Inthermodynamics, theenthalpy of mixing (alsoheat of mixing andexcess enthalpy) is theenthalpy liberated or absorbed from asubstance uponmixing.^[1] When a substance orcompound is combined with any other substance or compound, the enthalpy of mixing is the consequence of the new interactions between the two substances or compounds.^[1] This enthalpy, if releasedexothermically, can in an extreme case cause an explosion.

Enthalpy of mixing can often be ignored in calculations for mixtures where otherheat terms exist, or in cases where themixture is ideal.^[2] Thesign convention is the same as forenthalpy of reaction: when the enthalpy of mixing is positive, mixing isendothermic, while negative enthalpy of mixing signifiesexothermic mixing. In ideal mixtures, the enthalpy of mixing is null. In non-ideal mixtures, thethermodynamic activity of each component is different from its concentration by multiplying with theactivity coefficient.

One approximation for calculating the heat of mixing isFlory–Huggins solution theory for polymer solutions.

For a liquid, enthalpy of mixing can be defined as follows^[2]

${\displaystyle H_{(mixture)}=\mathrm{\Delta }{H}_{mix}+\sum x_i{H}_i}$

Where:

• H_(mixture) is the total enthalpy of the system after mixing
• ΔH_mix is the enthalpy of mixing
• x_i is themole fraction of componenti in the system
• H_i is the enthalpy of componenti in pure form

Enthalpy of mixing can also be defined using Gibbs free energy of mixing

${\displaystyle \mathrm{\Delta }{G}_{mix}=\mathrm{\Delta }{H}_{mix}-T\mathrm{\Delta }{S}_{mix}}$

However,Gibbs free energy of mixing andentropy of mixing tend to be more difficult to determine experimentally.^[3] As such, enthalpy of mixing tends to be determined experimentally in order to calculate entropy of mixing, rather than the reverse.

Enthalpy of mixing is defined exclusively for the continuum regime, which excludes molecular-scale effects (However, first-principles calculations have been made for some metal-alloy systems such as Al-Co-Cr^[4] or β-Ti^[5]).

When two substances are mixed the resulting enthalpy is not an addition of the pure component enthalpies, unless the substances form an ideal mixture.^[6] The interactions between each set of molecules determines the final change in enthalpy. For example, when compound "x" has a strong attractive interaction with compound "y" the resulting enthalpy is exothermic.^[6] In the case of alcohol and its interactions with a hydrocarbon, the alcohol molecule participates in hydrogen bonding with other alcohol molecules, and these hydrogen bonding interactions are much stronger than alcohol-hydrocarbon interactions, which results in an endothermic heat of mixing.^[7]

Enthalpy of mixing is often determined experimentally using calorimetry. Abomb calorimeter is considered to be anisolated system with an insulated frame and a reaction chamber, and is used to transfer the heat of mixing into surrounding water whose temperature is measured. A typical solution would use the equation${\displaystyle H_{mixture}=\mathrm{\Delta }{H}_{mix}+\sum x_i{H}_i}$ (derived from the definition above) in conjunction with experimentally determined total-mixture enthalpies and tabulated pure species enthalpies, the difference being equal to enthalpy of mixing.

More complex models, such as theFlory-Huggins andUNIFAC models, allow prediction of enthalpies of mixing.Flory-Huggins is useful in calculating enthalpies of mixing for polymeric mixtures and considers a system from a multiplicity perspective.

Calculations of organic enthalpies of mixing can be made by modifyingUNIFAC using the equations^[8]

• ${\displaystyle \mathrm{\Delta }{H}_{mix}=\sum x_i\overline{\mathrm{\Delta }{H}_i}}$
• ${\displaystyle \overline{\mathrm{\Delta }{H}_i}=\sum _k{N}_{ki}(H_k-H_{ki}^{\ast })}$
• ${\displaystyle \frac{H_k}{R{T}^2}=Q_k(\frac{\sum _m\theta {\psi }_{mk}^{\prime }}{\sum _m\theta {\psi }_{mk}}-(\sum _m\frac{{\theta }_m{\psi }_k^{\prime }m}{\sum _n{\theta }_n{\psi }_{nm}}-\frac{{\theta }_m{\psi }_{km}(\sum _n{\theta }_n{\psi }_{nm}^{\prime })}{(\sum _n{\theta }_n{\psi }_{nm}{)}^2}))}$

Where:

• • ${\displaystyle x_i}$ = liquid mole fraction of i
  • ${\displaystyle \overline{\mathrm{\Delta }{H}_i}}$ = partial molar excess enthalpy of i
  • ${\displaystyle N_{ki}}$ = number of groups of type k in i
  • ${\displaystyle H_k}$ = excess enthalpy of group k
  • ${\displaystyle H_{ki}^{\ast }}$ = excess enthalpy of group k in pure i
  • ${\displaystyle Q_k}$ = area parameter of group k
  • ${\displaystyle {\theta }_m=\frac{Q_m{X}_m}{\sum _n{Q}_n{X}_n}}$ = area fraction of group m
  • ${\displaystyle X_m=\frac{\sum _i{x}_i{N}_{mi}}{\sum _i{x}_i\sum _k{N}_{ki}}}$ = mole fraction of group m in the mixture
    • ${\displaystyle {\psi }_{mn}=exp(-\frac{Z{a}_{mn}}{2T})}$
    • ${\displaystyle {\psi }_{mn}^{\ast }=\frac{\delta }{\delta T}({\psi }_{m}n)}$
  • ${\displaystyle Z=35.2-0.1272T+0.00014T^2}$ = Temperature dependent coordination number

It can be seen that prediction of enthalpy of mixing is incredibly complex and requires a plethora of system variables to be known. This explains why enthalpy of mixing is typically experimentally determined.

Theexcess Gibbs free energy of mixing can be related to the enthalpy of mixing by the use of theGibbs-Helmholtz equation:

${\displaystyle {\left(\frac{\mathrm{\partial }(\mathrm{\Delta }{G}^E/T)}{\mathrm{\partial }T}\right)}_p=-\frac{\mathrm{\Delta }{H}^E}{T^2}=-\frac{\mathrm{\Delta }{H}_{mix}}{T^2}}$

or equivalently

${\displaystyle {\left(\frac{\mathrm{\partial }(\mathrm{\Delta }{G}^E/T)}{\mathrm{\partial }(1/T)}\right)}_p=\mathrm{\Delta }{H}^E=\mathrm{\Delta }{H}_{mix}}$

In these equations, the excess and total enthalpies of mixing are equal because the ideal enthalpy of mixing is zero. This is not true for the corresponding Gibbs free energies however.

Anideal mixture is any in which the arithmetic mean (with respect to mole fraction) of the two pure substances is the same as that of the final mixture. Among other important thermodynamic simplifications, this means that enthalpy of mixing is zero:${\displaystyle \mathrm{\Delta }{H}_{mix,ideal}=0}$. Any gas that follows theideal gas law can be assumed to mix ideally, as can hydrocarbons and liquids with similar molecular interactions and properties.^[2]

Aregular solution or mixture has a non-zero enthalpy of mixing with an idealentropy of mixing.^[9][10] Under this assumption,${\displaystyle \mathrm{\Delta }{H}_{mix}}$ scales linearly with${\displaystyle X_1{X}_2}$, and is equivalent to the excessinternal energy.^[11]

The enthalpy of mixing for a ternary mixture can be expressed in terms of the enthalpies of mixing of the corresponding binary mixtures:${\displaystyle \mathrm{\Delta }{H}_{123}=(1-x_1{)}^2\mathrm{\Delta }{H}_{23}+(1-x_2{)}^2\mathrm{\Delta }{H}_{13}+(1-x_3{)}^2\mathrm{\Delta }{H}_{12}}$

Where:

• ${\displaystyle x_i}$ is the mole fraction of speciesi in the ternary mixture
• ${\displaystyle \mathrm{\Delta }{H}_{ij}}$ is the molar enthalpy of mixing of the binary mixture consisting of speciesi andj

This method requires that the interactions between two species are unaffected by the addition of the third species.${\displaystyle \mathrm{\Delta }{H}_{ij}}$ is then evaluated for a binary concentration ratio equal to the concentration ratio of speciesi toj in the ternary mixture (${\displaystyle x_i/x_j}$).^[12]

Intermolecular forces are the main constituent of changes in the enthalpy of a mixture. Stronger attractive forces between the mixed molecules, such ashydrogen-bonding,induced-dipole, anddipole-dipole interactions result in a lower enthalpy of the mixture and a release of heat.^[6] If strong interactions only exist between like-molecules, such as H-bonds between water in a water-hexane solution, the mixture will have a higher total enthalpy and absorb heat.

• Apparent molar property
• Enthalpy
• Enthalpy change of solution
• Excess molar quantity
• Entropy of mixing
• Calorimetry
• Miedema's Model

• Can. J. Chem. Eng. Duran Kaliaguine

• Enthalpy

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