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Enthalpy of mixing

The enthalpy of mixing, denoted as \Delta H^\text{mix}, is the change in enthalpy that occurs when two or more pure substances are combined at constant temperature and pressure to form a homogeneous mixture or solution, reflecting the energetic interactions between the components.^[1] This quantity is formally defined as \Delta H^\text{mix} = H - \sum_i n_i \bar{H}_i, where H is the enthalpy of the mixture, n_i is the number of moles of component i, and \bar{H}_i is the molar enthalpy of pure component i.^[2] In ideal mixtures, such as those of perfect gases or solutions with negligible intermolecular interactions, \Delta H^\text{mix} = 0, meaning no heat is absorbed or released during mixing.^[1] However, real mixtures often exhibit non-zero values: positive \Delta H^\text{mix} indicates endothermic mixing (e.g., due to repulsive interactions), while negative values signify exothermic mixing (e.g., from favorable attractions).^[3] The magnitude and sign of \Delta H^\text{mix} depend on the nature of the components and can be modeled using approaches like regular solution theory, where \Delta H^\text{mix} = V \phi_1 \phi_2 (\delta_1 - \delta_2)^2 for binary liquid mixtures, with V as the total volume, \phi_i as volume fractions, and \delta_i as solubility parameters capturing cohesive energies.^[4] In solid solutions, it arises from differences in pairwise interaction energies (A-A, B-B, A-B bonds) and is expressed as \Delta H^\text{mix} = 0.5 N z x_A x_B W, where N is the number of sites, z the coordination number, x_i the mole fractions, and W the interaction parameter.^[3] These models highlight how \Delta H^\text{mix} deviates from ideality due to molecular or atomic interactions, influencing properties like miscibility and phase stability.^[4] Enthalpy of mixing plays a crucial role in thermodynamics by contributing to the Gibbs free energy of mixing, \Delta G^\text{mix} = \Delta H^\text{mix} - T \Delta S^\text{mix}, which determines the spontaneity and equilibrium of mixing processes.^[1] Positive \Delta H^\text{mix} can oppose mixing unless compensated by entropic gains, leading to phase separation, while negative values promote solution formation.^[3] This concept is essential across disciplines: in chemical engineering for process design and heat balance calculations; in materials science for predicting alloy stability and polymer blend compatibility; and in geochemistry for modeling mineral solid solutions.^[4] Experimental measurement often involves calorimetry, providing data for thermodynamic databases used in simulations.^[5]

Definition and Fundamentals

Formal Definition

The enthalpy of mixing, denoted as \Delta [H](/page/H+)_{\text{mix}}, is a thermodynamic property defined as the change in enthalpy that occurs when pure components are combined to form a mixture at constant temperature and pressure, without any chemical reaction. This quantity represents the difference between the enthalpy of the resulting mixture and the sum of the enthalpies of the pure components in their initial states.^[6] The general expression for the total enthalpy of mixing in a multicomponent system is given by

\Delta H_{\text{mix}} = H_{\text{mixture}} - \sum_i n_i H_{i,\text{pure}},

where H_{\text{mixture}} is the enthalpy of the mixture, n_i is the number of moles of component i, and H_{i,\text{pure}} is the molar enthalpy of pure component i under the same conditions.^[6] It is commonly expressed on a per-mole basis as the molar enthalpy of mixing,

\Delta h_{\text{mix}} = \frac{\Delta H_{\text{mix}}}{n_{\text{total}}},

where n_{\text{total}} is the total number of moles in the mixture.^[6] The units for \Delta h_{\text{mix}} are typically joules per mole (J/mol) or kilojoules per mole (kJ/mol), and the process is evaluated under isobaric (constant pressure) and isothermal (constant temperature) conditions to isolate the mixing contribution.^[6] The enthalpy of mixing is equivalent to the heat of mixing measured directly via calorimetry, where heat transfer during mixing reveals whether the process is endothermic or exothermic.^[6] The concept developed within 19th- and early 20th-century solution thermodynamics.

Thermodynamic Relations

The enthalpy of mixing, \Delta H_\text{mix}, forms a fundamental component of the Gibbs free energy of mixing through the relation \Delta G_\text{mix} = \Delta H_\text{mix} - T \Delta S_\text{mix}, where T is the absolute temperature and \Delta S_\text{mix} is the entropy of mixing. This equation underscores the enthalpic contribution to the driving force for mixing, balancing energetic interactions against entropic effects to determine solution stability. In terms of partial molar quantities, the contribution for component i is \bar{g}_i^\text{mix} = \bar{h}_i^\text{mix} - T \bar{s}_i^\text{mix}, where the partial molar enthalpy \bar{h}_i^\text{mix} directly influences chemical potentials and phase behavior. These relations are central to phase equilibria, as \Delta G_\text{mix} governs the shape of the free energy surface, enabling predictions of miscibility gaps and coexistence curves via minimization principles.^[7] In non-ideal solution thermodynamics, the excess molar enthalpy h^E quantifies deviations from ideality and is defined as h^E = h - \sum x_i h_i, where h is the molar enthalpy of the mixture, x_i are the mole fractions, and h_i are the molar enthalpies of the pure components. For liquid mixtures, where the ideal enthalpy of mixing vanishes, h^E = \Delta H_\text{mix}, capturing the net energetic effects of unlike interactions. This excess property integrates with the enthalpies of solution and dilution: the enthalpy of solution measures the heat released or absorbed upon dissolving a solute to a specific concentration, while the enthalpy of dilution accounts for changes upon further solvent addition. Mathematically, \Delta H_\text{mix} for a binary mixture can be obtained as the difference between the integral enthalpy of solution at infinite dilution and the enthalpy of dilution to the desired composition, providing a pathway to compute mixing properties from solution calorimetric data.^[8]^[9] The partial molar excess enthalpy \bar{h}_i^E is derived from the total excess enthalpy H^E = n h^E as \bar{h}_i^E = \left( \frac{\partial (n h^E)}{\partial n_i} \right)_{T,P,n_j}, where n is the total moles and n_j are moles of other components. This partial quantity links to activity coefficients through the excess Gibbs energy, g^E = h^E - T s^E, yielding \bar{g}_i^E = RT \ln \gamma_i, with \bar{h}_i^E contributing via the temperature derivative \left( \frac{\partial \ln \gamma_i}{\partial T} \right)_P = -\frac{\bar{h}_i^E}{RT^2}. The Gibbs-Duhem equation, \sum x_i d\bar{\mu}_i = 0 at constant T and P, ensures consistency among these partial properties, constraining activity coefficients derived from enthalpic data and facilitating accurate modeling of non-ideal behaviors in phase equilibria.^[10] The temperature dependence of \Delta H_\text{mix} is given by \Delta C_{p,\text{mix}} = \left( \frac{\partial \Delta H_\text{mix}}{\partial T} \right)_P. This relation stems from Maxwell identities applied to the fundamental thermodynamic potentials and enables extraction of \Delta C_{p,\text{mix}} from \Delta H_\text{mix} measurements across temperatures, informing the thermal evolution of mixing energetics in equilibria. In ideal mixtures, \Delta H_\text{mix} = 0 simplifies \Delta G_\text{mix} to entropic dominance. Pressure effects on \Delta H_\text{mix} arise indirectly through the volume of mixing via \left( \frac{\partial \Delta H_\text{mix}}{\partial P} \right)_T = \Delta V_\text{mix} - T \left( \frac{\partial \Delta V_\text{mix}}{\partial T} \right)_P, but are typically secondary for condensed phases.^[11]

Theoretical Models for Mixtures

Ideal Mixtures

In ideal mixtures, the enthalpy of mixing, \Delta H_{\text{mix}}, is zero for all compositions, indicating that no net heat is absorbed or released when the pure components are mixed isothermally and isobarically. This condition arises because the process of mixing does not alter the internal energy of the system beyond what would be expected from the separate components.^[12]^[13] The concept of ideal mixtures rests on key assumptions about molecular behavior. Mixing occurs randomly, with molecules distributed without preference, and intermolecular interactions remain unchanged: the average energy of unlike-pair interactions (A-B) equals that of like-pair interactions (A-A and B-B) in the pure components. As a consequence, ideal mixtures obey Raoult's law, where the partial vapor pressure of each component is directly proportional to its mole fraction in the solution. These assumptions simplify the modeling of solution properties, treating the mixture as a statistical ensemble akin to an ideal gas.^[12]^[8]^[14] Thermodynamically, the zero enthalpy of mixing implies that the Gibbs free energy of mixing, \Delta G_{\text{mix}}, is determined solely by the entropic contribution: \Delta G_{\text{mix}} = -T \Delta S_{\text{mix}}. For an ideal mixture, this takes the form

\Delta G_{\text{mix}} = RT \sum_i x_i \ln x_i,

where R is the gas constant, T is the temperature, and x_i are the mole fractions of the components; the negative value of \Delta G_{\text{mix}} for $0 < x_i < 1 drives spontaneous mixing through increased configurational entropy.^[13] Representative examples include mixtures of structurally similar non-polar molecules, such as benzene and toluene, which exhibit nearly ideal behavior due to comparable intermolecular forces. Dilute ideal gases also approximate this model, as interactions are negligible at low densities. However, ideal mixtures apply only when components have similar molecular sizes and weak, comparable interactions; significant deviations arise at high concentrations or with dissimilar species, where interaction energies differ.^[15]^[16]^[7]^[17]

Regular and Non-Ideal Mixtures

Regular solution theory provides a foundational model for describing the enthalpy of mixing in binary liquid mixtures where the entropy of mixing follows ideal behavior, but the enthalpy deviates due to intermolecular interactions. In this framework, the molar enthalpy of mixing is given by

\Delta H_\text{mix} = x_1 x_2 \Omega,

where x_1 and x_2 are the mole fractions of the components, and \Omega is the interaction parameter that quantifies the energetic cost of unlike-pair interactions relative to like-pair interactions. A positive \Omega corresponds to endothermic mixing, where energy is absorbed to form the mixture, while a negative \Omega indicates exothermic mixing, where energy is released.^[18] This expression arises from a lattice model of the solution, in which molecules occupy sites on a regular lattice with coordination number z, the average number of nearest neighbors. The interaction parameter \Omega is derived as \Omega = z w_{12} - \frac{z w_{11} + z w_{22}}{2}, where w_{ij} represents the pairwise interaction energy between molecules of types i and j. This formulation assumes a random distribution of molecules on the lattice (mean-field approximation), leading to a quadratic dependence of \Delta H_\text{mix} on composition without volume changes or specific ordering effects.^[19] Many real mixtures exhibit more complex non-ideal behavior where \Delta H_\text{mix} does not conform to the simple symmetric quadratic form of regular solutions, often due to asymmetric interactions or higher-order terms. Extensions such as the Margules model introduce additional parameters to capture asymmetries in excess properties, expressing the excess Gibbs energy (and thus enthalpy via thermodynamic relations) as a power series in mole fractions. Similarly, the van Laar model accounts for strong deviations in activity coefficients, suitable for systems with significant positive deviations from ideality. The sign of \Delta H_\text{mix} reflects the balance of intermolecular forces: positive values occur in systems like oil and water, where weak dispersion forces between hydrocarbons cannot overcome strong hydrogen bonding in water, leading to immiscibility and endothermic mixing. In contrast, negative \Delta H_\text{mix} is observed in exothermic processes such as the hydration of concentrated sulfuric acid with water, where strong ion-dipole interactions release significant heat.^[6] For polymeric systems, the Flory-Huggins theory extends regular solution concepts to account for chain connectivity and size asymmetry in lattice models. The enthalpic contribution is incorporated via the dimensionless Flory-Huggins interaction parameter \chi, defined as \chi = \frac{z \Delta \epsilon}{kT}, where \Delta \epsilon = \epsilon_{12} - \frac{\epsilon_{11} + \epsilon_{22}}{2} is the effective interaction energy difference, k is Boltzmann's constant, and T is temperature; positive \chi > 0.5 typically signals phase separation due to unfavorable enthalpic mixing.^[20]

Calculations and Applications

Binary Mixture Calculations

Empirical fitting methods for the enthalpy of mixing in binary mixtures typically rely on calorimetric measurements to parameterize simple polynomial expressions that capture the composition dependence of ΔH_mix. These approaches use experimental data from isothermal calorimetry to determine adjustable parameters in models such as the two-parameter form:

\Delta H_\text{mix} = x_1 x_2 (A + B x_1)

where x_1 and x_2 = 1 - x_1 are the mole fractions of components 1 and 2, and A and B are temperature-dependent constants fitted to the data. This model, a variant of the Margules expansion, effectively describes asymmetric non-ideal behavior in many liquid binaries by accounting for deviations from ideality through the linear term in x_1. For instance, calorimetric studies on systems like Ga-Li alloys have employed similar fits to quantify exothermic mixing enthalpies, yielding parameters that reproduce measured values within experimental precision.^[21] Predictive methods for binary enthalpy of mixing often employ group contribution approaches, which estimate interaction parameters based on molecular structure without requiring mixture-specific experimental data. The UNIFAC model, for example, divides molecules into functional groups and uses predefined group-group interaction parameters to compute the excess enthalpy contribution, enabling predictions for a wide range of organic binaries. Developed initially for vapor-liquid equilibria, extensions of UNIFAC to enthalpies incorporate enthalpy-specific parameters, allowing estimation of ΔH_mix from combinatorial and residual terms in the excess Gibbs energy. This method has been validated for alkane-alcohol systems, where it correlates mixing enthalpies with average deviations of 10-20% from calorimetry.^[22]^[23] Computational tools like molecular dynamics (MD) simulations provide an alternative for calculating ΔH_mix in binary systems by directly evaluating changes in potential energy upon mixing. In MD, the enthalpy of mixing is obtained from the difference in average potential energies of pure components and the equilibrated mixture, simulated under constant pressure and temperature using force fields such as OPLS or ab initio methods. This approach is particularly useful for systems where experimental data is scarce, such as polymer blends or ionic liquids, as it captures atomic-level interactions leading to exothermic or endothermic behavior. For example, MD simulations of pseudo-binary deep eutectic solvents with water have reproduced strongly exothermic mixing profiles, attributing them to hydrogen bonding networks.^[24]^[25] A representative example is the ethanol-water binary mixture at 25°C and equimolar composition (x_ethanol = 0.5), where ΔH_mix ≈ -400 J/mol, indicating exothermic mixing due to favorable hydrogen bonding between unlike molecules. This value, derived from high-precision calorimetry, highlights the non-ideal nature of the system and serves as a benchmark for model validation.^[26]^[27] Predictions of ΔH_mix for non-ideal binary mixtures, whether from empirical fits, group contribution methods, or simulations, typically carry uncertainties of 5-20%, with higher errors in hydrogen-bonded systems due to complex intermolecular effects. These uncertainties arise from parameter sensitivity, force field approximations, or limited training data in predictive models, but they are often reduced through hybrid approaches combining calorimetry with simulations.^[28]

Multicomponent Mixture Extensions

The enthalpy of mixing for multicomponent systems extends the binary framework by incorporating pairwise interactions between all constituent components, assuming the dominant contributions arise from binary pairs while higher-order effects can be added as needed. In the pairwise approximation, the excess molar enthalpy of mixing is given by

\Delta H^\text{E}_\text{mix} = \sum_{i < j} x_i x_j B_{ij},

where x_i and x_j are the mole fractions of components i and j, and B_{ij} represents the temperature-dependent binary interaction parameter characterizing the energetic interaction between the pair (with B_{ii} = 0). This formulation builds on binary calculations as foundational elements, scaling them to higher dimensions while maintaining thermodynamic consistency through the Gibbs-Duhem relation. Such models are particularly effective for systems where ternary or higher interactions are minor, allowing predictions across composition space with a manageable number of parameters derived from binary data.^[29] For ternary mixtures, direct summation of subsystem binary enthalpies often fails to capture observed asymmetries, as \Delta H^\text{ternary}_\text{mix} \neq \sum \Delta H^\text{binary}_\text{mix} due to altered local compositions and non-additive effects in the mixed state. To address this, models introduce ternary correction terms, such as x_1 x_2 x_3 C_{123}, where C_{123} quantifies three-body interactions beyond pairwise contributions; this term adjusts for deviations in systems exhibiting significant non-ideality, like those with directional bonding or volume changes upon mixing. In alloy systems such as Cu-Ni-Zn, experimental calorimetry reveals negative enthalpies dominated by binary Cu-Zn and Ni-Zn attractions, but ternary parameters are essential to fit phase boundaries and mixing data above 600°C, as optimized via Redlich-Kister-Muggianu polynomials in thermodynamic assessments. These extensions highlight how multicomponent mixing introduces compositional asymmetries not predictable from binaries alone, necessitating fitted higher-order coefficients for accuracy.^[30]^[29] A key challenge in multicomponent extensions lies in the proliferation of parameters—for an n-component system, the number of unique B_{ij} pairs grows as n(n-1)/2, compounded by ternary (C_{ijk}) and higher terms, which demands extensive experimental data or computational optimization to avoid overfitting. Thermodynamic databases developed via the CALPHAD (CALculation of PHAse Diagrams) method mitigate this by systematically integrating assessed binary and limited ternary data into self-consistent models, enabling reliable predictions for complex alloys and solutions without exhaustive measurements. In practice, these models are implemented in process simulation software like Aspen Plus, where multicomponent enthalpy of mixing informs multi-phase equilibrium calculations for distillation, extraction, and reaction processes, ensuring accurate energy balances and phase splits in industrial designs.^[31]^[32]

Physical and Experimental Basis

Intermolecular Forces

The enthalpy of mixing, ΔH_mix, arises primarily from changes in intermolecular interactions when unlike molecules are brought together in a mixture, overriding the enthalpic contributions from volume changes which are often negligible for liquids. Dispersion forces, which are universal van der Waals attractions arising from temporary dipoles in all molecules, contribute to ΔH_mix by favoring mixing when the dispersion interactions between unlike pairs are comparable to those between like pairs, as seen in nonpolar hydrocarbon blends. Dipole-dipole forces, present in polar molecules, can lead to either positive or negative ΔH_mix depending on whether the unlike-pair dipole alignments are weaker or stronger than like-pair ones, while hydrogen bonding, a particularly strong directional interaction in molecules like water or alcohols, often dominates in aqueous systems and promotes exothermic mixing when new hydrogen bonds form between solute and solvent.^[33] In endothermic mixing, where ΔH_mix > 0, the interactions between unlike molecules are weaker than the average of the like-like interactions, requiring energy input to disrupt the stronger cohesive forces in the pure components. A classic example is the hexane-water system, where nonpolar hexane molecules interact weakly via dispersion forces with polar water molecules dominated by hydrogen bonding and dipole-dipole forces, resulting in immiscibility and a positive ΔH_mix that favors phase separation.^[34]^[33] Conversely, exothermic mixing occurs when ΔH_mix < 0 due to stronger attractions between unlike molecules than in the pure states, releasing energy as new bonds form. This is evident in acid-base pairs like sulfuric acid and water, where the hydration of protons and formation of hydronium ions through hydrogen bonding and ion-dipole interactions yield highly exothermic mixing, with ΔH_mix values as low as -74 kJ/mol for concentrated solutions.^[6] A quantitative connection between these forces and ΔH_mix emerges from mean-field approximations in lattice models, where the mixing enthalpy is approximated as the difference in pair interaction energies:

\Delta H_{\text{mix}} \approx \frac{1}{2} \sum n_{ij} (u_{ij} - u_{ii} - u_{jj})

Here, n_{ij} represents the number of unlike i-j pairs formed, and u_{ij}, u_{ii}, u_{jj} are the pairwise interaction energies (negative for attractive forces) for unlike and like pairs, respectively; positive values of (u_{ij} - u_{ii} - u_{jj}) indicate endothermic mixing.^[35] Molecular size and shape further modulate ΔH_mix by affecting packing efficiency in the mixture, where mismatched sizes or shapes can create voids that reduce favorable interactions and contribute to positive ΔH_mix, as larger disparities lead to less efficient space filling and weaker overall attractions compared to the pure components.^[36]

Experimental Determination

The experimental determination of the enthalpy of mixing, ΔH_mix, primarily relies on direct calorimetric measurements that quantify the heat absorbed or released upon mixing components under controlled conditions. Isothermal titration calorimetry (ITC) is widely used to measure ΔH_mix at specific compositions by injecting one component into another in a thermally isolated cell, recording the heat flow to maintain constant temperature.^[37] This technique allows precise determination of partial molar enthalpies of mixing through successive titrations, with sensitivities down to microjoules, making it suitable for liquid mixtures at ambient pressures.^[38] For obtaining complete ΔH_mix curves across compositions, adiabatic calorimetry employs a vessel where the mixture is isolated from heat exchange with surroundings, enabling measurement of temperature changes to calculate total heat effects.^[39] This method integrates the heat capacity over the temperature rise post-mixing, providing excess enthalpies for binary and multicomponent systems, often at temperatures from 293 K to 363 K.^[40] Indirect methods derive ΔH_mix from vapor-liquid equilibrium (VLE) data by first obtaining excess Gibbs free energy (G^E) from activity coefficients, then estimating excess entropy (S^E) via heat capacity or configurational assumptions, and applying the relation H^E = G^E + T S^E.^[41] This approach is validated against direct measurements for consistency in thermodynamic models, particularly for systems where calorimetry is challenging.^[42] For supercritical mixtures under high pressure, flow calorimeters measure ΔH_mix by continuously mixing streams in a high-pressure cell, capturing heat effects across critical loci up to several hundred MPa.^[43] These setups, often using power-compensation designs, handle fluids like CO2 with organic solvents, yielding excess molar enthalpies with uncertainties below 2%.^[44] Compiled experimental data on ΔH_mix are accessible through databases such as the Dortmund Data Bank (DDB), which stores excess enthalpy records for over 30,000 binary mixtures, and the NIST Chemistry WebBook, providing thermophysical properties including mixing enthalpies for select fluids.^[45]^[46] Accurate measurements require stringent best practices, including temperature control to ±0.01 K to minimize baseline drifts and sample purity exceeding 99.5% to avoid impurity contributions to heat effects. Uncertainties typically range from 1-5% in modern setups, influenced by heat leaks and mixing efficiency, but can be reduced through calibration with known standards like water-ethanol mixtures.^[47] Since the 2000s, advancements in microcalorimeters, such as high-resolution flow microcalorimeters, have enabled measurements with sample volumes under 200 μL and resolutions better than 1 μW, enhancing precision for small-scale or reactive systems.^[48]^[49]

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Mar 1, 1996 · Simultaneous description of vapor-liquid equilibrium and excess enthalpies for methanol and ethanol binary mixtures with propanal. Full Record ...
[45]
[PDF] or THIS twmon is mwtrc - OSTI.GOV
Using a flow calorimetric procedure, heats of mixing (H^) were determined in temperature and pressure regions spanning critical loci for a large number of ...
[46]
Excess molar enthalpies for mixtures of supercritical CO2 and linalool
Aug 6, 2025 · An isothermal high-pressure flow calorimeter has been used to measure excess molar enthalpies (HmE) for mixtures of supercritical CO2 and N ...
[47]
HE – Excess Enthalpies in Non-Electrolyte Mixtures - DDBST
The h E data bank contains excess enthalpy (heats of mixing) data for a large number of binary and multicomponent mixtures.
[48]
Thermophysical Properties of Fluid Systems - the NIST WebBook
Accurate thermophysical properties are available for several fluids. These data include the following: Density; Cp; Enthalpy; Internal energy; Viscosity; Joule- ...
[49]
Enthalpy of mixing of liquid Co–Sn alloys - PMC - NIH
The integral molar enthalpy of mixing, ΔMixH, was calculated by summing the respective reaction enthalpies and division by the total molar amount of ...Missing: x_j B_ij
[50]
μFlowCal – High‐Resolution Differential Flow Microcalorimeter for ...
Mar 18, 2022 · The μFlowCal is a high precision microcalorimeter developed to measure accurately heats of mixing using small samples sizes (200 μL) in a wide variety of ...
[51]
High Precision Microcalorimetry: Apparatus, Procedures, and ... - NIH
Microcalorimetry is an important technique for the measurement of heats of reaction in solution. Since small amounts of solution (typically less than one mL) ...