Methanol: Association behaviour, third-law entropy analysis and determination of the enthalpy of formation

University of Groningen

Methanol

Graaf, Geert; Winkelman, Jos G M

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Fluid Phase Equilibria

DOI:

10.1016/j.fluid.2020.112851

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Graaf, G., & Winkelman, J. G. M. (2021). Methanol: Association behaviour, third-law entropy analysis and

determination of the enthalpy of formation. Fluid Phase Equilibria, 529, [112851].

https://doi.org/10.1016/j.fluid.2020.112851

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ContentslistsavailableatScienceDirect

Fluid

Phase

Equilibria

journalhomepage:www.elsevier.com/locate/fluid

Methanol:

association

behaviour,

third-law

entropy

analysis

and

determination

of

the

enthalpy

of

formation

G.H.

Graaf

a

_,

_J.G.M.

_Winkelman

b ,∗

a Graaf Independent Energy Advice, Parklaan 4, 9724 AL Groningen, the Netherlands

b Department of Chemical Engineering, ENTEG, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 20 June 2020 Revised 2 September 2020 Accepted 28 September 2020 Available online 2 October 2020

Keywords: Methanol Entropy Enthalpy of formation Association behaviour Chemical equilibrium

a

b

s

t

r

a

c

t

Chemicalequilibrium constants forthe methanolfrom CO/H2 reactioncalculated fromliterature

ther-modynamicparametersaretoohighas comparedtoexperimentalresults.Therefore,boththe entropy valueandtheenthalpyofformationvalueofmethanolwerereviewed.Arigorousanalysisofthe associa-tionbehaviourofmethanolvapourconfirmsthatdimersandcyclictetramersplayanimportantrole,but physicalnon-idealgasbehaviourmust alsobetakenintoaccount.Accuraterelationshipsforthe asso-ciationequilibriumconstantsandthephysicalcontributiontothesecondvirialcoefficientwerederived withtheuseofacombinedfitofmultipleexperimentaldatasourcesincludingheatcapacity,speedof sound,thermalconductivity,excessmolarenthalpyofmethanoland nitrogenand heatofvaporization. Additionally,hightemperaturesecondvirialcoefficientsandmeasuresfortheconsistenttemperature de-pendenciesofthe entropyandenthalpyofformation wereincludedinthe parameteroptimizationto supporttheaccuracyofthemodel.Allexperimentalresultsandsupportingdataweretakenorderived fromtheliterature.Theresultingvirialequationofstatewasusedtocalculatenewideal-gasentropyand enthalpyofformationvaluesofmethanol,whichnowturnouttobeconsistentwithvaluesderivedfrom experimentalchemicalequilibriumdata.Furthermore,thenewthird-lawentropyvalueturnsouttobe consistentwiththecurrentliteraturevalue.Anewenthalpyofformationvalueisrecommendedandan improvedchemicalequilibriumrelationshipforthemethanolfromCO/H2reactionispresented.

© 2020TheAuthor(s).PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

Abbreviations

AAD AverageabsolutedeviationAAD= 1

N∗ N  i=1

|

y_mod,i−y_exp,i y_exp,i

|

g Indicatesgasphase

l Indicatesliquidphase SCF Sensitivitycorrectionfactor VEoS Virialequationofstate VLE Vapourliquidequilibrium

1. Introduction

Recently we publisheda review on the chemical equilibria in methanolsynthesis[1] .Itturned outthataccurateideal-gas equi-libriumconstantrelationshipscouldbederivedfrom thermochem-icalbasicdatatakenfromtheliteratureincombinationwithsmall correctionsfortheGibbsenergyofCOandCH3OHtomatchthese

∗_{Corresponding author: Tel.: + 31-503634484 (secr.) Fax: + 31-503634479.}

E-mail address: j.g.m.winkelman@rug.nl (J.G.M. Winkelman).

relationshipswithan extensivedatabase ofexperimental equilib-riumconstants. TheseGibbs energy correctionsturned out tobe smallerthantheuncertaintiesofthecorresponding literature val-ues.

However, literature Gibbs energy values are derived from the correspondingliteratureenthalpyofformationandentropyvalues. Sincewehavenot adaptedenthalpyvalues,ourcorrectionsofthe Gibbs energy valuescan be translateddirectly to changes in the entropyvaluesforCOandCH3OH.Theresultingideal-gasentropy

values are: 197.83 J mol−1 K−1 (p0 _{= 1 bar; T = 298.15 K) for}

CO, basedon thecollectedexperimental equilibriumconstantsof thewater-gasshiftreaction and238.03Jmol−1 K−1 (p0 _{= 1bar;}

T₌298.15K)forCH3OH,basedonthecollectedequilibrium

con-stantsof themethanolfromCO/H2 reaction.ForCOthis

“experi-mentalchemical equilibrium” result is consistentwith the corre-sponding entropy literature value, but forCH3OH this is not the

case. Here, the literature entropy value equals 239.81 ± 0.17 J mol−1 K−1[2] .Infact,Chao etal.[3] reportanevenlower uncer-taintyof_{± 0.09}Jmol−1 K−1.Cravenetal.[4] reportexperimental uncertaintiesof ± 0.45 J mol−1 K−1 at200 K and ± 0.29 J mol

https://doi.org/10.1016/j.fluid.2020.112851

Nomenclature

A HelmholtzenergykJmol−1 a0– a7 Parametersofeq. (22)

a_Bph Parameterofeq. (27)

B Second virial coefficient pressureexplicit VEoSm3

mol−1

B’ SecondvirialcoefficientvolumeexplicitVEoSPa−1 bBph Parameterofeq. (27)

b0 – b6 Parametersofeq. (23)

C Third virial coefficient pressure explicit VEoS m6

mol−2

C’ ThirdvirialcoefficientvolumeexplicitVEoSPa−2 Cp HeatcapacityatconstantpressureJmol−1K−1 CV HeatcapacityatconstantvolumeJmol−1K−1

c Soundvelocityms−1 c0 – c3 Parametersofeq. (24)

D Fourth virial coefficient pressure explicit VEoS m9

mol−3

D’ FourthvirialcoefficientvolumeexplicitVEoSPa−3 d0– d4 Parametersofeq. (25)

E Fifth virial coefficient pressure explicit VEoS m12

mol−4

E’ FifthvirialcoefficientvolumeexplicitVEoSPa−4 F Sixth virial coefficient pressure explicit VEoS m15

mol−5

F’ SixthvirialcoefficientvolumeexplicitVEoSPa−5 g0– g4 Parametersofeq. (28)

H EnthalpykJ

Hm MolarenthalpykJmol−1

h0,h2 Parametersofeq. (29)

Kp1 Equilibriumconstant (CH3OH fromCO/H2 reaction)

Pa−2 orbar−2

K2 Equilibriumconstant(CH3OHdimerization)Pa−1

K3 Equilibriumconstant(CH3OHtrimerization)Pa−2

K4 Equilibriumconstant(CH3OHtetramerization)Pa−3

K5 Equilibrium constant (CH3OH pentamerization)

Pa−4

K6 Equilibrium constant (CH3OH hexamerization)

Pa−5

Mw Molecularweightgmol−1

p PressurePaorbar

PD_ij Pressure∗BinarydiffusioncoefficientPam2 _s−1

R Universalgasconstant(8.314463)Jmol−1K−1 S (Absolute)entropyJK−1

Sm MolarentropyJmol−1K−1

S2

rel Averagesumofsquaresofrelativeresiduals

T TemperatureK

uc Combinedstandarduncertainty

ui Individualstandarduncertainty

Vm Molarvolumem3 mol−1

Z Compressibilityfactor

β

1 –

β

7 ParametersinKp1°-relationship(eq. (38) )

δ

i Individualstandarduncertaintyofinputelements

࢞H EnthalpychangekJmol−1

ζ

Approachtooptimumfit

λ

ThermalconductivitycoefficientJm−1 s−1K−1

λ

1 Thermal conductivity coefficient atzero pressure J

m−1s−1K−1

ρ

Molardensitymolm−3

φ

Average isothermal Joule Thomson coefficient J mol−1 Pa−1 divided by the pressure difference as definedby[27]

Subscripts

A Association

app Apparent

CH3OH Indicatescomponentmethanol

c Criticalpoint exp Experimentalvalue exptd Expectedvalue

f Offormation

IGL Indicatesidealgasconditions k Indicatesparameternumber L Indicatesliquidphase

m Molar

mix Indicatesmixture mod Modelresult

N2 Indicatescomponentnitrogen

opt Optimum

ph Physical

r Reducedconditionsrelativetothecriticalpoint rel Relative

sat Atsaturationconditions

vap Vapour

Superscripts

E Indicatesexcess

0 Indicatesidealgasconditionsandat1bar

K−1 at300 - 1000K. Our“experimental” entropy value isa fac-tor0.993lower.Althoughthismaystillberegardedasarelatively smalldifference, one should realizethat theimpact on the equi-librium constant is quite significant. Using the literature entropy value, thecalculated equilibriumconstants ofthe methanolfrom CO/H2reactionareapproximately1.2timestoohigh.

The relevance of this comparisonof entropy values is further presentedin Fig. 1 .Here, we show the0.99confidence regionof methanol entropy and enthalpy values derived from our chemi-calequilibriumanalysis[1] incombinationwiththefittedentropy value(caseHfixed),theoptimumcombinationofentropyand en-thalpy and an alternative fit of the enthalpy of formation value (caseSfixed).Furthermore,we alsoshow theliterature values in-cluding the corresponding uncertainties. Here we assume uncer-tainties of 0.29 J mol K−1 for entropy based on [4] and 0.42 kJ mol−1 forenthalpyof formationbasedon [5] .It isclearthat the resultingpictureisinconsistent:noentropy-enthalpycombination can be derived from the chemical equilibrium analysis that fits withtheliteratureresults.

Theobjectiveofthispaperistoclarifytheobserveddifference aspresented inFig. 1 .For thispurpose,we will carefully review the third-lawentropy analysisincludingthe influence ofthe for-mation ofassociated methanolclustersin thevapour phase. Fur-thermore,wewill alsoreview theenthalpy offormationvalue of methanol,becausetheobserveddifferencemightbecausedbythis parameteraswell.

Ideally, this clarification should result in improved entropy and/orenthalpy offormationvaluesofmethanolanda consistent updateofthecomparisonpresentedinFig. 1 .Itwillalsoallowus to confirm orfurther improve thechemical equilibrium relation-shipofthemethanolfromCO/H2 reaction,whichisimportantfor

theaccuratedesignofmethanolsynthesisprocesses.

2. Literaturereview

2.1. Entropy

Themethanolliteratureentropyvalueisderived from spectro-scopicdataincombinationwithmolecular modellingand

statisti-Fig. 1. Entropy and enthalpy of formation of methanol (g, 298.15 K).

Literature values and results derived from chemical equilibrium constants (CO + 2 H 2 ⇔ CH 3 OH).

calthermodynamics[2] .Priortothatpublicationthemajor uncer-taintyforthespectroscopicentropyvaluewascausedbythe diffi-cultquantificationofthehinderedrotationalongtheC–Oaxisand uncertaintiesaboutthevibrationalfrequencies.Chenetal.[2] con-cludedthatvariousinvestigations inthisfieldhadreachedan im-proved accuracy leveljustifying a recalculation ofthe thermody-namic properties of methanol, including the entropy. Moreover, various literature sources showed that non-ideal gas behaviour should be taken intoaccount when calculating thethird-law en-tropyvalue.Themostimportantpublicationinthisrespectisfrom Weltner&Pitzer[6] ,who concludedfromheatcapacity measure-mentsthatmethanolvapourisamixtureofmonomers,dimersand tetramers (hydrogen bonding, the 1–2–4 model). Theywere able to estimate thecorresponding temperature-dependantassociation equilibriumconstantsfromtheir experimentalresultsin combina-tionwithexperimental datafromother sources.Subsequently,the corresponding virial coefficientswere derived fromthese equilib-rium constants, allowing for the calculation of the non-idealgas entropy-change. Weltner & Pitzer already concluded that the re-sultingthird-lawentropyvaluecorrespondedwellwiththe “spec-troscopic” value including thequantification oftorsionalrotation, although thelattersubjectwasstillsomewhat uncertaininthose days.Chenetal.reviewedtheworkofWeltner&Pitzerandseveral other publications on methanol association in the vapour phase andreachedthesameconclusionmorefirmly.Theycalculatedthe “spectroscopic” thermodynamicpropertiesandshowedthatthe re-sultingentropy-valuesagreereasonablywellwiththird-lawvalues based on the1–2–4 associationmodel ofWeltner & Pitzer.Chen et al. [2] report Sm0(CH3OH, g, 298.15 K) = 239.70 J mol−1 K−1

atareferencepressureof1atm,correspondingwiththereported valueofChaoetal.[3] of239.81Jmol−1 K−1at1bar,basedona similarreview.

Since the works of [2] and [3] no updates on Sm0(CH3OH,

g, 298.15 K) derived from spectroscopic data have been made to our knowledge. On the other hand, an increasing popular-ity of ab initio quantum mechanical calculations is visible in the literature. It can be seen that these entropy-results are somewhat lower than the “spectroscopic” value. Barone [7] re-ported calculated Sm0(CH3OH, g, 298.15 K)-values of 239.24 and

239.45 J mol−1 K−1 (corrected to p0 _{= 1 bar). More recently,}

Umer&Leonhard[8] havereportedan ideal-gasentropy-valueof 239.46Jmol−1K−1.

A closer look at the reported third law Sm0(CH3OH, g,

298.15 K)-values based on the association model of Weltner & Pitzer shows a somewhat different temperature-dependency as comparedtothe“spectroscopic” valuesasreportedbyChenetal. Moreover, Weltner & Pitzer have quantified their 1–2–4 associa-tionmodelbased onheat capacitymeasurementsin aT-rangeof 345.6 K – 521.2 K. Extrapolating the resulting virial equation of state(VEoS)to298.15Kthereforeintroduces anextrauncertainty in the calculation of the third law entropy-value. In this respect weaskourselveswhethertheVEoScanbeimprovedbyincluding atemperaturedependentdescriptionofphysical(inadditionto as-sociation)non-idealgasbehaviour.

OurpreliminaryanalysisshowedthatslightlyretuningtheVEoS as determined by Weltner & Pitzer using P

ρ

T-experiments from Cheam et al.[9] results in lower entropy-values as compared to thecurrentliterature value andreasonablyinlinewiththemore recent quantum mechanical results [7 ,8 ]. Cheam et al. measured the vapour density in terms of apparent molecular weight as a function ofpressureat 298.15Kwiththe useof anaccurate mi-crobalance.Theexperimentalset-upwasconstructedinawaythat adsorption effectsofmethanolare (assumedtobe)compensated. Weconcludethatitisworthwhiletoreviewthethirdlawentropy

valueandespeciallythenecessarycorrectionfornon-idealgas be-haviour.

2.2. Enthalpyofformation

With regard toenthalpy offormation Ruscic[10] has recently reporteda࢞fHm0(CH3OH,g,298.15K)-valueof−200.71± 0.16kJ

mol−1,whichseemstobemoreaccuratethanotherliterature val-ues. Moreover,thisvalue narrows the gapbetweenliterature en-tropydataandtheentropy-valuederivedfromexperimentalKp10

-data. Nevertheless, a significant portion of thisgap still remains, since theexperimentalKp1-dataincombinationwiththe ࢞fHm0

-valuereportedbyRuscicleadtoanentropy-valueofapproximately 238.6Jmol−1 K−1.

A more detailed analysis of literature _࢞fHm0(CH3OH, g,

298.15K)-valuesshowsthat thesearebasedonthefollowing pri-mary sources. The first source is Rossini[11] and dealswiththe heatofcombustionofgaseousmethanol,whilethesecondsource (Chao& Rossini) [12] dealswiththeheat ofcombustionofliquid methanol. It turns out that literature ࢞fHm0-values are in many

casesbasedon (anaverageresultof) bothprimary sources. How-ever,࢞fHm0-valuesderivedfromthesetwoprimarysourcesdiffer

byapproximately0.84kJmol−1andthevalueofRuscic[10] is rea-sonablyinlinewithRossini[11] ascanbederivedfromtheworks ofDomalski[13] andWilhoitetal.[5] .Iftheenthalpyofformation isderivedfromChao&Rossini[12] avalueof−201.4kJmol−1 re-sults,whichisclearlyincompatiblewithourchemicalequilibrium analysis(seeFig. 1 ). Itmustbenotedhoweverthat noclarity ex-istswithregard tothe observeddifference:bothprimary sources are consideredto bereliableandtheratiooftheunderlyingheat of combustion values is close to unity (1.001). Nevertheless, the result of Ruscic [10] andour own chemical equilibrium analysis strongly indicate that Rossini [11] should be preferredover Chao andRossini[12] .

2.3. Experimentaldatarelevantforquantifyingtheassociation behaviourofmethanolvapour

Based on the preliminary analysis described in the previous section we conclude that correcting for non-ideal gas behaviour is an essentialpartof anaccurate third-lawentropy analysisand quantifying the necessary virial coefficients should preferably be basedonagreaterandmorediversesetofexperimentaldatathan just theheat capacitydata ofWeltner & Pitzer.Moreover, thisis also truefor an accurate determination ofthe ideal-gasenthalpy offormationvalue.

Nevertheless, we regard the experimental data of Weltner & Pitzertobereliable,sincetheseareconsistentwithsimilar exper-imentaldatafromothersources[14 –17 ].

TheexperimentalvapourdensitydataofCheametal.have nei-ther beenconfirmedorquestioned byother researchers asfar as we know.Bich etal. [18] have presented an overviewof experi-mentally based second virial coefficients formethanol vapour as a functionoftemperature. Theiroverviewshowsthat experimen-taldatadivergesignificantlyatlowertemperatures(T_<400K).In thecaseofp

ρ

Tmeasurementstheunderlyingcauseisbelievedto be adsorption ofmethanol vapour.However, Bich etal. havenot reviewedtheresultsofCheametal.intheirpublication.

An analysis ofother literature sources containingrelevant ex-perimentaldatashowsthatthefollowingsourcesmayalsobeused inanupdateofthethird-lawentropyanalysisandforthe determi-nationofthecorrespondingideal-gasenthalpyofformationvalue. Like heat capacity also the thermal conductivity of methanol vapour is stronglydependanton temperatureandpressurebased on the formation of association clusters. Reliable experimental thermal conductivity data have been reported by Frurip et al.

[19] and these authors were able to tune the 1–2–4 association modelwiththeir experimental datausingtheButler-Brokaw the-ory[20–22] .Unfortunately,some extramodelparameters haveto befittedorestimatedaswell (thermalconductivitycoefficientsat zeropressureandtherelevantdiffusioncoefficients).Thermal con-ductivity ofmethanol vapour has also been investigatedby [23] . However, theseexperimental dataare notconsistent withthe re-sultsofFruripetal.andareconsidered tobe lessreliable(Sykioti etal.,[24] ).

Massuccietal., [25] havemeasuredthe excessmolarenthalpy of nitrogen and methanol vapour in a flow calorimeter. Also, in thiscasethe1–2–4modelwasappliedsuccessfully,butmodelling theobservedheateffectsisrelativelycomplexbecauseitrequires (amongst others) an estimation of the cross-second virial coeffi-cient(N2– CH3OH).

Boyesetal.[26] havemeasuredthespeedofsoundofmethanol vapouratvarioustemperaturesandpressures.Thepressure depen-dencyoftheir experimentaldataismostlikelyrelatedto associa-tionbehaviour.

Francis&Phutela[27] measuredtheisothermalJoule-Thomson effectofmethanolandethanolvapour.Theyconcludedthatthe1– 2–4modelisnotcapableofdescribingtheirexperimentaldata. In-stead ofusinga dimerizationequilibriumconstant, theseauthors used (and tuned) a relationship for the second virial coefficient as a function of temperature in combination with the tetramer-ization equilibrium constant. Bich et al. [18] indicate that these experimental data maybe less reliable based ona later publica-tion of Francis [28] dealing with an update of isothermal Joule-Thomson coefficientsofethanolvapour.Theselatterresultsdiffer significantly from the results of the earlierpublication, which is explainedbythefactthatanimprovedflowcalorimeterwasused. Finally,theenthalpyofvaporizationcanbecalculatedwiththe useoftheClapeyronequation (eq. (1) )andcomparedwith exper-imentalvalues.

v

apH=

dpvap

dT

(

1/

ρ

vap,sat− 1/

ρ

L,sat

)

T (1)

Formethanolaccurate experimental vapour pressuredata, en-thalpy ofvaporization data andsaturated liquiddensity data are available. The necessary saturatedvapour densities can be calcu-latedwithanappropriateequationofstate.

From thisliterature review wedraw thefollowingconclusions andresearchquestions.

a Althoughthe1–2–4modelseems tobe thepreferred associa-tionmodel,nounambiguousconfirmation isavailablefora fi-nalconclusionregardingthisstatement. Atleastthe tuningof the model parameters, especially for the second virial coeffi-cient and the dimerization equilibrium, is an aspect that de-servesfurtherattentionandanalysis.

b It is unclear whether the experimental data of Cheam et al.

[9] could contribute toan accurate third-lawentropyanalysis. Theirdataare obtainedat298.15K,which isideal inthis re-spect.It is worthwhile to investigate whether the association modelcanbetunedwiththecombinedexperimentsofCheam etal.andthevariousotherexperimentaldata.

c Aproperassociation modelshould resultina consistent tem-perature dependency of the entropy values derived from the third-lawanalysisandthismustalso holdfortheenthalpy of formation.

3. Modeldescription

3.1. Dimerization

Aspointedoutintheprevioussectionitisworthwhileto anal-yse the aspect of dimerization in detail. Since the dimerization

equilibriumconstant ismainlyreflectedinthe secondvirial coef-ficient,thequestioniswhether“physical” (i.e.notrelatedto asso-ciation)non-idealgasbehaviourshouldbeincludedinthemodel. In thiscontext itisusefultotake alookatothercompounds ex-hibitingdimerizationlikewaterandammonia.Fortheseandother compounds [29 ,30 ]havecomparedexperimentalsecond virial co-efficients withvaluescalculated fromthe Berthelotequation [31] . TheyshowthatexperimentalB-valuesaremorenegativethanthe calculatedonesandthedifferencesareattributedtodimerization. In fact, similar comparisons for compounds that do not exhibit dimerization (e.g.trimethyl amine andtriethyl amine) show that experimentalandBerthelotB-valuescorrespondreasonablywell.

We have analysed this approach and conclude that in many casesBerthelotB-valuesascalculatedwitheq. (2) giveafair reflec-tion of the physical non-idealgas behaviour. However, we found that thisapproach doesnot predict correctBph-valuesfor

associ-atingcompoundswhenthecriticalpropertiesofthecompoundof interest are used.Correct Bph-valuescan be determined by using

properly tuned effectivecritical properties (Tc andpc). These

ef-fectivecriticalpropertiesmayberegardedtoreflectahypothetical homomorphofthecompoundofinterest.

B pc R Tc = 9 128− 54/128 T2 r (2) Fig. 2 shows the results of water vapour, where the second virialcoefficientisplottedasafunctionoftemperaturebothforan accurate representationof experimentalresults [32] andthe split indimerizationandphysicaleffects.Here,thedimerizationresults weretakenfromWormald[33] ,whostudiedtheheatofmixingof (water+ nitrogen)and(water +oxygen).The differencebetween thetotalsecondvirialcoefficientandthepartrelatedto

dimeriza-tion(BA=-RTK2)thusequalsBphandthecalculatedBph-valuesare

accurately described witheq. (2) when usingTc = 351.17 Kand

pc = 5.348MPa.It isinteresting to realizethat thesevalues

rea-sonablyresemblethecriticalpropertiesoffluoromethane,whichis indeeda possiblehomomorph forwater.Poling etal. [34] report Tc = 315.0 K,pc =5.548MPa andZc = 0.240 forfluoromethane.

Here, itshould be noted that the Berthelotequation isbased on Zc = 9/23 ≈ 0.281as pointedout by Mathias [35] .Furthermore,

eq. (2) isapplicableatreducedtemperaturesaboveapproximately 0.8[35] .Thisconditionisfulfilledforouranalysisofwatervapour ascanbeseeninFig. 2 .

From this analysis we conclude that “physical” non-ideal gas behaviourcannotbe neglectedandmustbeincorporatedin mod-ellingtheeffectsofassociationbehaviourone.g. thepressure de-pendencyofheatcapacity.

3.2. Virialequationofstate

Tomodeltheexperimental datawiththeexceptionofthermal conductivitydatawewillusethevirialequationofstatebasedon the 1–2–4association model andtaking intoaccount the effects ofphysicalnon-idealgasbehaviouraswell.Here,thesecondvirial coefficient,B,isasummationofB_Ph (reflectingphysicalnon-ideal gasbehaviour)andBA(reflectingdimerization:BA=-RTK2).Higher

virialcoefficientsarebasedonassociationonly,assumingthatthe description of physical non-ideal gas behaviour does not require theuseofhighervirialcoefficients.Thisassumptionisreasonable, becauseourstudyonlydealswithlowpressureconditions.A sim-plifiedVEoSthenresultsbyusingB=BA+Bph andcalculatingD

(thefourthvirialcoefficient)fromthetetramerizationequilibrium

constant.However,Woolley[36] hasshownthattheexact transla-tionofassociationequilibriumconstantstovirialcoefficientsisin factmorecomplex.WewillusetheapproachofWoolleybut trun-catetheVEoSafterthesixthvirialcoefficientforpracticalreasons. Numericalsimulations showedthattheuseofhighervirial coeffi-cients canbe neglected. In fact, theimpact ofthe 5thand espe-ciallythe6thvirialcoefficientisalreadyverysmall.Thefollowing equationsareused.

PressureexplicitVEoS:

B=Bph− K2RT (3) C=



4K2 2− 2K3



(

RT

)

2 (4) D=



−20K3 2+18K2K3− 3K4



(

RT

)

3 (5) E=



112K4 2+18K32− 144K3K22+32K2K4− 4K5



(

RT

)

4 (6) F =



−672K5 2+1120K23K3− 315K2K32− 280K22K4 +60K3K4+ 50K2K5− 5K6



(

RT

)

5 (7)

VolumeexplicitVEoS:

B=Bph/

(

RT

)

− K2 (8) C=



3K2 2− 2K3



(9) D=



−10K3 2+12K2K3− 3K4



(10) E=



35K4 2+10K32− 60K3K22+20K2K4− 4K5



(11) F=



−126K5 2+280K23K3− 105K2K32− 105K22K4 +30K3K4+30K2K5− 5K6



(12)

The pressuredependencyofCp canbe calculatedwiththe

fol-lowingequations[37] incombinationwiththeVEoS.

Cp− C0p=



∂



H− H0



∂

T



P (13) H− H0₌



_{A− A}0



_+T



_{S− S}0



_+RT

₍

_{Z− 1}

₎

₍₁₄₎ A− A0_{= −}V_∫ ∞



p−RT V



dV− RT ln

V V0

(15) S− S0₌ V_∫ ∞



∂

p

∂

T

V −R V



dV+Rln

V V0

(16)

Vapour density expressed as apparent molecular weight fol-lowsdirectlyfromthecompressibilityfactorascalculatedwiththe VEoS.

Mw,app= Mw/Z (17)

Thespeedofsoundwascalculatedwiththefollowingequations incombinationwiththeVEoS[38] .

c=



Cp CV

₁ Mw

 ∂

P

∂ρ

T



0.5 (18) CV =−T



∂

2_A

∂

T2

ρ (19) Cp=CV+ T

(

∂

P/

∂

T

)

2_ρ

ρ

2

(

∂

_P/

∂

ρ

)

T (20)

For the calculation of the excess molar enthalpy we follow a slightly different approach as used by [25] . Eqs. (14) , (15) and

(16) were firstused to calculate theenthalpy departuresof pure methanolvapourandpurenitrogen. Next,theenthalpydeparture was calculated for the equimolar mixture. The excess molar en-thalpywascalculatedwithEq. (21) .

HmE=

(

H− Ho

)

mix− 0.5



(

H− Ho

₎

CH3OH+

(

H− H o

₎

N2



(21)

Forthe modellingofthe isothermal Joule-Thomsoncoefficient wereferto[27 ,28 ].Forthemodellingofthethermalconductivity coefficientswefollowtheapproachreportedby[19] .

3.3. Third-lawentropyanalysisanddeterminationoftheenthalpyof formation

Inthissectionwefirstpresentthecalculationframeworkofthe thirdlawentropyofmethanol.Here,wedonotlimitourselvesto thecalculation ofthe“direct” Sm0(CH3OH,g, 298.15K)-value,but

wewillalsocalculateentropy-valuesathighertemperatures. Sub-sequently,thesevaluescanbeconvertedto298.15Kwiththeuse of the ideal-gasheat capacity asa function of temperature. This methodprovidesacheckwhetherthecalculationframeworkyields consistent resultsover arange oftemperatures.Ideally, all calcu-latedSm0(CH3OH, g, 298.15K)-valuesshould be identical,

regard-lesswhether298.15Koranothertemperaturewaschosenforthe transitionfromliquidtovapour.

Forthiscalculation framework the following accurate ingredi-entsarerequired.

a Entropyvalueofliquidmethanolat298.15K.

b Heatofvaporizationofmethanolasafunctionoftemperature. c Vapourpressureofmethanolasafunctionoftemperature. d Heatcapacityofliquidmethanolasafunctionoftemperature. e Ideal-gasheatcapacity ofmethanolasa function of

tempera-ture.

f Virialcoefficientsasafunctionoftemperature.

Non-ideal behaviour of the liquid phase (entropy departure) was analysed with the Reference EoS of de Reuck & Craven

[38] andturnedouttobenegligiblefortheconditionsstudied. Ada.Entropyvalueofliquidmethanolat298.15K.

Hereweusethevaluereportedby[39] ,127.19Jmol−1K−1.The reporteduncertaintyequals0.21Jmol−1K−1 [5] .

Adb.Heatofvaporizationofmethanol.

Experimental values (T = 273 K – 400 K) were collected fromseveralsources[6 ,14 ,40–53 ].Experimentalvaluesobtainedat highertemperatures were not used, becausethese are less accu-rate[38] andnotrelevant forourinvestigation.Subsequently the mostaccuratedatawereselected(fordetailsthereaderisreferred totheSupplementaryData)andfittedwiththefollowingequation

[54] .AnoverviewofthefittedparametersofEqs. (22) -(25) isgiven inTable 1 . ln



v

apH a7

= ln

(

1− Tr

)

6  k=0 akTrk (22)

Theconsistencyofthedataandthegoodnessoffitareexcellent (102∗_AAD=0.08).

Adc.Vapourpressureofmethanol.

Experimentalvapour pressures(T= 273K– 400 K)were col-lected fromseveralsources [55-87] .Experimental valuesatlower andhighertemperatureswerenotusedforaccuracyreasons[38] .

Table 1

Parameter values Eqs. (22) - 25 .

k a k ( eq. (22) ) b k ( eq. (23) ) c k ( eq. (24) ) d k ( eq. (25) )

0 −1.35783768 10 +2 −8.517543 10 +0 7.862139 10 +0 _{7.1870815 10} +0 1 1.24374169 10 +3 _{−4.129730 10} +0 3.628199 10 +0 _{- 1.5624003 10} +1 2 −4.72308840 10 +3 −4.323712 10 +1 - 1.582552 10 +1 3.2041794 10 +1 3 9.52988005 10 +3 _{1.503914 10} +2 _{2.610463 10} +1 _{- 2.1506775 10} +1 4 −1.07762280 10 +4 −4.485880 10 +2 _{5.2126400 10} +0 5 6.47641896110 +3 6.126436 10 +2 6 −1.61638777 10 +3 −3.833550 10 +2 7 4.42785340 10 +1 Tc = 512.6 K; p c = 8.1035 MPa [38] .

Except for a 7 (kJ mol −1 ) all parameters are dimensionless.

Again, we selected the most accurate data (see Supplementary Datafordetails)tobefittedasapolynomialfunctionof(1-Tr).

ln

pvap pc

= 6  k=0 bk

(

1− Tr

)

k (23)

Theconsistencyofthedataandthegoodnessoffitareexcellent (102∗_AAD=0.09).

Add.Heatcapacityofliquidmethanol.

Hereweusetherelationshipfortheliquidheatcapacityat sat-urationasreportedby[88] ,whichmatchestheexperimentaldata of [39] very well (102∗_{AAD = 0.08). The results were fitted asa}

polynomialfunctionofTr(T=280K– 362K). CL,sat/R= 3  k=0 ck

_T Tc

k (24)

Ade.Ideal-gasheatcapacityofmethanol.

Here we useCp0-data asreported by [89] in thetemperature

range298.15– 600K.Thesedatawerefittedasapolynomial func-tion of Tr. Cp0-data taken from other sources are almost

identi-cal.Forinstance[34] usesthesamedataas[89] ,while[38] report slightlylowervalues.

C0 p/R= 4  k=0 dk

_T Tc

k (25)

Adf.Virialcoefficientsasafunctionoftemperature.

Virialcoefficientscan becalculated fromtheassociation equi-libriumconstantsoftheassociationmodelincombinationwiththe secondvirialcoefficientforphysicalnon-idealgasbehaviouras de-scribedintheprevioussection.

The sameingredients can also be used tobuild a comparable calculationframeworkfortheenthalpyofformation.Here,wehave to choose a “starting”-value for࢞fHm0(CH3OH, l, 298.15 K).

Wil-hoitetal.[5] havepresentedacomprehensiveapproachregarding thisitem, buttheir conclusion isbased on theaverage resultsof Rossini[11] andChao & Rossini[12] asstatedbefore.Wealready concludedthattheheatofcombustionofChao&Rossinileadsto an incompatible enthalpy of formation value. Therefore, we cor-rectedthe _࢞fHm0(CH3OH, l, 298.15 K)- value of Wilhoitetal. to

−238.614kJ mol−1 correspondingwith[11] .Thisstartingvalue is almostequaltothecorrespondingresultofDomalski[13] .

4. Determinationofthemodelparameters

Preliminarycalculationsshowedthattuningthe1–2–4 associa-tionmodelonheatcapacitydataisdifficultbecauseofthe domi-nanteffectoftetramerization.Infact,thishasalsobeenreported

Fig. 4. Heat capacity data of Weltner & Pitzer [6] and model results. The experimental data at the highest temperature were not used for the parameter optimization.

Fig. 6. Sound velocity data of Boyes et al. [26] and model results.

by[6] .Toovercomethisproblem,wedecidedtobasethe parame-ter optimizationonanobjectivefunctioncomposedofthe follow-ingelements.

a Fitofexperimentalheatcapacitydata[6 ,14 ].

b Fitofexperimentalthermalconductivitycoefficients[19] . c Fit of experimental apparent molecular weight data [9] (p <

115kPa).

d Fitofexperimentalspeedofsounddata[26] .

e Fitofexperimentalexcessmolarenthalpy(N2andCH3OH)data

[25] .

f FitofexperimentalisothermalJoule-Thomsoncoefficients[27] . g Fit of enthalpy of vaporization data from eq. (22) using the

Clapeyronequation.

h Fitofliterature[38] secondvirialcoefficientsathigher temper-atures.

iEntropyconsistencywithrespecttotemperature. j Enthalpyconsistencywithrespecttotemperature.

Hereitemsa.-g.arebasedonadirectcomparisonwith exper-imental informationanditemsh.-j.areprimarily usedtoensure (thermodynamic)consistency.

Item a. dealswith the experimental data of Weltner& Pitzer

[6] andCounsell & Lee [14] .The data of [6] at the highest tem-perature (521.2 K) were not used for fitting the parameters, be-cause herethe pressure dependency ofthe heat capacityis very small. Heatcapacitydata of[15–17] are not usedfor the param-eter fit,because these dataare believed to be slightlyless accu-rate.Wealsodecidedtoexcludetheexcessmolarenthalpydataof

[25] at thehighesttemperatures (398.2K and423.2K)fromthe fittingprocedure.Also,inthiscasethepressuredependencyeffect isvery small.Furthermore,a fewoutlierswere identifiedand

ex-Table 2

Enthalpy of vaporization of methanol.

T / (K) vap H / (kJ mol −1 ) 298.15 37.434 306 37.060 313 36.702 321 36.264 329 35.797 337 35.302

cludedfromtheparameter fit([25] :T ₌ 343.2K,p₌ 99.80kPa;

[19] :T=366.6K,p=1431Torr).

Itemg.isbasedonresultsofEq. (22) inthetemperaturerange 298.15– 337K.Theuppertemperaturelimitwaschosen,because the corresponding saturation pressure is roughly 0.1 MPa corre-sponding to themaximum pressure of mostof theexperimental data.Thelowertemperaturelimitwaschosenbecauseofaccuracy considerations. Belowthistemperaturethenumber ofunderlying experimental vapour pressure and enthalpy of vaporization data isverysmall.The(pseudo-experimental)enthalpy ofvaporization valuesarelistedinTable 2 .Modelvaluesoftheenthalpyof vapor-izationwerecalculatedwiththeClapeyronequation(eq. (1) )using dpvap/dT-valuesderivedfromeq. (23) and

ρ

L,sat-values takenfrom

[38] .

Item h. is based on second virial coefficients taken from de Reuck&Craven[38] .Athighertemperaturesthesedatamaybe re-gardedasafairlyaccuratereflectionofexperimentaldata.A com-parison wasmadewithsecond virial coefficientsfromBichetal.

Fig. 7. Excess molar enthalpy of methanol-nitrogen data of Massucci et al. [25] and model results. The experimental data at the two highest temperatures were not used for the parameter optimization.

Fig. 8. Thermal conductivity data of Frurip et al. [19] at lower temperatures and model results.

range450– 500K,butthedataof[38] resultedinaslightlybetter overallmodel-fit.Thefollowingdatawereused:Table 3 .

Items i. and j. take into account that an adequate model should yield (almost) identical Sm0(CH3OH, g, 298.15 K)- and

࢞fHm0(CH3OH,g,298.15K)-valuesforarangeofliquidtovapour

transitiontemperatures(Tvap). Herewe usedatemperaturerange

of T ₌ 298.15 K– 337 K for the samereasons as explained un-deritemf.Thistemperaturerangewasdividedin100equidistant intervals, resultingin 101Tvap-values.ComparingSm0(CH3OH,

di-Table 3

High temperature 2nd virial coefficients of methanol [38] .

T / (K) - B /( RT ) / (Pa −1 ) 450 6.8555 10 −8 460 6.1705 10 −8 470 5.5735 10 −8 480 5.0539 10 −8 490 4.5998 10 −8 500 4.2047 10 −8

rect Sm0(CH3OH, g, 298.15K)-value(obtainedatTvap= 298.15K)

thus providesa measurefortheentropyconsistency.Obviously,a similar consistency calculation can be made for the enthalpy of formation.

The objective function elements are based on the (averaged) sum of squares of relative residuals (S2

rel). Relative residuals (

(ymod-yexp)/yexp )werechosentomitigatethedominantinfluence

oftetramerization.Furthermore,weusesensitivitycorrection fac-torstoprovideabalancedcombinationofthevariouselementsin the overall objective function. These sensitivitycorrection factors were derived asfollows.First the S2

rel-valuesofelementsa. toi.

were calculated assuming ideal gas behaviour (worst case). Sub-sequently,theoptimalfitcanbedeterminedforeachelement in-dividually. From thesedata we define linearrelationshipsforthe expectedS2

rel-valueofeachelementasafunctionof

ζ

,definedas

theapproachtooptimum(eq. (26) ).

S2

rel,exptd= S2rel,IGL−



S2

rel,IGL− S2rel,opt



ζ

(26)

Setting

ζ

atadesiredvaluebetween0and1(e.g.0.995),these relationshipsyieldthecorresponding expectedS2

rel-values,which

canbeusedtodefinesensitivitycorrectionfactors.Preliminary cal-culations showedthatthe optimalfitforeachindividual element yieldsS2

rel-valuesclosetozero,whichallowsforareasonable

sim-plificationby assuming S2

rel,opt = 0.Table 4 givesan overviewof

theresultingsensitivitycorrectionfactors.

Theparameteroptimizationrequiresthesplitoftheoverall sec-ondvirialcoefficientinB_PhandBA.Forthispurpose,BPhwasbased

ontheBerthelotexpressionforthesecondvirialcoefficientbut us-ingadjustableparameters.

Bph= aBph+ bBphT

−2 ₍₂₇₎

The second virial coefficient of nitrogen was fitted with

Eq. (28) basedonresultsderivedfromSpanetal.[90] inthe tem-peraturerange283K– 523K. BN2= 4  k=0 gk

₄₀₃ T

k (28) g0=8.21841 g1=9.94747×10+1 Table 4

Sensitivity correction factors (SCF). Item S 2 rel (worst case) S 2 rel,exptd ( ζ= 0.995) SCF a) Cp 2.73 10 −2 1.36 10 −4 1 λ 1.20 10 −2 _{6.01 10} −5 _2.27 Mw 4.39 10 −5 2.19 10 −7 6.21 10 +2 c 5.92 10 −5 _{2.96 10} −7 _{4.61 10} +2 HE m 1 5.00 10 −3 2.72 10 −2 φ 1 5.00 10 −3 _{2.72 10} −2 vap H 1.636 10 −3 8.18 10 −6 1.67 10 +1 - B /( RT ) 1 5.00 10 −3 _{2.72 10} −2 Sm0 (CH 3 OH, g, 298.15 K) 1.25 10 −5 1.28 10 −8 1.07 10 +4 ࢞f H m0 (CH 3 OH, g, 298.15 K) 3.72 10 −6 3.69 10 −9 3.75 10 +4 a) SCF relative to C

p ; SCF(item) = S 2rel,exptd (Cp) / S 2rel,exptd (item).

g2 =−1.80909×10+2

g3 =1.06833×10+2

g4 =−2.41839×10+1

Thecross-secondvirialcoefficientofmethanol-nitrogenwas fit-tedwithEq. (29) basedonexperimentaldatafrom[91 ,92 ].Again, theadjustableBerthelotexpressionforthesecondvirialcoefficient turnsouttobeuseful.

BCH3OH−N2 = h0+ h2T

−2 ₍₂₉₎

h0 =3.4166×10–5m3mol−1

h2 =−10.4593m3mol−1K2

Themodellingofthermalconductivitycoefficientsrequiresthe accurateknowledgeofthebinarydiffusioncoefficients (monomer-dimerandmonomer-tetramer) asfunctionsoftemperature. Here, we used themodel publishedby Fruripetal.[19] ,multiplied by separatefittingparametersforeachbinarydiffusioncoefficient re-lationship.Furthermore,zeropressurethermalconductivity coeffi-cientshavetobeoptimizedaswell,althoughreasonableinitial val-uescan be estimatedby extrapolating theexperimental

λ

-values towardszeropressureforeachisotherm.Inordertominimizethe numberofextrafittingparameters weused asimple,linear rela-tionshipwithT2_{astheindependentvariable.Thissimplemethod}

wasvalidatedwiththe

λ

1-relationshipof[24] fortherelevant

tem-peraturerange.

Includingthermalconductivityinfittingthemodelthus impli-catesa significant increase offittingparameters. Despitethis ob-vious disadvantage we decided to do so, because it results in a clearersplitof theoverall secondvirial coefficient inBPh andBA,

since thermal conductivity is not B_Ph–dependent contrary to the otherexperimentaldata.

Preliminarycalculationsshowedthatnotallexperimental data are compatible. This was caused by the data of Cheam et al.

[9] and, asexpected, Francis& Phutela[27] .Allother dataturned out toresultina goodandconsistentfit. Therefore,weconclude thattheexperimental dataof[9] mustsufferfromsystematic de-viations,leadingtotoosmallapparentmolecularweights.The fol-lowingresultswereobtainedwithoutthedataofCheametal.and Francis&Phutela.

Bph/



m3_mol−1



₌



2.070010−4− 72.1757



T

(

K

)

−2



(30) K2/



Pa−1



=2.0149× 10−10exp



19075.0 RT/



Jmol−1





(31) K4/



Pa−3



=7.284710−34exp



103226.1 RT/



Jmol−1





(32) PD1,2/



Pam2_s−1



_{=9.4638 10}−6



T

(

K

)

1.9877 (33) PD1,4/



Pam2_s−1



_{=3.7554 10}−6



T

(

K

)

1.9877 (34)

λ

1/



Jm−1s−1K−1



= 6.304810−6+ 3.282310−10



T

(

K

)

2 (35) S0m

(

CH3OH,g,298.15K

)

=239.96



Jmol−1K−1



(36)



fHm0

(

CH3OH,g,298.15K

)

= −200.55



kJmol−1



(37) Table 5 showsthe fittingresultsfor all model-items.To bring thisinto perspectivewe havealso includedthe corresponding

ζ

-values (approach to optimal fit) for all items. We conclude that

Fig. 9. Thermal conductivity data of Frurip et al. [19] at higher temperatures and model results. Table 5 Model fit. Item 10 2∗_{AAD S} 2 rel ζ Cp 0.48 5.04 10 −5 0.998 λ 0.41 2.99 10 −5 _0.999 c 0.04 2.04 10 −7 _0.997 HE m 2.32 8.03 10 −4 0.999 vap H 0.13 3.12 10 −6 0.998 - B /( RT ) 3.78 1.91 10 −3 _0.998 Sm0 (CH 3 OH, g, 298.15 K) 0.009 1.18 10 −8 0.995 ࢞f H m0 (CH 3 OH, g, 298.15 K) 0.004 1.92 10 −9 0.997

thepresentedmodelgivesagoodandconsistentdescriptionofall itemsinvolvedinthemodelfit.

Basedonadditionalmodelcalculationswecannotruleoutthat contributionsfromotherclustersthandimersandtetramersmight play a role,butwe foundthat such modelextensionshavebuta small effect on the resultingentropy and enthalpy-values. More-over,includingotherclustersinthemodelrequiresextra parame-tersto beoptimized resultinginslowconvergenceofthe param-eteroptimization,significantdependencyoninitialparameter val-uesandinsomecasesphysicallyunrealisticresults.

At highertemperatures (T > 400K) theexperimental data do notdiffermuchfromthecorrespondingidealgasvalues.We there-fore concludethat ourmodel yields the mostaccurate non-ideal gasdescriptioninatemperaturerangeof280K– 400Kand fur-thermoreatpressures upto100kPabasedonthemajorityofthe experimentaldata.

Fig. 3 shows the second virial coefficient ofmethanol (model calculations) as a function of temperature including the split in Bph andBA. The resulting pictureis comparablewith Fig. 2

(wa-ter vapour),butinthe caseof methanoltherelative contribution ofBphisevensomewhatlarger,confirmingthenecessitytoinclude

Bph intheVEoS.Furthermore,ourmodelresultsforBcorrespond

reasonablywellwith[38] ,especiallyatlowertemperatures.

Figs. 4–10 show that the model yields a quite accurate de-scription of the experimental data. Furthermore, the resulting

࢞fHm0(CH3OH, g, 298.15K)-value is somewhathigher (less

neg-ative)thantheliterature valuetakenfrom[89] andreasonablyin linewithRuscic[10] .TheresultingSm0(CH3OH,g,298.15K)-value

almost equals the literature value and is in fact slightly higher, contrarytoourinitialexpectations.

Fig. 11 showsthecomparisonbetweenliterature Cp-valuesnot

usedinthefittingprocedureandthecorrespondingmodel predic-tions.Asexpected,thesepredictionsareconsistentwiththe exper-imentalresults.

5. Consistencyanalysis

Theresultingideal-gasentropyandenthalpy valuesare239.96 (± 0.28)Jmol−1 K−1 and

−200.55 (± 0.69) kJ mol−1, respectively. For details regarding the estimated uncertainties (based on the methods described in

[93] )thereaderisreferredtotheSupplementaryData.

Fig. 12 showstheseresultsincomparisonwiththe results de-rivedfromchemicalequilibriumdata.Now,all resultsturnout to be reasonably consistent taking into account the various uncer-tainties. Based on thisanalysis therelationship [1] forKp1° as a

function of temperature can be (slightly) improved by using the following entropy- andenthalpy-values: 239.81 J mol−1 K−1 and −200.06kJmol−1,respectively(CaseSfixedinFigs. 1 and12 ).

TheoptimizedKp1°-relationshipisdefinedas:

ln



K p0 1/



bar−2



= 1 RT/



Jmol−1





6  k=1

(

β

k

(

T

(

K

)

)

k−1 +

β

7T/

(

K

)

ln



T

(

K

)



(38)

β

1=7.34745×10+4

β

2=1.91037×10+2

β

3=3.2443×10−2

β

4=7.0432×10−6

β

5=-5.6053×10−9

β

6=1.0344×10−12

β

7=-6.4364×10+1

Fig. 10. Experimental enthalpy of vaporization data and model results.

Fig. 12. Entropy and enthalpy of formation of methanol (g, 298.15 K).

Results of this analysis and results derived from chemical equilibrium constants (CO + 2 H 2 ⇔ CH 3 OH).

Here

β

1 and

β

2 resultfrom the new entropy-and

enthalpy-values. The other parameters were calculated from [89] and thus remain unchangedascomparedto theprevious relationship

[1] .

Eq. (38) yields the following results at200, 250 and 300 °C: 1.73×10−2_{, 1.65×10}−3 _{and 2.32×10}−4 _bar−2_{. Compared with the}

results presented in [1] these Kp1°-results are slightly different.

The new Kp1°-values are slightly lower than values reported in

[1] at200°Cand250°Cbyfactors0.976and0.998respectively.At 300°CthenewKp1°-valueisslightlyhigherbyafactor1.017.

Nev-ertheless,theadaptationismeaningfulfromaconsistencypointof viewandalsoresultsinaslightaccuracyimprovement.

6. Conclusions

Animprovedthirdlawentropyandenthalpyofformation anal-ysis is presented based on modelling the temperature and pres-suredependency ofseveralexperimental data, includingheat ca-pacity, heat of mixing (methanol andnitrogen), thermal conduc-tivity, speed ofsound andenthalpy ofvaporization. Furthermore, consistenttemperaturedependenciesofbothentropyandenthalpy weretakenintoaccountaswellashigh-temperaturesecondvirial coefficients.Alldataweretakenorderivedfromtheliterature.

Inlinewithmostotherliteratureonthissubjectwefoundthat a model based on dimerization and tetramerization of methanol moleculesiscapableofdescribingtheexperimentalfindings. How-ever,wealsofoundthatphysicalnon-idealgasbehaviourincluding temperaturedependencymustbetakenintoaccountaswell.Here, a physicalcontribution to the second virial coefficient was suffi-cient to obtain accurate model results.The novelty ofthis study liesinthedevelopmentofahighlyaccuratevirialequationofstate

for low pressure methanol vapour resulting in an accurate fit of variousexperimentaldatasources.

Theresultingthirdlawentropyvalueequals239.96Jmol−1K−1 (T ₌298.15K;p0 _{= 0.1MPa; idealgas) withan estimated}

uncer-tainty of ± 0.28 Jmol−1 K−1. Thisentropy value isonly slightly higherthanthecurrentliteraturevalue.

For the enthalpy of formation we recommend a value of −200.55kJmol−1(T=298.15K;idealgas)basedonheatof com-bustion data from Rossini [11] , further analysis of Wilhoit et al.

[5] and corrections based on our model for non-ideal gas be-haviour as described in this paper. Here, we estimate an uncer-taintyof± 0.69kJmol−1.

These entropy and enthalpy values turn out to be consistent withresultsderived fromexperimental chemicalequilibriumdata formethanolsynthesis.An improvedrelationshipforKp1° results

fromouranalysisenablingreliableandaccurate chemical equilib-riumcalculationsforthemethanolsynthesisprocess.

This research did not receive any specific grant from funding agenciesinthepublic,commercial,ornot-for-profitsectors.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Supplementarymaterials

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.fluid.2020.112851 .

CRediTauthorshipcontributionstatement

G.H. Graaf:Conceptualization, Methodology, Writing - original draft.J.G.M. Winkelman:Conceptualization,Methodology,Writing -review&editing.

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