Enthalpy of mixing

Short description: Change in enthalpy during the mixture of substances

In thermodynamics, the enthalpy of mixing (also heat of mixing and excess enthalpy) is the enthalpy liberated or absorbed from a substance upon mixing.[1] When a substance or compound 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 released exothermically, can in an extreme case cause an explosion.

Enthalpy of mixing can often be ignored in calculations for mixtures where other heat terms exist, or in cases where the mixture is ideal.[2] The sign convention is the same as for enthalpy of reaction: when the enthalpy of mixing is positive, mixing is endothermic, while negative enthalpy of mixing signifies exothermic mixing. In ideal mixtures, the enthalpy of mixing is null. In non-ideal mixtures, the thermodynamic activity of each component is different from its concentration by multiplying with the activity coefficient.

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

## Contents

* 1 Formal definition
* 2 Calculations
* 3 Relation to the Gibbs free energy of mixing
* 4 Ideal and regular mixtures
* 5 Mixing binary mixtures to form ternary mixtures
* 6 Intermolecular forces
* 7 See also
* 8 References
* 9 External links

## Formal definition

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

H(mixture)=ΔHmix+∑xiHi

Where:

* H(mixture) is the total enthalpy of the system after mixing
* ΔHmix is the enthalpy of mixing
* xi is the mole fraction of component i in the system
* Hi is the enthalpy of component i in pure form

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

ΔGmix=ΔHmix−TΔSmix

However, Gibbs free energy of mixing and entropy 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]

## Calculations

Enthalpy of mixing is often determined experimentally using calorimetry. A bomb calorimeter is considered to be an isolated 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 Hmixture=ΔHmix+∑xiHi (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 the Flory-Huggins and UNIFAC 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 modifying UNIFAC using the equations[8]

* ΔHmix=∑xiΔHi‾
* ΔHi‾=∑kNki(Hk−Hki*)
* HkRT2=Qk(∑mθψ'mk∑mθψmk−(∑mθmψ'km∑nθnψnm−θmψkm(∑nθnψ'nm)(∑nθnψnm)2))

Where:

*     * xi = liquid mole fraction of i
* ΔHi‾ = partial molar excess enthalpy of i
* Nki = number of groups of type k in i
* Hk = excess enthalpy of group k
* Hki* = excess enthalpy of group k in pure i
* Qk = area parameter of group k
* θm=QmXm∑nQnXn = area fraction of group m
* Xm=∑ixiNmi∑ixi∑kNki = mole fraction of group m in the mixture
* ψmn=exp(−Zamn2T)
* ψmn*=δδT(ψmn)
* Z=35.2−0.1272T+0.00014T2 = 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.

## Relation to the Gibbs free energy of mixing

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

(∂(ΔGE/T)∂T)p=−ΔHET2=−ΔHmixT2

or equivalently

(∂(ΔGE/T)∂(1/T))p=ΔHE=ΔHmix

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.

## Ideal and regular mixtures

An ideal 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: ΔHmix,ideal=0. Any gas that follows the ideal gas law can be assumed to mix ideally, as can hydrocarbons and liquids with similar molecular interactions and properties.[2]

A regular solution or mixture has a non-zero enthalpy of mixing with an ideal entropy of mixing.[9][10] Under this assumption, ΔHmix scales linearly with X1X2, and is equivalent to the excess internal energy.[11]

## Mixing binary mixtures to form ternary mixtures

The enthalpy of mixing for a ternary mixture can be expressed in terms of the enthalpies of mixing of the corresponding binary mixtures: ΔH123=(1−x1)2ΔH23+(1−x2)2ΔH13+(1−x3)2ΔH12

Where:

* xi is the mole fraction of species i in the ternary mixture
* ΔHij is the molar enthalpy of mixing of the binary mixture consisting of species i and j

This method requires that the interactions between two species are unaffected by the addition of the third species. ΔHij is then evaluated for a binary concentration ratio equal to the concentration ratio of species i to j in the ternary mixture (xi/xj).[12]

## Intermolecular forces

Intermolecular forces are the main constituent of changes in the enthalpy of a mixture. Stronger attractive forces between the mixed molecules, such as hydrogen-bonding, induced-dipole, and dipole-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.

## See also

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

## References

1. ↑ 1.0 1.1 Carlson, Phillip (2002). Hazardous Chemicals Handbook (2nd ed.). Elsevier. p. 52. ISBN 978-0-7506-4888-2.
2. ↑ 2.0 2.1 2.2 Sinnot, Ray K (2009). Chemical Engineering Design - SI Edition (5th ed.). Elsevier. p. 95. ISBN 978-0-7506-8551-1.
3. ↑ Lin, Shu-Kun (1996). "Gibbs paradox of entropy of mixing: experimental facts, its rejection and the theoretical consequences". Electronic Journal of Theoretical Chemistry 1: 135–150. doi:10.1002/ejtc.27. http://www.mdpi.org/lin/ejtc027.pdf.
4. ↑ Liu, Xuan L.; Gheno, Thomas; Lindahl, Bonnie B.; Lindwall, Greta; Gleeson, Brian; Liu, Zi-Kui (2015-04-13). "First-Principles Calculations, Experimental Study, and Thermodynamic Modeling of the Al-Co-Cr System". PLOS ONE 10 (4). doi:10.1371/journal.pone.0121386. ISSN 1932-6203. PMID 25875037. Bibcode: 2015PLoSO..1021386L.
5. ↑ Chandran, Mahesh; Subramanian, P. R.; Gigliotti, Michael F. (2013-02-15). "First principles calculation of mixing enthalpy of β-Ti with transition elements". Journal of Alloys and Compounds 550: 501–508. doi:10.1016/j.jallcom.2012.10.141.
6. ↑ 6.0 6.1 6.2 CB,378-2590,224-2707, Richard Rowley,350. "Heat_of_Mixing". http://www.et.byu.edu/~rowley/ChEn273/Topics/Energy_Balances/Nonreacting_Systems/Nonideal_Mixtures/Heat_of_Mixing.htm.
7. ↑ Savini, C. G.; Winterhalter, D. R.; Kovach, L. H.; Van Ness, H. C. (1966-01-01). "Endothermic Heats of Mixing by Isothermal Dilution Calorimetry.". Journal of Chemical & Engineering Data 11 (1): 40–43. doi:10.1021/je60028a009. ISSN 0021-9568.
8. ↑ Dang, Dinh; Tassios, Dimitrios P. (1986-01-01). "Prediction of enthalpies of mixing with a UNIFAC model". Industrial & Engineering Chemistry Process Design and Development 25 (1): 22–31. doi:10.1021/i200032a004. ISSN 0196-4305.
9. ↑ Atkins, Peter; de Paula, Julio (2010). Atkins' Physical Chemistry. Oxford University Press. p. 167. ISBN 978-0-19-954337-3. https://books.google.com/books?id=BV6cAQAAQBAJ&pg=PA167.
10. ↑ Rock, Peter A. (1969). Chemical Thermodynamics: Principles and Applications. Macmillan. p. 263. https://books.google.com/books?id=G0vDvgEACAAJ.
11. ↑ Vidal, Jean (2003). Thermodynamics - Applications in Chemical Engineering and the Petroleum Industry. Editions Technip. p. 232. ISBN 978-2-7108-0800-8.
12. ↑ Kohler, F. (1960). "Zur Berechnung der thermodynamischen Daten eines ternären Systems aus den zugehörigen binären Systemen" (in de). Monatshefte für Chemie 91 (4): 738. doi:10.1007/BF00899814.

## External links

* Can. J. Chem. Eng. Duran Kaliaguine

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