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Enthalpy of mixing

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.

Formal definition

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

 

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 pure i

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

 

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 calculated experimentally using calorimetry methods. A bomb calorimeter is created to be an isolated system. With an insulated frame and a reaction chamber, a bomb calorimeter is used to transfer heat of a reaction or mixing into surrounding water which is then calculated for temperature. A typical solution would use the equation   (derived from the definition above) in conjunction 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]

  •  
  •  
  •  

Where:

    •   = liquid mole fraction of i
    •   = partial molar excess enthalpy of i
    •   = number of groups of type k in i
    •   = excess enthalpy of group k
    •   = excess enthalpy of group k in pure i
    •   = area parameter of group k
    •   = area fraction of group m
    •   = mole fraction of group m in the mixture
      •  
      •  
    •   = 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 ușe of the Gibbs-Helmholtz equation:

 

or equivalently

 

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:  . 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,   scales linearly with  , and is equivalent to the excess internal energy.[11]

Mixing binary mixtures to form ternary mixtures

The heat of mixing for binary mixtures to form ternary one can be expressed as a function of mixing ratios of binary mixtures:  [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

References

  1. ^ a b Carlson, Phillip (2002). Hazardous Chemicals Handbook (2nd ed.). Elsevier. p. 52. ISBN 978-0-7506-4888-2.
  2. ^ a b c 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" (PDF). Electronic Journal of Theoretical Chemistry. 1: 135–150. doi:10.1002/ejtc.27.
  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): e0121386. Bibcode:2015PLoSO..1021386L. doi:10.1371/journal.pone.0121386. ISSN 1932-6203. PMC 4395364. PMID 25875037.
  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. ^ a b c CB,378-2590,224-2707, Richard Rowley,350. "Heat_of_Mixing". www.et.byu.edu. Retrieved 2017-02-22.{{cite web}}: CS1 maint: multiple names: authors list (link)
  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 9780199543373.
  10. ^ Rock, Peter A. (1969). Chemical Thermodynamics: Principles and Applications. Macmillan. p. 263.
  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". Monatshefte für Chemie (in German). 91 (4): 738. doi:10.1007/BF00899814.

External links

  • Can. J. Chem. Eng. Duran Kaliaguine

enthalpy, mixing, thermodynamics, enthalpy, mixing, also, heat, mixing, excess, enthalpy, enthalpy, liberated, absorbed, from, substance, upon, mixing, when, substance, compound, combined, with, other, substance, compound, enthalpy, mixing, consequence, intera. 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 linksFormal definition EditFor a liquid enthalpy of mixing can be defined as follows 2 H m i x t u r e D H m i x x i H i displaystyle H mixture Delta H mix sum x i H i Where H mixture is the total enthalpy of the system after mixing DHmix is the enthalpy of mixing xi is the mole fraction of component i in the system Hi is the enthalpy of pure iEnthalpy of mixing can also be defined using Gibbs free energy of mixingD G m i x D H m i x T D S m i x displaystyle Delta G mix Delta H mix T Delta S mix 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 b 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 EditEnthalpy of mixing is often calculated experimentally using calorimetry methods A bomb calorimeter is created to be an isolated system With an insulated frame and a reaction chamber a bomb calorimeter is used to transfer heat of a reaction or mixing into surrounding water which is then calculated for temperature A typical solution would use the equation H m i x t u r e D H m i x x i H i displaystyle H mixture Delta H mix sum x i H i derived from the definition above in conjunction 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 D H m i x x i D H i displaystyle Delta H mix sum x i overline Delta H i D H i k N k i H k H k i displaystyle overline Delta H i sum k N ki H k H ki H k R T 2 Q k m 8 ps m k m 8 ps m k m 8 m ps k m n 8 n ps n m 8 m ps k m n 8 n ps n m n 8 n ps n m 2 displaystyle H k over RT 2 Q k biggl sum m theta psi mk over sum m theta psi mk biggl sum m theta m psi k m over sum n theta n psi nm theta m psi km sum n theta n psi nm over sum n theta n psi nm 2 biggr biggr Where x i displaystyle x i liquid mole fraction of i D H i displaystyle overline Delta H i partial molar excess enthalpy of i N k i displaystyle N ki number of groups of type k in i H k displaystyle H k excess enthalpy of group k H k i displaystyle H ki excess enthalpy of group k in pure i Q k displaystyle Q k area parameter of group k 8 m Q m X m n Q n X n displaystyle theta m Q m X m over sum n Q n X n area fraction of group m X m i x i N m i i x i k N k i displaystyle X m sum i x i N mi over sum i x i sum k N ki mole fraction of group m in the mixture ps m n e x p Z a m n 2 T displaystyle psi mn exp biggl Za mn over 2T biggr ps m n d d T ps m n displaystyle psi mn delta over delta T psi m n Z 35 2 0 1272 T 0 00014 T 2 displaystyle Z 35 2 0 1272T 0 00014T 2 Temperature dependent coordination numberIt 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 EditThe excess Gibbs free energy of mixing can be related to the enthalpy of mixing by the ușe of the Gibbs Helmholtz equation D G E T T p D H E T 2 D H m i x T 2 displaystyle left frac partial Delta G E T partial T right p frac Delta H E T 2 frac Delta H mix T 2 or equivalently D G E T 1 T p D H E D H m i x displaystyle left frac partial Delta G E T partial 1 T right p Delta H E 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 Ideal and regular mixtures EditAn 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 D H m i x i d e a l 0 displaystyle Delta H mix 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 D H m i x displaystyle Delta H mix scales linearly with X 1 X 2 displaystyle X 1 X 2 and is equivalent to the excess internal energy 11 Mixing binary mixtures to form ternary mixtures EditThe heat of mixing for binary mixtures to form ternary one can be expressed as a function of mixing ratios of binary mixtures D H 123 1 x 1 2 D H 23 1 x 2 2 D H 13 1 x 3 2 D H 12 displaystyle Delta H 123 1 x 1 2 Delta H 23 1 x 2 2 Delta H 13 1 x 3 2 Delta H 12 12 Intermolecular forces EditIntermolecular 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 EditApparent molar property Enthalpy Enthalpy change of solution Excess molar quantity Entropy of mixing Calorimetry Miedema s ModelReferences Edit a b Carlson Phillip 2002 Hazardous Chemicals Handbook 2nd ed Elsevier p 52 ISBN 978 0 7506 4888 2 a b c Sinnot Ray K 2009 Chemical Engineering Design SI Edition 5th ed Elsevier p 95 ISBN 978 0 7506 8551 1 Lin Shu Kun 1996 Gibbs paradox of entropy of mixing experimental facts its rejection and the theoretical consequences PDF Electronic Journal of Theoretical Chemistry 1 135 150 doi 10 1002 ejtc 27 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 e0121386 Bibcode 2015PLoSO 1021386L doi 10 1371 journal pone 0121386 ISSN 1932 6203 PMC 4395364 PMID 25875037 Chandran Mahesh Subramanian P R Gigliotti Michael F 2013 02 15 First principles calculation of mixing enthalpy of b Ti with transition elements Journal of Alloys and Compounds 550 501 508 doi 10 1016 j jallcom 2012 10 141 a b c CB 378 2590 224 2707 Richard Rowley 350 Heat of Mixing www et byu edu Retrieved 2017 02 22 a href Template Cite web html title Template Cite web cite web a CS1 maint multiple names authors list link 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 amp Engineering Data 11 1 40 43 doi 10 1021 je60028a009 ISSN 0021 9568 Dang Dinh Tassios Dimitrios P 1986 01 01 Prediction of enthalpies of mixing with a UNIFAC model Industrial amp Engineering Chemistry Process Design and Development 25 1 22 31 doi 10 1021 i200032a004 ISSN 0196 4305 Atkins Peter de Paula Julio 2010 Atkins Physical Chemistry Oxford University Press p 167 ISBN 9780199543373 Rock Peter A 1969 Chemical Thermodynamics Principles and Applications Macmillan p 263 Vidal Jean 2003 Thermodynamics Applications in Chemical Engineering and the Petroleum Industry Editions Technip p 232 ISBN 978 2 7108 0800 8 Kohler F 1960 Zur Berechnung der thermodynamischen Daten eines ternaren Systems aus den zugehorigen binaren Systemen Monatshefte fur Chemie in German 91 4 738 doi 10 1007 BF00899814 External links EditCan J Chem Eng Duran Kaliaguine Retrieved from https en wikipedia org w index php title Enthalpy of mixing amp oldid 1108780165, wikipedia, wiki, book, books, library,

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