Thermodynamic Modeling with Equations of State: Present Challenges with Established Methods

Estimating metastable thermodynamic properties by isochoric extrapolation from stable states

The description of metastable fluids, those in local but not global equilibrium, remains an important problem of thermodynamics, and it is crucial for many industrial applications and all first order phase transitions. One way to estimate their properties is by extrapolation from nearby stable states. This is often done isothermally, in terms of a virial expansion for gases or a Taylor expansion in density for liquids. This work presents evidence that an isochoric expansion of pressure at a given temperature is superior to an isothermal density expansion. Two different isochoric extrapolation strategies are evaluated, one best suited for vapors and one for liquids. Both are exact for important model systems, including the van der Waals equation of state. Moreover, we present a simple method to evaluate all the coefficients of the isochoric expansion directly from a simulation in the canonical ensemble. Using only the properties of stable states, the isochoric extrapolation methods reproduce simulation results with Lennard-Jones potentials, mostly within their uncertainties. The isochoric extrapolation methods are able to predict deeply metastable pressures accurately even from temperatures well above the critical. Isochoric extrapolation also predicts a mechanical stability limit, i.e., the thermodynamic spinodal. For water, the liquid spinodal pressure is predicted to be monotonically decreasing with decreasing temperature, in contrast to the re-entrant behavior predicted by the direct extension of the reference equation of state.

Cryogenic Process Optimization for Natural Gas Purification: Predictive Modeling of Methane-CO2 Solid-Vapor Phase Equilibrium Using Response Surface Methodology

This research combines industrial engineering principles with chemical process modeling to explore the capture of CO2 from natural gas under cryogenic conditions. The study specifically investigates the Solid-Vapor (S-V) phase equilibrium in a methane-carbon dioxide (CH4-CO2) system. The study employs Response Surface Methodology (RSM) to develop a robust model for predicting phase behavior in industrial gas separation processes. The model is validated using experimental data, offering enhanced operational insights into cryogenic CO2 capture in industrial applications. The developed RSM model is particularly valuable as it can predict the mole fractions of methane and CO2 at various temperatures and pressures in the solid-vapor region of phase equilibrium, where limited experimental data make it difficult to estimate these components accurately. The key contribution of this study is to validate the RSM model's available experimental data, and the model can further be used to predict the process conditions at which high methane composition (yCH4) can be achieved. The developed model showed good agreement when the results were compared with previous experimental studies. The utilization of chemical engineering data to forecast previously unknown conditions in gas separation processes broadens the scope of industrial process optimization in this work.

Revised Enskog theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities, and thermal conductivities

Since the 1920s, the Enskog solutions to the Boltzmann equation have provided a route to predicting the transport properties of dilute gas mixtures. At higher densities, predictions have been limited to gases of hard spheres. In this work, we present a revised Enskog theory for multicomponent mixtures of Mie fluids, where the Barker-Henderson perturbation theory is used to calculate the radial distribution function at contact. With parameters of the Mie-potentials regressed to equilibrium properties, the theory is fully predictive for transport properties. The presented framework offers a link between the Mie potential and transport properties at elevated densities, giving accurate predictions for real fluids. For mixtures of noble gases, diffusion coefficients from experiments are reproduced within ±4%. For hydrogen, the predicted self-diffusion coefficient is within 10% of experimental data up to 200 MPa and at temperatures above 171 K. Binary diffusion coefficients of the CO2/CH4 mixture from simulations are reproduced within 20% at pressures up to 14.7 MPa. Except for xenon in the vicinity of the critical point, the thermal conductivity of noble gases and their mixtures is reproduced within 10% of the experimental data. For other molecules than noble gases, the temperature dependence of the thermal conductivity is under-predicted, while the density dependence appears to be correctly predicted. Predictions of the viscosity are within ±10% of the experimental data for methane, nitrogen, and argon up to 300 bar, for temperatures ranging from 233 to 523 K. At pressures up to 500 bar and temperatures from 200 to 800 K, the predictions are within ±15% of the most accurate correlation for the viscosity of air. Comparing the theory to an extensive set of measurements of thermal diffusion ratios, we find that 49% of the model predictions are within ±20% of the reported measurements. The predicted thermal diffusion factor differs by less than 15% from the simulation results of Lennard-Jones mixtures, even at densities well exceeding the critical density.

Development of thermodynamically consistent machine-learning equations of state: Application to the Mie fluid

A procedure for deriving thermodynamically consistent data-driven equations of state (EoS) for fluids is presented. The method is based on fitting the Helmholtz free energy using artificial neural networks to obtain a closed-form relationship between the thermophysical properties of fluids (FE-ANN EoS). As a proof-of-concept, an FE-ANN EoS is developed for the Mie fluids, starting from a database obtained by classical molecular dynamics simulations. The FE-ANN EoS is trained using first- (pressure and internal energy) and second-order (e.g., heat capacities, Joule-Thomson coefficients) derivative data. Additional constraints ensure that the data-driven model fulfills thermodynamically consistent limits and behavior. The results for the FE-ANN EoS are shown to be as accurate as the best available analytical model while being developed in a fraction of the time. The robustness of the "digital" equation of state is exemplified by computing physical behavior it has not been trained on, for example, fluid phase equilibria. Furthermore, the model's internal consistency is successfully assessed using Brown's characteristic curves.

Classical density functional theory for interfacial properties of hydrogen, helium, deuterium, neon, and their mixtures

We present a classical density functional theory (DFT) for fluid mixtures that is based on a third-order thermodynamic perturbation theory of Feynman-Hibbs-corrected Mie potentials. The DFT is developed to study the interfacial properties of hydrogen, helium, neon, deuterium, and their mixtures, i.e., fluids that are strongly influenced by quantum effects at low temperatures. White Bear fundamental measure theory is used for the hard-sphere contribution of the Helmholtz energy functional, and a weighted density approximation is used for the dispersion contribution. For mixtures, a contribution is included to account for non-additivity in the Lorentz-Berthelot combination rule. Predictions of the radial distribution function from DFT are in excellent agreement with results from molecular simulations, both for pure components and mixtures. Above the normal boiling point and 5% below the critical temperature, the DFT yields surface tensions of neon, hydrogen, and deuterium with average deviations from experiments of 7.5%, 4.4%, and 1.8%, respectively. The surface tensions of hydrogen/deuterium, para-hydrogen/helium, deuterium/helium, and hydrogen/neon mixtures are reproduced with a mean absolute error of 5.4%, 8.1%, 1.3%, and 7.5%, respectively. The surface tensions are predicted with an excellent accuracy at temperatures above 20 K. The poor accuracy below 20 K is due to the inability of Feynman-Hibbs-corrected Mie potentials to represent the real fluid behavior at these conditions, motivating the development of new intermolecular potentials. This DFT can be leveraged in the future to study confined fluids and assess the performance of porous materials for hydrogen storage and transport.

Local equilibrium in liquid phase shock waves

We have assessed the assumption of local thermodynamic equilibrium in a shock wave by comparing local thermodynamic data generated with nonequilibrium molecular dynamics (NEMD) simulations with results from corresponding equilibrium simulations. The shock had a Mach number approximately equal to 2 in a Lennard-Jones spline liquid. We found that the local equilibrium assumption holds perfectly well behind the wave front, and is a very good approximation in the front itself. This was supported by calculations of the excess entropy production in the shock front with four different methods that use the local equilibrium assumption in different ways. Two of the methods assume local equilibrium between excess thermodynamic variables by treating the shock as an interface in Gibbs's sense. The other two methods are based on the local equilibrium assumption in a continuous description of the shock front. We show for the shock studied in this work that all four methods give excess entropy productions that are in excellent agreement, with an average variance of 3.5% for the nonequilibrium molecular dynamics (NEMD) simulations. In addition, we solved the Navier-Stokes (N-S) equations numerically for the same shock wave using an equilibrium equation of state (EoS) based on a recently developed perturbation theory. The results for the density, pressure, and temperature profiles agree well with the profiles from the NEMD simulations. For instance, the shock waves generated in the two simulations travel with almost the same speed; the average absolute Mach-number deviation of the N-S simulations relative to NEMD is 2.6% in the investigated time interval.

Equation of state for the Mie (λr,6) fluid with a repulsive exponent from 11 to 13

An empirical multi-parameter equation of state in terms of the reduced Helmholtz energy is presented for the Mie (λ_r-6) fluid with a repulsive exponent λ_r from 11 to 13. The equation is fitted to an extensive dataset from molecular dynamics simulation as well as the second and third thermal virial coefficients. It is comprehensively compared with the SAFT-VR model and is a more accurate description of the considered fluid class. The equation is valid for reduced temperatures T/T_c from 0.55 to 4.5 and for reduced pressures of up to p/p_c = 265. A good extrapolation behavior and the occurrence of a single Maxwell loop down to the vicinity of the triple point temperature are realized.

New Method of NPOH Equation-Based to Estimate the Physicochemical Properties of Noncyclic Alkanes

Changes in various physicochemical properties (P _(n)) of noncyclic alkanes can be roughly classified as linear and nonlinear changes. In our previous study, the NPOH equation was proposed to express nonlinear changes in the properties of organic homologues. Until now, there has been no general equation to express nonlinear changes in the properties of noncyclic alkanes involving linear and branched alkane isomers. This work, on the basis of NPOH equation, proposes a general equation to express nonlinear changes in the physicochemical properties of noncyclic alkanes, including a total of 12 properties, boiling point, critical temperature, critical pressure, acentric factor, heat capacity, liquid viscosity, and flash point, named as the "NPNA equation", as follows: ln(P _(n)) = a + b(n - 1) + c(S _CNE) + d (ΔAOEI) + f(ΔAIMPI), where a, b, c, and f are coefficients, and P _(n) represents the property of the alkane with n carbon atom number. n, S _CNE, ΔAOEI, and ΔAIMPI are number of carbon atoms, sum of carbon number effects, average odd-even index difference, and average inner molecular polarizability index difference, respectively. The obtained results show that various nonlinear changes in the properties of noncyclic alkanes can be expressed by the NPNA equation. Nonlinear and linear change properties of noncyclic alkanes can be correlated with four parameters, n, S _CNE, ΔAOEI, and ΔAIMPI. The NPNA equation has the advantages of uniform expression, usage of fewer parameters, and high estimation accuracy. Furthermore, using the above four parameters, a quantitative correlation equation can be established between any two properties of noncyclic alkanes. Employing the obtained equations as model equations, the property data of noncyclic alkanes, involving 142 critical temperatures, 142 critical pressures, 115 acentric factors, 116 flash points, 174 heat capacities, 142 critical volumes, and 155 gas enthalpies of formation, a total of 986 values, were predicted, which have not be experimentally measured. NPNA equation not only can provide a simple and convenient estimation or prediction method for the properties of noncyclic alkanes but also can provide new perspectives for studying quantitative structure-property relationships of branched organic compounds.

A prediction model to predict the thermodynamic conditions of gas hydrates

Gas Hydrate modelling has gained huge attention in the past decade due to its increase in usage for various energy as well as environmental applications at an industrial scale. As the experimental approach is highly expensive and time-consuming, modelling is the best way to predict the conditions before the actual applications at industrial scales. The commercial software currently existing uses the equation of states (EOS) to predict the thermodynamic conditions of gas hydrates. But, in certain cases, the prediction by using EOS fails to predict the hydrate conditions accurately. Therefore, there arose a need for an accurate prediction model to estimate the hydrate formation conditions. So, in this work, an accurate prediction model has been proposed to predict the thermodynamic equilibrium conditions of the gas hydrate formation. The performance of prediction accuracy for the proposed model is compared with those of the SRK equation of state and Peng Robinson (PR) Equation of state. It was observed that in most of the cases the proposed model has predicted the thermodynamic conditions more accurately than the PR and SRK equation of state. This work helps in understanding the limitations of EOS for the prediction hydrate conditions. Also, the current work helps in strengthening the conventional statistical modelling technique to predict the hydrate conditions for a broader range.

Thermodynamic Equilibrium Study of Anaerobic Digestion through Helmholtz Equation of State

The growing attention regarding a more sustainable future, and thus into energy recovery and waste reduction technologies, has intensified the interest towards processes which allow to exploit waste and biomasses to generate energy, such as the anaerobic digestion. Improving the efficiency of this industrial application is crucial to increase methane production, and is essential from the economic, environmental and safety point of view. This study focuses on the thermodynamic modelling of a steady-state reactor as a flash unit, in order to determine the best operating conditions to produce the maximum amount of pure bio-methane. To this purpose, a new hybrid approach based on the Peng–Robinson cubic equation of state and on the Multi-Parameter Helmholtz-Energy EoS has been proposed. The simulations, performed using the developed algorithm at temperatures between 20 and 55 °C and at pressure values between 0.3 atm and 1.5 atm, point out that the fugacity of the mixture evaluated with the proposed technique is much more accurate and reliable than the one calculated with the PR EoS. In addition, this research has shown not only that the purity and the production of the biogas can be optimised by working at mesophilic conditions and at pressure between 1 atm and 1.5 atm, but also that it is not convenient to operate in a temperature range of 42 °C–45 °C, since about 20 % more H2S goes into the exiting biogas, reducing the CH4 amount and raising the post-treatment costs. Lastly, it has been seen that there is a significant water content in the vapour phase (∼5 %wt.), and this is a factor to be taken into account in order to improve the process.

Equation of state for confined fluids

Fluids confined in small volumes behave differently than fluids in bulk systems. For bulk systems, a compact summary of the system’s thermodynamic properties is provided by equations of state. However, there is currently a lack of successful methods to predict the thermodynamic properties of confined fluids by use of equations of state, since their thermodynamic state depends on additional parameters introduced by the enclosing surface. In this work, we present a consistent thermodynamic framework that represents an equation of state for pure, confined fluids. The total system is decomposed into a bulk phase in equilibrium with a surface phase. The equation of state is based on an existing, accurate description of the bulk fluid and uses Gibbs’ framework for surface excess properties to consistently incorporate contributions from the surface. We apply the equation of state to a Lennard-Jones spline fluid confined by a spherical surface with a Weeks–Chandler–Andersen wall-potential. The pressure and internal energy predicted from the equation of state are in good agreement with the properties obtained directly from molecular dynamics simulations. We find that when the location of the dividing surface is chosen appropriately, the properties of highly curved surfaces can be predicted from those of a planar surface. The choice of the dividing surface affects the magnitude of the surface excess properties and its curvature dependence, but the properties of the total system remain unchanged. The framework can predict the properties of confined systems with a wide range of geometries, sizes, interparticle interactions, and wall–particle interactions, and it is independent of ensemble. A targeted area of use is the prediction of thermodynamic properties in porous media, for which a possible application of the framework is elaborated.

Perturbation theories for fluids with short-ranged attractive forces: A case study of the Lennard-Jones spline fluid

It is generally not straightforward to apply molecular-thermodynamic theories to fluids with short-ranged attractive forces between their constituent molecules (or particles). This especially applies to perturbation theories, which, for short-ranged attractive fluids, typically must be extended to high order or may not converge at all. Here, we show that a recent first-order perturbation theory, the uv-theory, holds promise for describing such fluids. As a case study, we apply the uv-theory to a fluid with pair interactions defined by the Lennard-Jones spline potential, which is a short-ranged version of the LJ potential that is known to provide a challenge for equation-of-state development. The results of the uv-theory are compared to those of third-order Barker–Henderson and fourth-order Weeks–Chandler–Andersen perturbation theories, which are implemented using Monte Carlo simulation results for the respective perturbation terms. Theoretical predictions are compared to an extensive dataset of molecular simulation results from this (and previous) work, including vapor–liquid equilibria, first- and second-order derivative properties, the critical region, and metastable states. The uv-theory proves superior for all properties examined. An especially accurate description of metastable vapor and liquid states is obtained, which might prove valuable for future applications of the equation-of-state model to inhomogeneous phases or nucleation processes. Although the uv-theory is analytic, it accurately describes molecular simulation results for both the critical point and the binodal up to at least 99% of the critical temperature. This suggests that the difficulties typically encountered in describing the vapor–liquid critical region are only to a small extent caused by non-analyticity.

Liquid Hydrogen Spills on Water—Risk and Consequences of Rapid Phase Transition

Liquid hydrogen (LH2) spills share many of the characteristics of liquefied natural gas (LNG) spills. LNG spills on water sometimes result in localized vapor explosions known as rapid phase transitions (RPTs), and are a concern in the LNG industry. LH2 RPT is not well understood, and its relevance to hydrogen safety is to be determined. Based on established theory from LNG research, we present a theoretical assessment of an accidental spill of a cryogen on water, including models for pool spreading, RPT triggering, and consequence quantification. The triggering model is built upon film-boiling theory, and predicts that the mechanism for RPT is a collapse of the gas film separating the two liquids (cryogen and water). The consequence model is based on thermodynamical analysis of the physical processes following a film-boiling collapse, and is able to predict peak pressure and energy yield. The models are applied both to LNG and LH2, and the results reveal that (i) an LNG pool will be larger than an LH2 pool given similar sized constant rate spills, (ii) triggering of an LH2 RPT event as a consequence of a spill on water is very unlikely or even impossible, and (iii) the consequences of a hypothetical LH2 RPT are small compared to LNG RPT. Hence, we conclude that LH2 RPT seems to be an issue of only minor concern.

Industrial Requirements for Thermodynamic and Transport Properties: 2020

This paper reports the results of an investigation of industrial requirements for thermodynamic and transport properties carried out during the years 2019-2020. It is a follow-up of a similar investigation performed and published 10 years ago by the Working Party (WP) of Thermodynamics and Transport Properties of European Federation of Chemical Engineering (EFCE).1 The main goal was to investigate the advances in this area over the past 10 years, to identify the limitations that still exist, and to propose future R&D directions that will address the industrial needs. An updated questionnaire, with two new categories, namely, digitalization and comparison to previous survey/changes over the past 10 years, was sent to a broad number of experts in companies with a diverse activity spectrum, in oil and gas, chemicals, pharmaceuticals/biotechnology, food, chemical/mechanical engineering, consultancy, and power generation, among others, and in software suppliers and contract research laboratories. Very comprehensive answers were received by 37 companies, mostly from Europe (operating globally), but answers were also provided by companies in the USA and Japan. The response rate was about 60%, compared to 47% in the year 2010. The paper is written in such a way that both the majority and minority points of view are presented, and although the discussion is focused on needs and challenges, the benefits of thermodynamics and success stories are also reported. The results of the survey are thematically structured and cover changes, challenges, and further needs for a number of areas of interest such as data, models, systems, properties, and computational aspects (molecular simulation, algorithms and standards, and digitalization). Education and collaboration are discussed and recommendations on the future research activities are also outlined. In addition, a few initiatives, books, and reviews published in the past decade are briefly discussed. It is a long paper and, to provide the reader with a more complete understanding of the survey, many (anonymous) quotations (indicated with "..." and italics) from the industrial colleagues who have participated in the survey are provided. To help disseminate the specific information of interest only to particular industrial sectors, the paper has been written in such a way that the individual sections can also be read independently of each other.

Modeling the Physicochemical Properties of Natural Deep Eutectic Solvents

Natural deep eutectic solvents (NADES) are mixtures of naturally derived compounds with a significantly decreased melting point owing to specific interactions among the constituents. NADES have benign properties (low volatility, flammability, toxicity, cost) and tailorable physicochemical properties (by altering the type and molar ratio of constituents); hence, they are often considered to be a green alternative to common organic solvents. Modeling the relation between their composition and properties is crucial though, both for understanding and predicting their behavior. Several efforts have been made to this end. This Review aims at structuring the present knowledge as an outline for future research. First, the key properties of NADES are reviewed and related to their structure on the basis of the available experimental data. Second, available modeling methods applicable to NADES are reviewed. At the molecular level, DFT and molecular dynamics allow density differences and vibrational spectra to be interpreted, and interaction energies to be computed. Additionally, properties at the level of the bulk medium can be explained and predicted by semi-empirical methods based on ab initio methods (COSMO-RS) and equation of state models (PC-SAFT). Finally, methods based on large datasets are discussed: models based on group-contribution methods and machine learning. A combination of bulk-medium and dataset modeling allows qualitative prediction and interpretation of phase equilibria properties on the one hand, and quantitative prediction of melting point, density, viscosity, surface tension, and refractive index on the other. Multiscale modeling, combining molecular and macroscale methods, is expected to strongly enhance the predictability of NADES properties and their interaction with solutes, and thus yield truly tailorable solvents to accommodate (bio)chemical reactions.

Thermodynamic Stability of Volatile Droplets and Thin Films Governed by Disjoining Pressure in Open and Closed Containers

Distributed thin films of water and their coexistence with droplets are investigated using a capillary description based on a thermodynamic fundamental relation for the film Helmholtz energy, derived from disjoining pressure isotherms and an accurate equation of state. Gas-film and film-solid interfacial tensions are derived, and the latter has a dependence on film thickness. The resulting energy functionals from the capillary description are discretized, and stationary states are identified. The thermodynamic stability of configurations with thin films in systems that are closed (canonical ensemble) or connected to a particle reservoir (grand canonical ensemble) is evaluated by considering the eigenvalues of the corresponding Hessian matrices. The conventional stability criterion from the literature states that thin flat films are stable when the derivative of the disjoining pressure with respect to the film thickness is negative. This criterion is found to apply only in open systems. A closer inspection of the eigenvectors of the negative eigenvalues shows that condensation/evaporation destabilizes the film in an open system. In closed systems, thin films can be stable even though the disjoining pressure derivative is positive, and their stability is governed by mechanical instabilities of a similar kind to those responsible for spinodal dewetting in nonvolatile systems. The films are stabilized when their thickness and disjoining pressure derivative are such that the minimum unstable wavelength is larger than the container diameter. Droplets in coexistence with thin films are found to be unstable for all considered examples in open systems. In closed systems, they are found to be stable under certain conditions. The unstable droplets in both open and closed systems are saddle points in their respective energy landscapes. In the closed system, they represent the activation barrier for the transition between a stable film and a stable droplet. In the open system, the unstable droplets represent the activation barrier for the transition from a film into a bulk liquid phase. Thin films are found to be the equilibrium configuration up to a certain value of the total density in a closed system. Beyond this value, there is a morphological phase transition to stable droplet configurations.

Cluster integrals and virial coefficients for realistic molecular models

We present a concise, general, and efficient procedure for calculating the cluster integrals that relate thermodynamic virial coefficients to molecular interactions. The approach encompasses nonpairwise intermolecular potentials generated from quantum chemistry or other sources; a simple extension permits efficient evaluation of temperature and other derivatives of the virial coefficients. We demonstrate with a polarizable model of water. We argue that cluster-integral methods are a potent yet underutilized instrument for the development and application of first-principles molecular models and methods.

Equation of state and force fields for Feynman-Hibbs-corrected Mie fluids. II. Application to mixtures of helium, neon, hydrogen, and deuterium

We extend the statistical associating fluid theory of quantum corrected Mie potentials (SAFT-VRQ Mie), previously developed for pure fluids [Aasen et al., J. Chem. Phys. 151, 064508 (2019)], to fluid mixtures. In this model, particles interact via Mie potentials with Feynman-Hibbs quantum corrections of first order (Mie-FH1) or second order (Mie-FH2). This is done using a third-order Barker-Henderson expansion of the Helmholtz energy from a non-additive hard-sphere reference system. We survey existing experimental measurements and ab initio calculations of thermodynamic properties of mixtures of neon, helium, deuterium, and hydrogen and use them to optimize the Mie-FH1 and Mie-FH2 force fields for binary interactions. Simulations employing the optimized force fields are shown to follow the experimental results closely over the entire phase envelopes. SAFT-VRQ Mie reproduces results from simulations employing these force fields, with the exception of near-critical states for mixtures containing helium. This breakdown is explained in terms of the extremely low dispersive energy of helium and the challenges inherent in current implementations of the Barker-Henderson expansion for mixtures. The interaction parameters of two cubic equations of state (Soave-Redlich-Kwong and Peng-Robinson) are also fitted to experiments and used as performance benchmarks. There are large gaps in the ranges and properties that have been experimentally measured for these systems, making the force fields presented especially useful.

Tolman lengths and rigidity constants of multicomponent fluids: Fundamental theory and numerical examples

The curvature dependence of the surface tension can be described by the Tolman length (first-order correction) and the rigidity constants (second-order corrections) through the Helfrich expansion. We present and explain the general theory for this dependence for multicomponent fluids and calculate the Tolman length and rigidity constants for a hexane-heptane mixture by use of square gradient theory. We show that the Tolman length of multicomponent fluids is independent of the choice of dividing surface and present simple formulae that capture the change in the rigidity constants for different choices of dividing surface. For multicomponent fluids, the Tolman length, the rigidity constants, and the accuracy of the Helfrich expansion depend on the choice of path in composition and pressure space along which droplets and bubbles are considered. For the hexane-heptane mixture, we find that the most accurate choice of path is the direction of constant liquid-phase composition. For this path, the Tolman length and rigidity constants are nearly linear in the mole fraction of the liquid phase, and the Helfrich expansion represents the surface tension of hexane-heptane droplets and bubbles within 0.1% down to radii of 3 nm. The presented framework is applicable to a wide range of fluid mixtures and can be used to accurately represent the surface tension of nanoscopic bubbles and droplets.