent. The novel packings intensify the mass transfer in CO2 absorption and show better separation efficiency than traditional structured packings. In this work, the economic influence of applying the more efficient packing with a shorter absorber column is studied in a techno-economic analysis for a carbon capture unit at a coal-fired power plant. The baseline case includes CO2 capture unit in a supercritical pulverized coal power plant to generate 650 MWe (net) of electricity, where the additional CO2 capture unit leads to 63.4% increase of LCOE. While implementing UK PCC with traditional and novel packings, there will be 47.2% and 46.4% LCOE increase separately, both of which are lower than the baseline value. Applying more efficient packing and smaller absorber could further lessen the LCOE increase. The UK PCC with traditional packings reduces CO2 capture cost by 23.4% and the application of the advanced packings allows to the reduction increases to 24.4%.

60 APPLIED LIFE SCIENCES↗

Surface Doping Boosts Triplet Generation Yield in Perovskite-Sensitized Upconversion

Triplet–triplet annihilation-based photon upconversion (TTA-UC) can efficiently generate higher energy photons at low relative fluences. Bulk metal halide perovskites have offered promise in efficiently sensitizing molecular triplet states in the solid state, necessary for the integration of TTA-UC into device-based applications. Recent work focused on TTA-UC from a rubrene triplet annihilator sensitized by perovskite thin films has established relatively efficient charge extraction from the perovskite, forming the triplet exciton in rubrene. Yet, the specifics underpinning charge transfer at the perovskite/rubrene interface are not fully elucidated. Here, to improve device performance and study the properties governing charge transfer at the interface, various organic solvents are explored to treat the perovskite surface. Scanning tunneling microscopy and spectroscopy show a difference in the electronic band structure, where both n- and p-type terminated perovskite surfaces are observed depending on the solvent used. Supported by optical spectroscopy, the impact of the perovskite electronic structure is monitored, indicating that n-type perovskite sensitizers feature higher TTA-UC efficiencies due to favorable band bending resulting in efficient hole-mediated triplet formation. Overall, the tuning of the electronic structure of the perovskite sensitizer through solvent treatment is shown to be a key force in tuning the mechanism of efficient triplet generation.

36 MATERIALS SCIENCE↗

Spectroscopic Studies of Charge-Transfer Character and Photoresponses of F 4 TCNQ-Based Donor–Acceptor Complexes

F 4 TCNQ (2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane) is used widely as a hole-doping agent in photoresponsive organic semiconducting materials, yet relatively little is known about the photoresponses of the F 4 TCNQ· - anion generated via doping. Furthermore, there is still relatively little systematic exploration of how the properties of the local material or chemical environment impacts the driving force for generating these charge-transfer complexes. Here we present spectroscopic and photophysical studies of F 4 TCNQ in charge-transfer complexes (CTCs) with the electron donor N,N’-Diphenyl-N-N’-di-p-tolylbenzene-1,4-diamine (MPDA) both in dichloroethane solution and polystyrene matrices. Integer charge transfer (ICT) between donor and acceptor occurs readily in dichloroethane solvent to form F 4 TCNQ· - :MPDA + CTCs, due to a ~150 mV difference in MPDA + /MPDA and F 4 TCNQ/F 4 TCNQ· - reduction potentials. Ultrafast spectroscopic studies of the CTC as well as electrochemically generated F 4 TCNQ· - and MDPA + reveal that the photoresponses of these CTCs are dominated by that of the dopant anion, including rapid deactivation (800 fs) after excitation to the anion D 1 excited state, followed by slower (~10 ps) vibrational cooling in the anion D 0 state. Excitation to the higher-lying D 2 state results in a rapid relaxation to the D 1 state, in contrast to direct D 2 -D 0 relaxation previously observed for F 4 TCNQ· - in gas phase. CTCs embedded in polystyrene (PS) matrices are observed to lose their integer charge-transfer character upon evaporation of solvent, as evidenced by changes to electronic and vibrational absorption features associated with F 4 TCNQ· - . This change is attributed to the loss of solvent stabilization of the ion pair formed through the charge-transfer reaction. Ultrafast spectral measurements reveal that the photoresponses of the partial charge-transfer (PCT) species embedded in PS are still highly similar to those of the ICT species and unlike that of neutral F 4 TCNQ, implying the electronic properties of the PCT state are likewise dominated by properties of the reduced acceptor molecule. We conclude that excitation of ICT or PCT states introduces optical losses for photoresponses of doped organic semiconductor materials due to the large anion absorption cross section and its rapid, dissipative deactivation dynamics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Convergent spray process for environmentally friendly coatings

Conventional spray application processes have poor transfer efficiencies, resulting in an exorbitant loss in materials, solvents, and time. Also, with ever tightening Environmental Protection Agency (EPA) regulations and Occupational Safety and Health Administration requirements, the low transfer efficiencies have a significant impact on the quantities of materials and solvents that are released into the environment. High solids spray processes are also limited by material viscosities, thus requiring many passes over the surface to achieve a thickness in the 0.125 -inch range. This results in high application costs and a negative impact on the environment. Until recently, requirements for a 100 percent solid sprayable, environmentally friendly, lightweight thermal protection system that can be applied in a thick (greater than 0.125 inch) single-pass operation exceeded the capability of existing systems. Such coatings must be applied by hand lay-up techniques, especially for thermal and/or fire protection systems. The current formulation of these coatings has presented many problems such as worker safety, environmental hazards, waste, high cost, and application constraints. A system which can apply coatings without using hazardous materials would alleviate many of these problems. Potential applications include the aerospace thermal protective specialty coatings, chemical and petroleum industries that require fire-protection coatings that resist impact, chemicals, and weather. These markets can be penetrated by offering customized coatings applied by automated processes that are environmentally friendly.

Scarpa, Jack↗

Pilot scale testing of an advanced solvent in a 0.7 MWe post-combustion CO 2 capture unit

An advanced Amine Promoted Buffer Solution (APBS), APBS-CDRMax® (CDRMax), developed by Carbon Clean Solutions Limited (CCSL), was tested at the 0.7 MWe CO 2 capture facility at Kentucky Utilities E.W. Brown Generation Station using a heat-integrated two-staged stripping CO 2 capture process. The performance of the solvent was evaluated to determine operating conditions that maximized the cyclic capacity of the solvent and results in energy savings. The regeneration energy ranged from 2.9 to 3.3 GJ/ton CO 2 with 14 vol % (dry) CO 2 inlet and approximately 90 % capture. The difference in the reboiler specific heat duty at stripper pressures of 1.7 and 2.1 bar was minimal as similar amounts of water vapor were observed in the CO 2 product stream at stripper outlet. Recycling of product CO 2 increased the inlet CO 2 concentration to the absorber from 14 to 16 vol% which enhanced mass transfer from the gas to the solvent resulting in about 5 % reduction in the energy of regeneration at the lower stripper pressure. The solvent circulation rate was reduced by about 30 % relative to previous 30 wt% MEA campaign for the CO 2 target capture of 90 %. Finally, the reduced solvent make-up rate of CDRMax shows promise for capital and operating cost savings for post-combustion CO 2 capture.

42 ENGINEERING↗

High Li + Transference Number Electrolyte Enabled by Fluoride Acceptor for Low-Temperature Li-Ion Batteries

To enable wide-temperature operation of lithium-ion batteries (LIBs), new electrolyte formulations have been developed to enhance the performance, particularly at low temperatures. A key challenge lies in achieving both high ionic conductivity and a high lithium-ion transference number due to their inherent trade-off. In this study, we designed an electrolyte system comprising tris(pentafluorophenyl)borane (TPFPB), a fluoride acceptor, and LiF salt in ethylene carbonate (EC)-free solvents. TPFPB, with its electron-deficient boron center, facilitates fluoride transfer reactions that promote the dissociation of otherwise insoluble LiF. When methyl acetate (MA) was used as the solvent, the electrolyte exhibited a high transference number (t Li + = 0.85) and ionic conductivity (σ = 5.0 × 10 –3 S cm –1 ). The optimized electrolyte demonstrated excellent performance at −20 °C, with no evidence of lithium plating. This work presents a new strategy for electrolyte design by leveraging cation desolvation to achieve high-performance LIBs for low-temperature and high-power applications.

anions↗

An effective air–liquid contactor for CO 2 direct air capture using aqueous solvents

Here, the development of a cost-effective, corrosion resistant, high-flux direct air capture (HiDAC) contactor for a solvent-based direct air capture (DAC) process is reported. Literature technoeconomic analyses suggest that the air–liquid contactor can cost over 20 % of the overall DAC plant’s annualized capital costs. To bring down the overall cost of DAC, it is imperative that an effective contactor is developed. A hybrid contactor consisting of a commercial polyvinylchloride structured packing enhanced with stainless-steel 410 random packing has been developed to provide a high surface area for air–liquid contact. The contactor geometry, wettability, corrosion resistance, pressure drop, along with its CO 2 uptake efficiency, CO 2 uptake rates, and extended loading potential using potassium sarcosinate solutions are investigated. Results show that the HiDAC contactor has a relatively high specific surface area (885 m 2 /m 3 ), which allows for CO 2 uptake efficiencies of up to 75 % and capture rates of up to 550 g of CO 2 per day for a 0.3 × 0.25 × 0.3 m contactor. The contactor also exhibits high levels of wettability and corrosion resistance with amino acid-based DAC solvents. A CO 2 uptake model was developed, and modeling results are compared to experimental data to simulate and predict the performance of the contactor in a DAC process using potassium sarcosinate solvent. The system’s overall mass transfer coefficient and theoretical pressure drop were also calculated, and these results were compared to DAC data found in literature. The results presented indicate that the HiDAC contactor is well suited for DAC due to its high specific surface area, resistance to corrosion, and high degree of wettability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

The Role of Protons and Hydrides in the Catalytic Hydrogenolysis of Guaiacol at the Ruthenium Nanoparticle-water Interface

The mechanistic roles of free hydronium ions, surface hydrides, and interfacial protons during guaiacol hydrodeoxygenation (HDO) on ruthenium nanoparticles are established. As guaiacol adsorbs on Ru, it loses its strong aromaticity and undergoes a rapid H-shift from its hydroxyl to meta carbons (in relation to its hydroxyl group), leading enol and keto surface isomers to exist in chemical equilibrium. HDO occurs via a hydridic H-adatom (H*) attack to the enol, followed by a kinetically relevant C-O bond rupture step, during which water shuttles the hydroxyl proton, enabling its intramolecular attack to the methoxy, evolving a high charged [Ru(s)-(C6H5O-)…(H+)…OCH3]† transition state. The competing HYD begins with a rapid H* attack to the keto, before a second, kinetically relevant H* attack, without proton involvement. Water, despite shifting the thermodynamics towards the more polar surface keto, promotes HDO to a much greater extent than HYD, because of its dual catalytic roles—it mobilizes hydroxyl proton (Brønsted acid) to cleave the strong C-O bond, synchronizing with the Ru metal surface (base) function that stabilizes the resulting [Ru(s)-(C6H5O-)…(H+)…OCH3]† transition state, and the water layers solvate this charged transition state, further reducing its free energy. Free hydronium ions do catalyze a separate homogeneous enol-keto isomerization, but this reaction is kinetically unrelated to HDO catalysis. This mechanistic picture explains the strong effects of polar protic solvent in hydrodeoxygenation, highlighting (i) the requirements of surface hydrides and interfacial protons acting in tandem to complete a HDO turnover and (ii) the cooperative role of protic solvent and metal surface in breaking the aromaticity and stabilizing charged reactive precursors and transition states.

Shangguan, Junnan↗

Adiabatic criteria for outer-sphere bimolecular electron-transfer reactions

A model is presented for outer-sphere bimolecular electron-transfer reactions which is correct in the adiabatic, nonadiabatic, and intermediate dynamical regimes for an overdamped solvent coordinate. From this model, the conditions for the transfer to be adiabatic or nonadiabatic are deduced. The time-scale separations needed to adequately describe the process as an average over (distant dependent) unimolecular rates are described.

Onuchic, Jose Nelson↗

Sub-100-fs energy transfer in coenzyme NADH is a coherent process assisted by a charge-transfer state

Excitation energy transfer (EET) is a key photoinduced process in biological chromophoric assemblies. Here we investigate the factors which can drive EET into efficient ultrafast sub-ps regimes. We demonstrate how a coherent transport of electronic population could facilitate this in water solvated NADH coenzyme and uncover the role of an intermediate dark charge-transfer state. High temporal resolution ultrafast optical spectroscopy gives a 54 ± 11 fs time constant for the EET process. Nonadiabatic quantum dynamical simulations computed through the time-evolution of multidimensional wavepackets suggest that the population transfer is mediated by photoexcited molecular vibrations due to strong coupling between the electronic states. The polar aqueous solvent environment leads to the active participation of a dark charge transfer state, accelerating the vibronically coherent EET process in favorably stacked conformers and solvent cavities. Our work demonstrates how the interplay of structural and environmental factors leads to diverse pathways for the EET process in flexible heterodimers and provides general insights relevant for coherent EET processes in stacked multichromophoric aggregates like DNA strands.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Electrochemical properties of poly(ethylene oxide) electrolytes above the entanglement threshold

The ion transport in electrolytes depends on three transport coefficients, conductivity (κ), salt diffusion coefficient (D), and the cation transference number with respect to the solvent velocity ($t_+^0$), and the thermodynamic factor ($T_f$). Current methods for determining these parameters involve four separate experiments, and the coupled nature of the equations used to determine them generally results in large experimental uncertainty. We present data obtained from 64 independent polymer electrolytes comprising poly(ethylene oxide) (PEO) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) salt. The molecular weights of PEO ranged from 5 to 275 kg mol -1 ; these samples are all above the entanglement threshold. We minimize the experimental uncertainty in transport and thermodynamic measurements by exploiting the fact that ion transport in entangled polymer electrolytes should be independent of molecular weight. The dependence of $κ, D, t_+^0$, and $T_f$ as a function of salt concentration in the range 0.035 ≤ r ≤ 0.30 are presented with a 95% confidence interval, where r is the molar ratio of lithium ions to ethylene oxide monomer units. While κ, D, and $T_f$ are all positive as required by thermodynamic constraints, there is no constraint on the sign of $t_+^0$. We find that $t_+^0$ is negative in the salt concentration range of 0.093 ≤ r ≤ 0.189.

25 ENERGY STORAGE↗

Controlling Symmetry Breaking Charge Transfer in BODIPY Pairs

Symmetry breaking charge transfer (SBCT) is a process in which a pair of identical chromophores absorb a photon and use its energy to transfer an electron from one chromophore to the other, breaking the symmetry of the chromophore pair. This excited state phenomenon is observed in photosynthetic organisms where it enables efficient formation of separated charges that ultimately catalyze biosynthesis. SBCT has also been proposed as a means for developing photovoltaics and photocatalytic systems that operate with minimal energy loss. It is known that SBCT in both biological and artificial systems is in part made possible by the local environment in which it occurs, which can move to stabilize the asymmetric SBCT state. However, how environmental degrees of freedom act in concert with steric and structural constraints placed on a chromophore pair to dictate its ability to generate long-lived charge pairs via SBCT remain open topics of investigation. In this work we compare a broad series of dipyrrin dimers that are linked by distinct bridging groups to discern how the spatial separation and mutual orientation of linked chromophores and the structural flexibility of their linker each impact SBCT efficiency. Across this material set, we observe a general trend that SBCT is accelerated as the spatial separation between dimer chromophores decreases, consistent with the expectation that the electronic coupling between these units varies exponentially with their separation. However, one key observation is that the rate of charge recombination following SBCT was found to slow with decreasing interchromophore separation, rather than speed up. This stems from an enhancement of the dimers’ structural rigidity due to increasing steric repulsion as the length of their linker shrinks. This rigidity further inhibits charge recombination in systems where symmetry has already enforced zero HOMO-LUMO overlap. Additionally, for the forward transfer the active torsion is shown to increase LUMO-LUMO coupling, allowing for faster SBCT within bridging groups. Further, by understanding trends for how rates of SBCT and charge recombination depend on a dimer’s internal structure and their environment, we identify design guidelines for creating artificial systems for driving sustained light-induced charge separation. Such systems can find application in solar energy technologies and photocatalytic applications but can serve as a model for light-induced charge separation in biological systems.

14 SOLAR ENERGY↗

Divergence of Velocity Fields in Electrochemical Systems

The passage of current through electrochemical systems results in the development of concentration gradients in the electrolytic phase that can be modeled using concentrated solution theory. Application of this theory requires knowledge of three concentration-dependent transport coefficients, which are often taken to be conductivity, diffusion coefficient, and the cation transference number with respect to the solvent velocity. The governing diffusion equation for molar concentration contains two additional terms - the thermodynamic factor which is related to activity of the electrolytic species and the solvent velocity. The main advance in this paper is the derivation of an expression for the divergence of the solvent velocity. Solving this equation requires knowledge of the partial molar volume of the electrolyte. Analogous expressions are derived for the mass average and molar average velocities. These velocities occur naturally in the diffusion equation if concentration is expressed as weight fraction and mole fraction of the electrolytic species, respectively.

diffusion↗