University of Groningen Methanol Graaf, Geert; Winkelman ......G.H. Graaf and J.G.M. Winkelman Fluid...

17
University of Groningen Methanol Graaf, Geert; Winkelman, Jos G M Published in: Fluid Phase Equilibria DOI: 10.1016/j.fluid.2020.112851 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2021 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): 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 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 05-06-2021

Transcript of University of Groningen Methanol Graaf, Geert; Winkelman ......G.H. Graaf and J.G.M. Winkelman Fluid...

  • University of Groningen

    MethanolGraaf, Geert; Winkelman, Jos G M

    Published in:Fluid Phase Equilibria

    DOI:10.1016/j.fluid.2020.112851

    IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

    Document VersionPublisher's PDF, also known as Version of record

    Publication date:2021

    Link to publication in University of Groningen/UMCG research database

    Citation for published version (APA):Graaf, G., & Winkelman, J. G. M. (2021). Methanol: Association behaviour, third-law entropy analysis anddetermination of the enthalpy of formation. Fluid Phase Equilibria, 529, [112851].https://doi.org/10.1016/j.fluid.2020.112851

    CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

    Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

    Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

    Download date: 05-06-2021

    https://doi.org/10.1016/j.fluid.2020.112851https://research.rug.nl/en/publications/methanol(a6c89630-8c12-4306-8f9b-09977c53fee4).htmlhttps://doi.org/10.1016/j.fluid.2020.112851

  • Fluid Phase Equilibria 529 (2021) 112851

    Contents lists available at ScienceDirect

    Fluid Phase Equilibria

    journal homepage: 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

    Chemical equilibrium constants for the methanol from CO/H 2 reaction calculated from literature ther-

    modynamic parameters are too high as compared to experimental results. Therefore, both the entropy

    value and the enthalpy of formation value of methanol were reviewed. A rigorous analysis of the associa-

    tion behaviour of methanol vapour confirms that dimers and cyclic tetramers play an important role, but

    physical non-ideal gas behaviour must also be taken into account. Accurate relationships for the asso-

    ciation equilibrium constants and the physical contribution to the second virial coefficient were derived

    with the use of a combined fit of multiple experimental data sources including heat capacity, speed of

    sound, thermal conductivity, excess molar enthalpy of methanol and nitrogen and heat of vaporization.

    Additionally, high temperature second virial coefficients and measures for the consistent temperature de-

    pendencies of the entropy and enthalpy of formation were included in the parameter optimization to

    support the accuracy of the model. All experimental results and supporting data were taken or derived

    from the literature. The resulting virial equation of state was used to calculate new ideal-gas entropy and

    enthalpy of formation values of methanol, which now turn out to be consistent with values derived from

    experimental chemical equilibrium data. Furthermore, the new third-law entropy value turns out to be

    consistent with the current literature value. A new enthalpy of formation value is recommended and an

    improved chemical equilibrium relationship for the methanol from CO/H 2 reaction is presented.

    © 2020 The Author(s). Published by Elsevier B.V.

    This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

    A

    A

    g

    l

    S

    V

    V

    1

    m

    l

    i

    c

    r

    r

    s

    u

    c

    S

    G

    e

    v

    C

    t

    T

    s

    m

    s

    c

    h

    0

    bbreviations

    AD Average absolute deviation AAD = 1 N ∗N ∑

    i =1 | y mod,i −y exp,i y exp,i |

    Indicates gas phase

    Indicates liquid phase

    CF Sensitivity correction factor

    EoS Virial equation of state

    LE Vapour liquid equilibrium

    . Introduction

    Recently we published a review on the chemical equilibria in

    ethanol synthesis [1] . It turned out that accurate ideal-gas equi-

    ibrium constant relationships could be derived from thermochem-

    cal basic data taken from the literature in combination with small

    orrections for the Gibbs energy of CO and CH 3 OH to match these

    ∗ Corresponding author: Tel.: + 31-503634484 (secr.) Fax: + 31-503634479. E-mail address: [email protected] (J.G.M. Winkelman).

    m

    t

    u

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

    378-3812/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article

    elationships with an extensive database of experimental equilib-

    ium constants. These Gibbs energy corrections turned out to be

    maller than the uncertainties of the corresponding literature val-

    es.

    However, literature Gibbs energy values are derived from the

    orresponding literature enthalpy of formation and entropy values.

    ince we have not adapted enthalpy values, our corrections of the

    ibbs energy values can be translated directly to changes in the

    ntropy values for CO and CH 3 OH. The resulting ideal-gas entropy

    alues are: 197.83 J mol −1 K −1 ( p 0 = 1 bar; T = 298.15 K) forO, based on the collected experimental equilibrium constants of

    he water-gas shift reaction and 238.03 J mol −1 K −1 ( p 0 = 1 bar; = 298.15 K) for CH 3 OH, based on the collected equilibrium con- tants of the methanol from CO/H 2 reaction. For CO this “experi-

    ental chemical equilibrium” result is consistent with the corre-

    ponding entropy literature value, but for CH 3 OH this is not the

    ase. Here, the literature entropy value equals 239.81 ± 0.17 J ol −1 K −1 [2] . In fact, Chao et al. [3] report an even lower uncer-

    ainty of ± 0.09 J mol −1 K −1 . Craven et al. [4] report experimental ncertainties of ± 0.45 J mol −1 K −1 at 200 K and ± 0.29 J mol

    under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

    https://doi.org/10.1016/j.fluid.2020.112851http://www.ScienceDirect.comhttp://www.elsevier.com/locate/fluidhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.fluid.2020.112851&domain=pdfhttp://creativecommons.org/licenses/by/4.0/mailto:[email protected]://doi.org/10.1016/j.fluid.2020.112851http://creativecommons.org/licenses/by/4.0/

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    K

    t

    s

    l

    v

    C

    p

    m

    c

    v

    t

    (

    c

    t

    m

    r

    c

    w

    a

    t

    m

    t

    m

    p

    a

    u

    t

    s

    t

    2

    2

    Nomenclature

    A Helmholtz energy kJ mol −1

    a 0 – a 7 Parameters of eq. (22)

    a Bph Parameter of eq. (27)

    B Second virial coefficient pressure explicit VEoS m 3

    mol −1

    B ’ Second virial coefficient volume explicit VEoS Pa −1

    b Bph Parameter of eq. (27)

    b 0 – b 6 Parameters of eq. (23)

    C Third virial coefficient pressure explicit VEoS m 6

    mol −2

    C ’ Third virial coefficient volume explicit VEoS Pa −2

    C p Heat capacity at constant pressure J mol −1 K −1

    C V Heat capacity at constant volume J mol −1 K −1

    c Sound velocity m s −1

    c 0 – c 3 Parameters of eq. (24)

    D Fourth virial coefficient pressure explicit VEoS m 9

    mol −3

    D ’ Fourth virial coefficient volume explicit VEoS Pa −3

    d 0 – d 4 Parameters of eq. (25)

    E Fifth virial coefficient pressure explicit VEoS m 12

    mol −4

    E ’ Fifth virial coefficient volume explicit VEoS Pa −4

    F Sixth virial coefficient pressure explicit VEoS m 15

    mol −5

    F ’ Sixth virial coefficient volume explicit VEoS Pa −5

    g 0 – g 4 Parameters of eq. (28)

    H Enthalpy kJ

    H m Molar enthalpy kJ mol −1

    h 0 , h 2 Parameters of eq. (29)

    K p1 Equilibrium constant (CH 3 OH from CO/H 2 reaction)

    Pa −2 or bar −2

    K 2 Equilibrium constant (CH 3 OH dimerization) Pa −1

    K 3 Equilibrium constant (CH 3 OH trimerization) Pa −2

    K 4 Equilibrium constant (CH 3 OH tetramerization) Pa −3

    K 5 Equilibrium constant (CH 3 OH pentamerization)

    Pa −4

    K 6 Equilibrium constant (CH 3 OH hexamerization)

    Pa −5

    M w Molecular weight g mol −1

    p Pressure Pa or bar

    PD ij Pressure ∗ Binary diffusion coefficient Pa m 2 s −1

    R Universal gas constant (8.314463) J mol −1 K −1

    S (Absolute) entropy J K −1

    S m Molar entropy J mol −1 K −1

    S 2 rel Average sum of squares of relative residuals

    T Temperature K

    u c Combined standard uncertainty

    u i Individual standard uncertainty

    V m Molar volume m 3 mol −1

    Z Compressibility factor

    β1 – β7 Parameters in K p 1 °-relationship ( eq. (38) ) δi Individual standard uncertainty of input elements �H Enthalpy change kJ mol −1

    ζ Approach to optimum fit λ Thermal conductivity coefficient J m −1 s −1 K −1

    λ1 Thermal conductivity coefficient at zero pressure J m −1 s −1 K −1

    ρ Molar density mol m −3

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

    s

    2

    Subscripts

    A Association

    app Apparent

    CH 3 OH Indicates component methanol

    c Critical point

    exp Experimental value

    exptd Expected value

    f Of formation

    IGL Indicates ideal gas conditions

    k Indicates parameter number

    L Indicates liquid phase

    m Molar

    mix Indicates mixture

    mod Model result

    N 2 Indicates component nitrogen

    opt Optimum

    ph Physical

    r Reduced conditions relative to the critical point

    rel Relative

    sat At saturation conditions

    vap Vapour

    Superscripts

    E Indicates excess

    0 Indicates ideal gas conditions and at 1 bar

    −1 at 300 - 1000 K. Our “experimental” entropy value is a fac- or 0.993 lower. Although this may still be regarded as a relatively

    mall difference, one should realize that the impact on the equi-

    ibrium constant is quite significant. Using the literature entropy

    alue, the calculated equilibrium constants of the methanol from

    O/H 2 reaction are approximately 1.2 times too high.

    The relevance of this comparison of entropy values is further

    resented in Fig. 1 . Here, we show the 0.99 confidence region of

    ethanol entropy and enthalpy values derived from our chemi-

    al equilibrium analysis [1] in combination with the fitted entropy

    alue (case H fixed), the optimum combination of entropy and en-

    halpy and an alternative fit of the enthalpy of formation value

    case S fixed). Furthermore, we also show the literature values in-

    luding the corresponding uncertainties. Here we assume uncer-

    ainties of 0.29 J mol K −1 for entropy based on [4] and 0.42 kJ ol −1 for enthalpy of formation based on [5] . It is clear that the

    esulting picture is inconsistent: no entropy-enthalpy combination

    an be derived from the chemical equilibrium analysis that fits

    ith the literature results.

    The objective of this paper is to clarify the observed difference

    s presented in Fig. 1 . For this purpose, we will carefully review

    he third-law entropy analysis including the influence of the for-

    ation of associated methanol clusters in the vapour phase. Fur-

    hermore, we will also review the enthalpy of formation value of

    ethanol, because the observed difference might be caused by this

    arameter as well.

    Ideally, this clarification should result in improved entropy

    nd/or enthalpy of formation values of methanol and a consistent

    pdate of the comparison presented in Fig. 1 . It will also allow us

    o confirm or further improve the chemical equilibrium relation-

    hip of the methanol from CO/H 2 reaction, which is important for

    he accurate design of methanol synthesis processes.

    . Literature review

    .1. Entropy

    The methanol literature entropy value is derived from spectro-

    copic data in combination with molecular modelling and statisti-

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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).

    c

    t

    c

    u

    c

    p

    n

    v

    s

    t

    W

    m

    t

    t

    e

    t

    c

    r

    e

    s

    t

    a

    d

    o

    a

    s

    b

    e

    a

    v

    s

    g

    t

    i

    t

    s

    p

    2

    U

    2

    2

    P

    c

    M

    t

    3

    s

    i

    w

    a

    s

    a

    C

    t

    r

    t

    f

    c

    a

    al thermodynamics [2] . Prior to that publication the major uncer-

    ainty for the spectroscopic entropy value was caused by the diffi-

    ult quantification of the hindered rotation along the C –O axis and

    ncertainties about the vibrational frequencies. Chen et al. [2] con-

    luded that various investigations in this field had reached an im-

    roved accuracy level justifying a recalculation of the thermody-

    amic properties of methanol, including the entropy. Moreover,

    arious literature sources showed that non-ideal gas behaviour

    hould be taken into account when calculating the third-law en-

    ropy value. The most important publication in this respect is from

    eltner & Pitzer [6] , who concluded from heat capacity measure-

    ents that methanol vapour is a mixture of monomers, dimers and

    etramers (hydrogen bonding, the 1–2–4 model). They were able

    o estimate the corresponding temperature-dependant association

    quilibrium constants from their experimental results in combina-

    ion with experimental data from other sources. Subsequently, the

    orresponding virial coefficients were derived from these equilib-

    ium constants, allowing for the calculation of the non-ideal gas

    ntropy-change. Weltner & Pitzer already concluded that the re-

    ulting third-law entropy value corresponded well with the “spec-

    roscopic” value including the quantification of torsional rotation,

    lthough the latter subject was still somewhat uncertain in those

    ays. Chen et al. reviewed the work of Weltner & Pitzer and several

    ther publications on methanol association in the vapour phase

    nd reached the same conclusion more firmly. They calculated the

    spectroscopic” thermodynamic properties and showed that the re-

    ulting entropy-values agree reasonably well with third-law values

    ased on the 1–2–4 association model of Weltner & Pitzer. Chen

    t al. [2] report S m 0 (CH 3 OH, g, 298.15 K) = 239.70 J mol −1 K −1

    t a reference pressure of 1 atm, corresponding with the reported

    alue of Chao et al. [3] of 239.81 J mol −1 K −1 at 1 bar, based on aimilar review.

    W

    3

    Since the works of [2] and [3] no updates on S m 0 (CH 3 OH,

    , 298.15 K) derived from spectroscopic data have been made

    o our knowledge. On the other hand, an increasing popular-

    ty of ab initio quantum mechanical calculations is visible in

    he literature. It can be seen that these entropy-results are

    omewhat lower than the “spectroscopic” value. Barone [7] re-

    orted calculated S m 0 (CH 3 OH, g, 298.15 K)-values of 239.24 and

    39.45 J mol −1 K −1 (corrected to p 0 = 1 bar). More recently, mer & Leonhard [8] have reported an ideal-gas entropy-value of

    39.46 J mol −1 K −1 . A closer look at the reported third law S m

    0 (CH 3 OH, g,

    98.15 K)-values based on the association model of Weltner &

    itzer shows a somewhat different tem perature-dependency as

    ompared to the “spectroscopic” values as reported by Chen et al.

    oreover, Weltner & Pitzer have quantified their 1–2–4 associa-

    ion model based on heat capacity measurements in a T-range of

    45.6 K – 521.2 K. Extrapolating the resulting virial equation of

    tate (VEoS) to 298.15 K therefore introduces an extra uncertainty

    n the calculation of the third law entropy-value. In this respect

    e ask ourselves whether the VEoS can be improved by including

    temperature dependent description of physical (in addition to as-

    ociation) non-ideal gas behaviour.

    Our preliminary analysis showed that slightly retuning the VEoS

    s determined by Weltner & Pitzer using P ρT -experiments from heam et al. [9] results in lower entropy-values as compared to

    he current literature value and reasonably in line with the more

    ecent quantum mechanical results [ 7 , 8 ]. Cheam et al. measured

    he vapour density in terms of apparent molecular weight as a

    unction of pressure at 298.15 K with the use of an accurate mi-

    robalance. The experimental set-up was constructed in a way that

    dsorption effects of methanol are (assumed to be) compensated.

    e conclude that it is worthwhile to review the third law entropy

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    v

    h

    2

    r

    m

    u

    t

    d

    s

    v

    2

    2

    m

    h

    (

    m

    c

    e

    b

    s

    o

    i

    s

    a

    i

    a

    o

    r

    s

    a

    2

    b

    s

    i

    q

    b

    j

    a

    o

    P

    i

    t

    w

    m

    a

    t

    t

    b

    r

    p

    i

    n

    v

    o

    t

    [

    m

    o

    b

    z

    d

    H

    s

    e

    o

    t

    t

    (

    c

    v

    d

    t

    e

    2

    s

    u

    a

    i

    e

    t

    T

    s

    e

    u

    i

    t

    a

    l

    a

    3

    3

    y

    alue and especially the necessary correction for non-ideal gas be-

    aviour.

    .2. Enthalpy of formation

    With regard to enthalpy of formation Ruscic [10] has recently

    eported a �f H m 0 (CH 3 OH, g, 298.15 K) -value of −200.71 ± 0.16 kJ

    ol −1 , which seems to be more accurate than other literature val- es. Moreover, this value narrows the gap between literature en-

    ropy data and the entropy-value derived from experimental K p 1 0 -

    ata. Nevertheless, a significant portion of this gap still remains,

    ince the experimental K p 1 -data in combination with the �f H m 0 -

    alue reported by Ruscic lead to an entropy-value of approximately

    38.6 J mol −1 K −1 . A more detailed analysis of literature �f H m

    0 (CH 3 OH, g,

    98.15 K)-values shows that these are based on the following pri-

    ary sources. The first source is Rossini [11] and deals with the

    eat of combustion of gaseous methanol, while the second source

    Chao & Rossini) [12] deals with the heat of combustion of liquid

    ethanol. It turns out that literature �f H m 0 -values are in many

    ases based on (an average result of) both primary sources. How-

    ver, �f H m 0 -values derived from these two primary sources differ

    y approximately 0.84 kJ mol −1 and the value of Ruscic [10] is rea- onably in line with Rossini [11] as can be derived from the works

    f Domalski [13] and Wilhoit et al. [5] . If the enthalpy of formation

    s derived from Chao & Rossini [12] a value of −201.4 kJ mol −1 re-ults, which is clearly incompatible with our chemical equilibrium

    nalysis (see Fig. 1 ). It must be noted however that no clarity ex-

    sts with regard to the observed difference: both primary sources

    re considered to be reliable and the ratio of the underlying heat

    f combustion values is close to unity (1.001). Nevertheless, the

    esult of Ruscic [10] and our own chemical equilibrium analysis

    trongly indicate that Rossini [11] should be preferred over Chao

    nd Rossini [12] .

    .3. Experimental data relevant for quantifying the association

    ehaviour of methanol vapour

    Based on the preliminary analysis described in the previous

    ection we conclude that correcting for non-ideal gas behaviour

    s an essential part of an accurate third-law entropy analysis and

    uantifying the necessary virial coefficients should preferably be

    ased on a greater and more diverse set of experimental data than

    ust the heat capacity data of Weltner & Pitzer. Moreover, this is

    lso true for an accurate determination of the ideal-gas enthalpy

    f formation value.

    Nevertheless, we regard the experimental data of Weltner &

    itzer to be reliable, since these are consistent with similar exper-

    mental data from other sources [ 14 – 17 ].

    The experimental vapour density data of Cheam et al. have nei-

    her been confirmed or questioned by other researchers as far as

    e know. Bich et al. [18] have presented an overview of experi-

    entally based second virial coefficients for methanol vapour as

    function of temperature. Their overview shows that experimen-

    al data diverge significantly at lower temperatures ( T < 400 K). In

    he case of p ρT measurements the underlying cause is believed to e adsorption of methanol vapour. However, Bich et al. have not

    eviewed the results of Cheam et al. in their publication.

    An analysis of other literature sources containing relevant ex-

    erimental data shows that the following sources may also be used

    n an update of the third-law entropy analysis and for the determi-

    ation of the corresponding ideal-gas enthalpy of formation value.

    Like heat capacity also the thermal conductivity of methanol

    apour is strongly dependant on temperature and pressure based

    n the formation of association clusters. Reliable experimental

    hermal conductivity data have been reported by Frurip et al.

    4

    19] and these authors were able to tune the 1–2–4 association

    odel with their experimental data using the Butler-Brokaw the-

    ry [20–22] . Unfortunately, some extra model parameters have to

    e fitted or estimated as well (thermal conductivity coefficients at

    ero pressure and the relevant diffusion coefficients). Thermal con-

    uctivity of methanol vapour has also been investigated by [23] .

    owever, these experimental data are not consistent with the re-

    ults of Frurip et al. and are considered to be less reliable (Sykioti

    t al., [24] ).

    Massucci et al., [25] have measured the excess molar enthalpy

    f nitrogen and methanol vapour in a flow calorimeter. Also, in

    his case the 1–2–4 model was applied successfully, but modelling

    he observed heat effects is relatively complex because it requires

    amongst others) an estimation of the cross-second virial coeffi-

    ient (N 2 – CH 3 OH).

    Boyes et al. [26] have measured the speed of sound of methanol

    apour at various temperatures and pressures. The pressure depen-

    ency of their experimental data is most likely related to associa-

    ion behaviour.

    Francis & Phutela [27] measured the isothermal Joule-Thomson

    ffect of methanol and ethanol vapour. They concluded that the 1–

    –4 model is not capable of describing their experimental data. In-

    tead of using a dimerization equilibrium constant, these authors

    sed (and tuned) a relationship for the second virial coefficient

    s a function of temperature in combination with the tetramer-

    zation equilibrium constant. Bich et al. [18] indicate that these

    xperimental data may be less reliable based on a later publica-

    ion of Francis [28] dealing with an update of isothermal Joule-

    homson coefficients of ethanol vapour. These latter results differ

    ignificantly from the results of the earlier publication, which is

    xplained by the fact that an improved flow calorimeter was used.

    Finally, the enthalpy of vaporization can be calculated with the

    se of the Clapeyron equation ( eq. (1) ) and compared with exper-

    mental values.

    v ap H = d p v ap dT

    ( 1 / ρv ap, sat − 1 / ρL, sat ) T (1) For methanol accurate experimental vapour pressure data, en-

    halpy of vaporization data and saturated liquid density data are

    vailable. The necessary saturated vapour densities can be calcu-

    ated with an appropriate equation of state.

    From this literature review we draw the following conclusions

    nd research questions.

    a Although the 1–2–4 model seems to be the preferred associa-

    tion model, no unambiguous confirmation is available for a fi-

    nal conclusion regarding this statement. At least the tuning of

    the model parameters, especially for the second virial coeffi-

    cient and the dimerization equilibrium, is an aspect that de-

    serves further attention and analysis.

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

    [9] could contribute to an accurate third-law entropy analysis.

    Their data are obtained at 298.15 K, which is ideal in this re-

    spect. It is worthwhile to investigate whether the association

    model can be tuned with the combined experiments of Cheam

    et al. and the various other experimental data.

    c A proper association model should result in a consistent tem-

    perature dependency of the entropy values derived from the

    third-law analysis and this must also hold for the enthalpy of

    formation.

    . Model description

    .1. Dimerization

    As pointed out in the previous section it is worthwhile to anal-

    se the aspect of dimerization in detail. Since the dimerization

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    e

    fi

    c

    I

    h

    c

    e

    T

    c

    I

    d

    e

    c

    t

    t

    a

    i

    p

    f

    h

    v

    a

    i

    w

    (

    t

    t

    a

    p

    s

    i

    T

    H

    Z

    e

    0

    a

    b

    e

    p

    3

    c

    t

    o

    c

    g

    v

    d

    t

    b

    p

    (

    quilibrium constant is mainly reflected in the second virial coef-

    cient, the question is whether “physical” (i.e. not related to asso-

    iation) non-ideal gas behaviour should be included in the model.

    n this context it is useful to take a look at other compounds ex-

    ibiting dimerization like water and ammonia. For these and other

    ompounds [ 29 , 30 ] have compared experimental second virial co-

    fficients with values calculated from the Berthelot equation [31] .

    hey show that experimental B -values are more negative than the

    alculated ones and the differences are attributed to dimerization.

    n fact, similar comparisons for compounds that do not exhibit

    imerization (e.g. trimethyl amine and triethyl amine) show that

    xperimental and Berthelot B -values correspond reasonably well.

    We have analysed this approach and conclude that in many

    ases Berthelot B -values as calculated with eq. (2) give a fair reflec-

    ion of the physical non-ideal gas behaviour. However, we found

    hat this approach does not predict correct B ph -values for associ-

    ting compounds when the critical properties of the compound of

    nterest are used. Correct B ph -values can be determined by using

    roperly tuned effective critical properties ( T c and p c ). These ef-

    ective critical properties may be regarded to reflect a hypothetical

    omomorph of the compound of interest.

    B p c

    R T c = 9

    128 − 54 / 128

    T 2 r (2)

    Fig. 2 shows the results of water vapour, where the second

    irial coefficient is plotted as a function of temperature both for an

    ccurate representation of experimental results [32] and the split

    n dimerization and physical effects. Here, the dimerization results

    ere taken from Wormald [33] , who studied the heat of mixing of

    water + nitrogen) and (water + oxygen). The difference between he total second virial coefficient and the part related to dimeriza-

    Fig. 2. Second virial coefficient of wat

    5

    ion ( B A = - RTK 2 ) thus equals B ph and the calculated B ph -values areccurately described with eq. (2) when using T c = 351.17 K and c = 5.348 MPa. It is interesting to realize that these values rea- onably resemble the critical properties of fluoromethane, which is

    ndeed a possible homomorph for water. Poling et al. [34] report

    c = 315.0 K, p c = 5.548 MPa and Z c = 0.240 for fluoromethane.ere, it should be noted that the Berthelot equation is based on

    c = 9/23 ≈ 0.281 as pointed out by Mathias [35] . Furthermore, q. (2) is applicable at reduced temperatures above approximately

    .8 [35] . This condition is fulfilled for our analysis of water vapour

    s can be seen in Fig. 2 .

    From this analysis we conclude that “physical” non-ideal gas

    ehaviour cannot be neglected and must be incorporated in mod-

    lling the effects of association behaviour on e.g. the pressure de-

    endency of heat capacity.

    .2. Virial equation of state

    To model the experimental data with the exception of thermal

    onductivity data we will use the virial equation of state based on

    he 1–2–4 association model and taking into account the effects

    f physical non-ideal gas behaviour as well. Here, the second virial

    oefficient, B , is a summation of B Ph (reflecting physical non-ideal

    as behaviour) and B A (reflecting dimerization: B A = - RTK 2 ). Higher irial coefficients are based on association only, assuming that the

    escription of physical non-ideal gas behaviour does not require

    he use of higher virial coefficients. This assumption is reasonable,

    ecause our study only deals with low pressure conditions. A sim-

    lified VEoS then results by using B = B A + B ph and calculating D the fourth virial coefficient) from the tetramerization equilibrium

    er as a function of temperature.

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    c

    t

    f

    c

    N

    c

    c

    e

    B

    C

    D

    E

    F

    B

    C

    D

    E

    F

    l

    C

    H

    A

    S

    l

    V

    M

    i

    c

    C

    C

    s

    (

    m

    w

    t

    H

    w

    c

    3

    f

    t

    t

    w

    s

    o

    m

    c

    l

    l

    t

    e

    w

    [

    r

    f

    h

    r

    m

    t

    [

    i

    l

    (

    l

    a

    onstant. However, Woolley [36] has shown that the exact transla-

    ion of association equilibrium constants to virial coefficients is in

    act more complex. We will use the approach of Woolley but trun-

    ate the VEoS after the sixth virial coefficient for practical reasons.

    umerical simulations showed that the use of higher virial coeffi-

    ients can be neglected. In fact, the impact of the 5th and espe-

    ially the 6th virial coefficient is already very small. The following

    quations are used.

    Pressure explicit VEoS:

    = B ph − K 2 RT (3)

    = (4 K 2 2 − 2 K 3

    )( RT )

    2 (4)

    = (−20 K 3 2 + 18 K 2 K 3 − 3 K 4

    )( RT )

    3 (5)

    = (112 K 4 2 + 18 K 2 3 − 144 K 3 K 2 2 + 32 K 2 K 4 − 4 K 5

    )( RT )

    4 (6)

    = (−672 K 5 2 + 1120 K 3 2 K 3 − 315 K 2 K 2 3 − 280 K 2 2 K 4

    + 60 K 3 K 4 + 50 K 2 K 5 − 5 K 6 )( RT )

    5 (7)

    Volume explicit VEoS:

    ′ = B ph / ( RT ) − K 2 (8)

    ′ = (3 K 2 2 − 2 K 3

    )(9)

    ′ = (−10 K 3 2 + 12 K 2 K 3 − 3 K 4

    )(10)

    ′ = (35 K 4 2 + 10 K 2 3 − 60 K 3 K 2 2 + 20 K 2 K 4 − 4 K 5

    )(11)

    ′ = (−126 K 5 2 + 280 K 3 2 K 3 − 105 K 2 K 2 3 − 105 K 2 2 K 4

    + 30 K 3 K 4 + 30 K 2 K 5 − 5 K 6 )

    (12)

    The pressure dependency of C p can be calculated with the fol-

    owing equations [37] in combination with the VEoS.

    p − C 0 p = (

    ∂ (H − H 0

    )∂T

    ) P

    (13)

    − H 0 = (A − A 0

    )+ T

    (S − S 0

    )+ RT ( Z − 1 ) (14)

    − A 0 = −V

    ∫ ∞

    [ p − RT

    V

    ] dV − RT ln

    (V

    V 0

    )(15)

    − S 0 = V

    ∫ ∞

    [(∂ p

    ∂T

    )V

    − R V

    ]dV + R ln

    (V

    V 0

    )(16)

    Vapour density expressed as apparent molecular weight fol-

    ows directly from the compressibility factor as calculated with the

    EoS.

    w, app = M w /Z (17) The speed of sound was calculated with the following equations

    n combination with the VEoS [38] .

    = [(

    C p

    C V

    )(1

    M w

    )(∂P

    ∂ρ

    )T

    ]0 . 5 (18)

    V = −T (

    ∂ 2 A

    ∂ T 2

    (19)

    p = C V + T ( ∂ P/∂ T )

    2 ρ

    ρ2 ( ∂ P/∂ ρ) (20)

    T

    6

    For the calculation of the excess molar enthalpy we follow a

    lightly different approach as used by [25] . Eqs. (14) , (15) and

    16) were first used to calculate the enthalpy departures of pure

    ethanol vapour and pure nitrogen. Next, the enthalpy departure

    as calculated for the equimolar mixture. The excess molar en-

    halpy was calculated with Eq. (21) .

    E m = ( H − H o ) mix − 0 . 5

    [( H − H o ) C H 3 OH + ( H − H o ) N 2

    ](21)

    For the modelling of the isothermal Joule-Thomson coefficient

    e refer to [ 27 , 28 ]. For the modelling of the thermal conductivity

    oefficients we follow the approach reported by [19] .

    .3. Third-law entropy analysis and determination of the enthalpy of

    ormation

    In this section we first present the calculation framework of the

    hird law entropy of methanol. Here, we do not limit ourselves to

    he calculation of the “direct” S m 0 (CH 3 OH, g, 298.15 K)-value, but

    e will also calculate entropy-values at higher temperatures. Sub-

    equently, these values can be converted to 298.15 K with the use

    f the ideal-gas heat capacity as a function of temperature. This

    ethod provides a check whether the calculation framework yields

    onsistent results over a range of temperatures. Ideally, all calcu-

    ated S m 0 (CH 3 OH, g, 298.15 K)-values should be identical, regard-

    ess whether 298.15 K or another temperature was chosen for the

    ransition from liquid to vapour.

    For this calculation framework the following accurate ingredi-

    nts are required.

    a Entropy value of liquid methanol at 298.15 K.

    b Heat of vaporization of methanol as a function of temperature.

    c Vapour pressure of methanol as a function of temperature.

    d Heat capacity of liquid methanol as a function of temperature.

    e Ideal-gas heat capacity of methanol as a function of tempera-

    ture.

    f Virial coefficients as a function of temperature.

    Non-ideal behaviour of the liquid phase (entropy departure)

    as analysed with the Reference EoS of de Reuck & Craven

    38] and turned out to be negligible for the conditions studied.

    Ad a. Entropy value of liquid methanol at 298.15 K.

    Here we use the value reported by [39] , 127.19 J mol −1 K −1 . Theeported uncertainty equals 0.21 J mol −1 K −1 [5] .

    Ad b. Heat of vaporization of methanol.

    Experimental values ( T = 273 K – 400 K) were collected rom several sources [ 6 , 14 , 40–53 ]. Experimental values obtained at

    igher temperatures were not used, because these are less accu-

    ate [38] and not relevant for our investigation. Subsequently the

    ost accurate data were selected (for details the reader is referred

    o the Supplementary Data) and fitted with the following equation

    54] . An overview of the fitted parameters of Eqs. (22) - (25) is given

    n Table 1 .

    n

    (�v ap H

    a 7

    )= l n ( 1 − T r )

    6 ∑ k =0

    a k T k

    r (22)

    The consistency of the data and the goodness of fit are excellent

    10 2 ∗AAD = 0.08).

    Ad c. Vapour pressure of methanol.

    Experimental vapour pressures ( T = 273 K – 400 K) were col- ected from several sources [55-87] . Experimental values at lower

    nd higher temperatures were not used for accuracy reasons [38] .

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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

    T c = 512.6 K; p c = 8.1035 MPa [38] . Except for a 7 (kJ mol

    −1 ) all parameters are dimensionless.

    A

    D

    l

    (

    u

    o

    p

    C

    r

    t

    c

    s

    C

    l

    s

    s

    c

    t

    h

    t

    R

    c

    a

    r

    −a

    4

    t

    n

    gain, we selected the most accurate data (see Supplementary

    ata for details) to be fitted as a polynomial function of (1- T r ).

    n

    (p v ap

    p c

    )=

    6 ∑ k =0

    b k ( 1 − T r ) k (23)

    The consistency of the data and the goodness of fit are excellent

    10 2 ∗AAD = 0.09). Ad d. Heat capacity of liquid methanol.

    Here we use the relationship for the liquid heat capacity at sat-

    ration as reported by [88] , which matches the experimental data

    f [39] very well (10 2 ∗AAD = 0.08). The results were fitted as aolynomial function of T r ( T = 280 K – 362 K).

    L,sat /R = 3 ∑

    k =0 c k

    (T

    T c

    )k (24)

    Ad e. Ideal-gas heat capacity of methanol.

    Fig. 3. Second virial coefficient of metha

    7

    Here we use C p 0 -data as reported by [89] in the temperature

    ange 298.15 – 600 K. These data were fitted as a polynomial func-

    ion of T r . C p 0 -data taken from other sources are almost identi-

    al. For instance [34] uses the same data as [89] , while [38] report

    lightly lower values.

    0 p /R =

    4 ∑ k =0

    d k

    (T

    T c

    )k (25)

    Ad f. Virial coefficients as a function of temperature.

    Virial coefficients can be calculated from the association equi-

    ibrium constants of the association model in combination with the

    econd virial coefficient for physical non-ideal gas behaviour as de-

    cribed in the previous section.

    The same ingredients can also be used to build a comparable

    alculation framework for the enthalpy of formation. Here, we have

    o choose a “starting”-value for �f H m 0 (CH 3 OH, l, 298.15 K). Wil-

    oit et al. [5] have presented a comprehensive approach regarding

    his item, but their conclusion is based on the average results of

    ossini [11] and Chao & Rossini [12] as stated before. We already

    oncluded that the heat of combustion of Chao & Rossini leads to

    n incompatible enthalpy of formation value. Therefore, we cor-

    ected the �f H m 0 (CH 3 OH, l, 298.15 K)- value of Wilhoit et al. to

    238.614 kJ mol −1 corresponding with [11] . This starting value is lmost equal to the corresponding result of Domalski [13] .

    . Determination of the model parameters

    Preliminary calculations showed that tuning the 1–2–4 associa-

    ion model on heat capacity data is difficult because of the domi-

    ant effect of tetramerization. In fact, this has also been reported

    nol as a function of temperature.

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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. 5. Heat capacity data of Counsell & Lee [14] and model results.

    8

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

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

    b

    t

    i

    i

    (

    [

    p

    c

    s

    e

    r

    [

    fi

    i

    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

    c

    [

    2

    t

    s

    d

    c

    e

    i

    v

    i

    d

    [

    R

    g

    p

    [

    y [6] . To overcome this problem, we decided to base the parame-

    er optimization on an objective function composed of the follow-

    ng elements.

    a Fit of experimental heat capacity data [ 6 , 14 ].

    b Fit of experimental thermal conductivity coefficients [19] .

    c Fit of experimental apparent molecular weight data [9] ( p <

    115 kPa).

    d Fit of experimental speed of sound data [26] .

    e Fit of experimental excess molar enthalpy (N 2 and CH 3 OH) data

    [25] .

    f Fit of experimental isothermal Joule-Thomson coefficients [27] .

    g Fit of enthalpy of vaporization data from eq. (22) using the

    Clapeyron equation.

    h Fit of literature [38] second virial coefficients at higher temper-

    atures.

    i Entropy consistency with respect to temperature.

    j Enthalpy consistency with respect to temperature.

    Here items a. - g. are based on a direct comparison with exper-

    mental information and items h. - j. are primarily used to ensure

    thermodynamic) consistency.

    Item a. deals with the experimental data of Weltner & Pitzer

    6] and Counsell & Lee [14] . The data of [6] at the highest tem-

    erature (521.2 K) were not used for fitting the parameters, be-

    ause here the pressure dependency of the heat capacity is very

    mall. Heat capacity data of [15–17] are not used for the param-

    ter fit, because these data are believed to be slightly less accu-

    ate. We also decided to exclude the excess molar enthalpy data of

    25] at the highest temperatures (398.2 K and 423.2 K) from the

    tting procedure. Also, in this case the pressure dependency effect

    s very small. Furthermore, a few outliers were identified and ex-

    9

    luded from the parameter fit ( [25] : T = 343.2 K, p = 99.80 kPa;19] : T = 366.6 K, p = 1431 Torr).

    Item g. is based on results of Eq. (22) in the temperature range

    98.15 – 337 K. The upper temperature limit was chosen, because

    he corresponding saturation pressure is roughly 0.1 MPa corre-

    ponding to the maximum pressure of most of the experimental

    ata. The lower temperature limit was chosen because of accuracy

    onsiderations. Below this temperature the number of underlying

    xperimental vapour pressure and enthalpy of vaporization data

    s very small. The (pseudo-experimental) enthalpy of vaporization

    alues are listed in Table 2 . Model values of the enthalpy of vapor-

    zation were calculated with the Clapeyron equation ( eq. (1) ) using

    p vap /d T -values derived from eq. (23) and ρL, sat -values taken from 38] .

    Item h. is based on second virial coefficients taken from de

    euck & Craven [38] . At higher temperatures these data may be re-

    arded as a fairly accurate reflection of experimental data. A com-

    arison was made with second virial coefficients from Bich et al.

    18] , showing a good agreement (10 2 ∗AAD < 3) in the temperature

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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.

    r

    o

    s

    t

    o

    d

    i

    v

    ange 450 – 500 K, but the data of [38] resulted in a slightly better

    verall model-fit. The following data were used: Table 3 .

    Items i. and j. take into account that an adequate model

    hould yield (almost) identical S m 0 (CH 3 OH, g, 298.15 K)- and

    H m 0 (CH 3 OH, g, 298.15 K)- values for a range of liquid to vapour

    f

    10

    ransition temperatures ( T vap ). Here we used a temperature range

    f T = 298.15 K – 337 K for the same reasons as explained un- er item f. This temperature range was divided in 100 equidistant

    ntervals, resulting in 101 T vap -values. Comparing S m 0 (CH 3 OH, g)-

    alues obtained at other T vap –values than 298.15 K with the di-

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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

    r

    t

    s

    f

    s

    (

    o

    t

    t

    w

    w

    s

    d

    e

    t

    S

    r

    c

    c

    y

    p

    t

    o

    o

    i

    B

    E

    p

    B

    T

    S

    t

    t

    t

    B

    a

    d

    w

    s

    l

    c

    u

    t

    n

    t

    w

    p

    c

    v

    c

    s

    o

    a

    [

    o

    t

    v

    l

    F

    B

    K

    K

    ect S m 0 (CH 3 OH, g, 298.15 K)-value (obtained at T vap = 298.15 K)

    hus provides a measure for the entropy consistency. Obviously, a

    imilar consistency calculation can be made for the enthalpy of

    ormation.

    The objective function elements are based on the (averaged)

    um of squares of relative residuals (S 2 rel ). Relative residuals (

    y mod -y exp )/y exp ) were chosen to mitigate the dominant influence

    f tetramerization. Furthermore, we use sensitivity correction fac-

    ors to provide a balanced combination of the various elements in

    he overall objective function. These sensitivity correction factors

    ere derived as follows. First the S 2 rel -values of elements a. to i.

    ere calculated assuming ideal gas behaviour (worst case). Sub-

    equently, the optimal fit can be determined for each element in-

    ividually. From these data we define linear relationships for the

    xpected S 2 rel -value of each element as a function of ζ, defined as he approach to optimum ( eq. (26) ).

    2 rel,exptd = S 2 rel, IGL −

    (S 2 rel,IGL − S 2 rel,opt

    )ζ (26)

    Setting ζ at a desired value between 0 and 1 (e.g. 0.995), these elationships yield the corresponding expected S 2 rel -values, which

    an be used to define sensitivity correction factors. Preliminary cal-

    ulations showed that the optimal fit for each individual element

    ields S 2 rel -values close to zero, which allows for a reasonable sim-

    lification by assuming S 2 rel,opt = 0. Table 4 gives an overview of he resulting sensitivity correction factors.

    The parameter optimization requires the split of the overall sec-

    nd virial coefficient in B Ph and B A . For this purpose, B Ph was based

    n the Berthelot expression for the second virial coefficient but us-

    ng adjustable parameters.

    ph = a B ph + b B ph T −2 (27) The second virial coefficient of nitrogen was fitted with

    q. (28) based on results derived from Span et al. [90] in the tem-

    erature range 283 K – 523 K.

    N 2 = 4 ∑

    k =0 g k

    (403

    T

    )k (28)

    g 0 = 8.21841 g 1 = 9.94747 ×10 + 1

    able 4

    ensitivity correction factors (SCF).

    Item

    S 2 rel (worst

    case)

    S 2 rel,exptd ( ζ = 0.995) SCF a)

    C p 2.73 10 −2 1.36 10 −4 1

    λ 1.20 10 −2 6.01 10 −5 2.27 M w 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

    H E 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

    S m 0 (CH 3 OH, g, 298.15 K) 1.25 10

    −5 1.28 10 −8 1.07 10 + 4

    �f H m 0 (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 2 rel,exptd (Cp) / S 2 rel, exptd (item).

    P

    P

    λ

    S

    t

    v

    11

    g 2 = −1.80909 ×10 + 2 g 3 = 1.06833 ×10 + 2 g 4 = −2.41839 ×10 + 1

    The cross-second virial coefficient of methanol-nitrogen was fit-

    ed with Eq. (29) based on experimental data from [ 91 , 92 ]. Again,

    he adjustable Berthelot expression for the second virial coefficient

    urns out to be useful.

    C H 3 OH−N 2 = h 0 + h 2 T −2 (29) h 0 = 3.4166 ×10 –5 m 3 mol −1 h 2 = −10.4593 m 3 mol −1 K 2

    The modelling of thermal conductivity coefficients requires the

    ccurate knowledge of the binary diffusion coefficients (monomer-

    imer and monomer-tetramer) as functions of temperature. Here,

    e used the model published by Frurip et al. [19] , multiplied by

    eparate fitting parameters for each binary diffusion coefficient re-

    ationship. Furthermore, zero pressure thermal conductivity coeffi-

    ients have to be optimized as well, although reasonable initial val-

    es can be estimated by extrapolating the experimental λ-values owards zero pressure for each isotherm. In order to minimize the

    umber of extra fitting parameters we used a simple, linear rela-

    ionship with T 2 as the independent variable. This simple method

    as validated with the λ1 -relationship of [24] for the relevant tem- erature range.

    Including thermal conductivity in fitting the model thus impli-

    ates a significant increase of fitting parameters. Despite this ob-

    ious disadvantage we decided to do so, because it results in a

    learer split of the overall second virial coefficient in B Ph and B A ,

    ince thermal conductivity is not B Ph –dependent contrary to the

    ther experimental data.

    Preliminary calculations showed that not all experimental data

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

    9] and, as expected, Francis & Phutela [27] . All other data turned

    ut to result in a good and consistent fit. Therefore, we conclude

    hat the experimental data of [9] must suffer from systematic de-

    iations, leading to too small apparent molecular weights. The fol-

    owing results were obtained without the data of Cheam et al. and

    rancis & Phutela.

    ph / (m 3 mo l −1

    )=

    ( 2 . 0700 10 −4 − 72 . 1757

    (T

    ( K )

    )−2 ) (30)

    2 / (P a −1

    )= 2 . 0149 × 10 −10 exp

    ( 19075 . 0

    RT / (J mo l −1

    ))

    (31)

    4 / (P a −3

    )= 7 . 2847 10 −34 e xp

    ( 103226 . 1

    RT / (J mo l −1

    ))

    (32)

    D 1 , 2 / (P a m 2 s −1

    )= 9 . 4638 10 −6

    (T

    ( K )

    )1 . 9877 (33)

    D 1 , 4 / (P a m 2 s −1

    )= 3 . 7554 10 −6

    (T

    ( K )

    )1 . 9877 (34)

    1 / (J m −1 s −1 K −1

    )= 6 . 3048 10 −6 + 3 . 2823 10 −10

    (T

    ( K )

    )2 (35)

    0 m ( C H 3 OH, g, 298 . 15 K ) = 239 . 96

    (J mo l −1 K −1

    )(36)

    f H 0 m ( C H 3 OH, g, 298 . 15 K ) = −200 . 55

    (kJ mo l −1

    )(37)

    Table 5 shows the fitting results for all model-items. To bring

    his into perspective we have also included the corresponding ζ- alues (approach to optimal fit) for all items. We conclude that

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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

    C p 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 H E 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 S m

    0 (CH 3 OH, g, 298.15 K) 0.009 1.18 10 −8 0.995

    �f H m 0 (CH 3 OH, g, 298.15 K) 0.004 1.92 10

    −9 0.997

    t

    i

    c

    p

    s

    o

    t

    e

    u

    n

    f

    g

    t

    e

    c

    B

    t

    o

    B

    r

    s

    �a

    l

    a

    c

    u

    t

    i

    5

    (

    t

    [

    r

    b

    t

    f

    f

    l

    he presented model gives a good and consistent description of all

    tems involved in the model fit.

    Based on additional model calculations we cannot rule out that

    ontributions from other clusters than dimers and tetramers might

    lay a role, but we found that such model extensions have but a

    mall effect on the resulting entropy and enthalpy-values. More-

    ver, including other clusters in the model requires extra parame-

    ers to be optimized resulting in slow convergence of the param-

    ter optimization, significant dependency on initial parameter val-

    es and in some cases physically unrealistic results.

    At higher temperatures ( T > 400 K) the experimental data do

    ot differ much from the corresponding ideal gas values. We there-

    ore conclude that our model yields the most accurate non-ideal

    as description in a temperature range of 280 K – 400 K and fur-

    hermore at pressures up to 100 kPa based on the majority of the

    xperimental data.

    Fig. 3 shows the second virial coefficient of methanol (model

    alculations) as a function of temperature including the split in

    ph and B A . The resulting picture is comparable with Fig. 2 (wa-

    er vapour), but in the case of methanol the relative contribution

    f B ph is even somewhat larger, confirming the necessity to include

    ph in the VEoS. Furthermore, our model results for B correspond

    easonably well with [38] , especially at lower temperatures.

    Figs. 4–10 show that the model yields a quite accurate de-

    cription of the experimental data. Furthermore, the resulting

    12

    f H m 0 (CH 3 OH, g, 298.15 K)- value is somewhat higher (less neg-

    tive) than the literature value taken from [89] and reasonably in

    ine with Ruscic [10] . The resulting S m 0 (CH 3 OH, g, 298.15 K) -value

    lmost equals the literature value and is in fact slightly higher,

    ontrary to our initial expectations.

    Fig. 11 shows the comparison between literature C p -values not

    sed in the fitting procedure and the corresponding model predic-

    ions. As expected, these predictions are consistent with the exper-

    mental results.

    . Consistency analysis

    The resulting ideal-gas entropy and enthalpy values are 239.96

    ± 0.28) J mol −1 K −1 and −200.55 ( ± 0.69) kJ mol −1 , respectively. For details regarding

    he estimated uncertainties (based on the methods described in

    93] ) the reader is referred to the Supplementary Data.

    Fig. 12 shows these results in comparison with the results de-

    ived from chemical equilibrium data. Now, all results turn out to

    e reasonably consistent taking into account the various uncer-

    ainties. Based on this analysis the relationship [1] for K p 1 ° as a unction of temperature can be (slightly) improved by using the

    ollowing entropy- and enthalpy-values: 239.81 J mol −1 K −1 and 200.06 kJ mol −1 , respectively (Case S fixed in Figs. 1 and 12 ).

    The optimized Kp 1 °-relationship is defined as:

    n (K p 0 1 /

    (ba r −2

    ))= 1

    RT / (J mo l −1

    )[

    6 ∑ k =1

    ( βk ( T

    ( K ) )

    k −1 + β7 T / ( K ) ln

    (T

    ( K )

    )] (38)

    β1 = 7.34745 ×10 + 4 β2 = 1.91037 ×10 + 2 β3 = 3.2443 ×10 −2 β4 = 7.0432 ×10 −6 β5 = - 5.6053 ×10 −9 β6 = 1.0344 ×10 −12 β = - 6.4364 ×10 + 1

    7

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

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

    Fig. 11. Heat capacity data ( p ≈ 100 kPa) of Sinke & De Vries [15] , Strömsöe et al. [16] , de Vries & Collins [17] and model predictions.

    13

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    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).

    v

    t

    [

    1

    r

    T

    [

    3

    e

    v

    6

    y

    s

    p

    t

    c

    w

    c

    a

    m

    e

    t

    a

    c

    l

    f

    v

    (

    t

    h

    − b

    [

    h

    t

    w

    f

    f

    r

    a

    D

    c

    i

    S

    f

    Here β1 and β2 result from the new entropy- and enthalpy- alues. The other parameters were calculated from [89] and

    hus remain unchanged as compared to the previous relationship

    1] .

    Eq. (38) yields the following results at 200, 250 and 300 °C: .73 ×10 −2 , 1.65 ×10 −3 and 2.32 ×10 −4 bar −2 . Compared with the esults presented in [1] these K p 1 °-results are slightly different. he new K p 1 °-values are slightly lower than values reported in 1] at 200 °C and 250 °C by factors 0.976 and 0.998 respectively. At 00 °C the new K p 1 °-value is slightly higher by a factor 1.017. Nev-rtheless, the adaptation is meaningful from a consistency point of

    iew and also results in a slight accuracy improvement.

    . Conclusions

    An improved third law entropy and enthalpy of formation anal-

    sis is presented based on modelling the temperature and pres-

    ure dependency of several experimental data, including heat ca-

    acity, heat of mixing (methanol and nitrogen), thermal conduc-

    ivity, speed of sound and enthalpy of vaporization. Furthermore,

    onsistent temperature dependencies of both entropy and enthalpy

    ere taken into account as well as high-temperature second virial

    oefficients. All data were taken or derived from the literature.

    In line with most other literature on this subject we found that

    model based on dimerization and tetramerization of methanol

    olecules is capable of describing the experimental findings. How-

    ver, we also found that physical non-ideal gas behaviour including

    emperature dependency must be taken into account as well. Here,

    physical contribution to the second virial coefficient was suffi-

    ient to obtain accurate model results. The novelty of this study

    ies in the development of a highly accurate virial equation of state

    14

    or low pressure methanol vapour resulting in an accurate fit of

    arious experimental data sources.

    The resulting third law entropy value equals 239.96 J mol −1 K −1

    T = 298.15 K; p 0 = 0.1 MPa; ideal gas) with an estimated uncer-ainty of ± 0.28 J mol −1 K −1 . This entropy value is only slightly igher than the current literature value.

    For the enthalpy of formation we recommend a value of

    200.55 kJ mol −1 ( T = 298.15 K; ideal gas) based on heat of com-ustion data from Rossini [11] , further analysis of Wilhoit et al.

    5] and corrections based on our model for non-ideal gas be-

    aviour as described in this paper. Here, we estimate an uncer-

    ainty of ± 0.69 kJ mol −1 . These entropy and enthalpy values turn out to be consistent

    ith results derived from experimental chemical equilibrium data

    or methanol synthesis. An improved relationship for K p 1 ° results rom our analysis enabling reliable and accurate chemical equilib-

    ium calculations for the methanol synthesis process.

    This research did not receive any specific grant from funding

    gencies in the public, commercial, or not-for-profit sectors.

    eclaration of competing interest

    The authors declare that they have no known competing finan-

    ial interests or personal relationships that could have appeared to

    nfluence the work reported in this paper.

    upplementary materials

    Supplementary material associated with this article can be

    ound, in the online version, at doi: 10.1016/j.fluid.2020.112851 .

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

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    C

    d

    -

    R

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    RediT authorship contribution statement

    G.H. Graaf: Conceptualization, Methodology, Writing - original

    raft. J.G.M. Winkelman: Conceptualization, Methodology, Writing

    review & editing.

    eferences

    [1] G.H. Graaf , J.G.M. Winkelman , Chemical equilibria in methanol synthesis in-

    cluding the water-gas shift reaction: a critical reassessment, Ind. Eng. Chem. Res. 55 (2016) 5854–5864 .

    [2] S.S. Chen , R.C. Wilhoit , B.J. Zwolinski , Thermodynamic properties of normal

    and deuterated methanols, J. Phys. Chem. Ref. Data 6 (1977) 105–112 . [3] J. Chao , K.R. Hall , K.N. Marsh , R.C. Wilhoit , Thermodynamic properties of key

    organic oxygen compounds in the carbon range C 1 to C 4 . Part 2. Ideal gasproperties, J. Phys. Chem. Ref. Data 15 (1986) 1369–1436 .

    [4] R.J.B. Craven , K.M. de Reuck , Ideal-gas and saturation properties of methanol, Int. J. Thermophys. 7 (1986) 541–552 .

    [5] R.C. Wilhoit , B.J. Zwolinski , Physical and thermodynamic properties of aliphatic

    alcohols, J. Phys. Chem. Ref. Data 2 (Suppl. 1) (1973) 1-44–1-54 . [6] W. Weltner Jr. , K.S. Pitzer , Methyl alcohol: the entropy, heat capacity and poly-

    merization equilibria in the vapour, and potential barrier to internal rotation, J. Am. Chem. Soc. 73 (1951) 2606–2610 .

    [7] V. Barone , Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation, J. Chem. Phys. 120 (2004) 3059–

    3065 .

    [8] M. Umer , K. Leonhard , Ab initio calculations of thermochemical properties of methanol clusters, J. Phys. Chem. A. 117 (2013) 1569–1582 .

    [9] V. Cheam , S.B. Farnham , S.D. Christian , Vapour phase association of methanol.Vapour density evidence for trimer formation, J. Phys. Chem. 74 (1970)

    4157–4159 . [10] B. Ruscic , Active thermochemical tables: sequential bond dissociation en-

    thalpies of methane, ethane, and methanol and the related thermochemistry, J. Phys. Chem. 199 (2015) 7810–7837 .

    [11] F.D. Rossini , The heats of combustion of methyl and ethyl alcohols, NBS J. Res.

    8 (1932) 119–139 RP405 . [12] J. Chao , F.D. Rossini , Heats of combustion, formation and isomerization of nine-

    teen alkanols, J. Chem. Eng. Data 10 (1965) 374–379 . [13] E.S. Domalski , Selected values of the heats of combustion and heats of forma-

    tion of organic compounds containing the elements C, H, N, O, P and S, J. Phys.Chem. Ref. Data 1 (1972) 221–277 .

    [14] J.F. Counsell , D.A. Lee , Thermodynamic properties of organic oxygen com-

    pounds. Vapour heat capacity and enthalpy of vaporization of methanol, J. Chem. Thermodyn. 5 (1973) 583–589 .

    [15] G.C. Sinke , T. De Vries , The heat capacity of organic vapors. VIII. Data for somealiphatic alcohols using an improved flow calorimeter requiring only 25 ml. of

    sample, J. Am. Chem. Soc 75 (1953) 1815–1818 . [16] E. Strömsöe , H.G. Rönne , A.L. Lydersen , Heat capacity of alcohol vapors at at-

    mospheric pressure, J. Chem. Eng. Data 15 (1970) 286–290 .

    [17] T. de Vries , B.T. Collins , The heat capacity of organic vapors. I. Methyl alcohol,J. Am. Chem. Soc. 63 (1941) 1343–1346 .

    [18] E. Bich , H. Hendle , A. Vogel , A new evaluation of p ρT measurements onthe methanol vapour-steam mixture and its pure components with correc-

    tions for adsorption and impurity effects, Fluid Phase Equilib 133 (1997) 129–144 .

    [19] D.J. Frurip , L.A. Curtiss , M. Blander , Thermal conductivity measurements and

    molecular association in a series of alcohol vapors: methanol, ethanol, iso- propanol and t-butanol, Int. J. Thermophys. 2 (1981) 115–132 .

    20] J.N. Butler , R.S. Brokaw , Thermal Conductivity of gas mixtures in chemical equilibrium, J. Chem. Phys. 26 (1957) 1636–1643 .

    [21] R.S. Brokaw , Thermal Conductivity of gas mixtures in chemical equilibrium, II. J. Chem. Phys. 32 (1960) 1005–1006 .

    22] L.A. Curtiss , D.J. Frurip , M. Blander , Studies of hydrogen bonding in the vapour

    phase by measurement of thermal conductivity and molecular orbital calcula- tions, 2,2,2,-trifluoroethanol, J. Am. Chem. Soc. 100 (1978) 79–86 .

    23] T.A. Renner , G.H. Kucera , M. Blander , A study of hydrogen bonding in methanolvapour by measurement of thermal conductivity, J. Chem. Phys 66 (1977)

    177–184 . 24] E.A. Sykioti , M.J. Assaei , M.L. Huber , R.A. Perkins , Reference correlation of the

    thermal conductivity of methanol from the triple point to 660 K and up to 245

    MPa, J. Phys. Chem. Ref. Data 42 (2013) 1–10 04310 . 25] M. Massucci , A.P. du’Gay , A.M. Diaz-Laviada , C.J. Wormald , Second virial co-

    efficient of methanol from measurements of the excess molar enthalpy of methanol-nitrogen, J. Chem. Soc., Faraday Trans. 88 (1992) 427–432 .

    26] S.J. Boyes , M.B. Ewing , A.R.H. Goodwin , Heat capacities and second virial coef-ficients for gaseous methanol determined from speed-of-sound measurements

    at temperatures between 280 and 360 K and pressures from 1.03 kPa to 80.5 kPa, J. Chem. Thermodyn. 24 (1992) 1151–1166 1992 .

    27] P.G. Francis , R.C. Phutela , The isothermal Joule-Thomson coefficient and equa-

    tion of state for methanol and ethanol vapours, J. Chem. Thermodyn. 11 (1979) 747–756 .

    28] P.G. Francis , A flow calorimeter for the measurement of the heat capacities and Joule-Thomson coefficients of condensable vapours, J. Chem. Thermodyn.

    22 (1990) 545–556 .

    15

    29] J.D. Lambert , E.D.T. Strong , The dimerization of ammonia and amines, Proc. R. Soc. London, Ser. A 200 (1063) (1950) 566–572 .

    30] J.S. Rowlinson , The second virial coefficients of polar gases, Trans. Faraday Soc. 45 (1949) 974–984 .

    [31] D. Berthelot , Sur les thermométres a gaz et sur la reduction de leurs indica-tions a l’echelle absolue des temperatures, Trav. Mem. Bur. Int. Poids Mes. 13

    (1907) 1–113 ; D. Berthelot, Sur le calcul de la compressibilité des gaz au voisi- nage de la pression atmosphérique au moyen des constantes critiques. Compt.

    Rendus, 144 (1907) 194 - 197 .

    32] A.H. Harvey , E.W. Lemmon , Correlation for the second virial coefficient of wa- ter, J. Phys. Chem. Ref. Data 33 (2004) 369–376 .

    33] C.J. Wormald , The heat of mixing and second cross virial coefficient of wa- ter + oxygen and water + nitrogen, J. Chem. Thermodyn. 41 (2009) 689–694 .

    34] B.E. Poling , J.M. Prausnitz , J.P. O’Connell , The Properties of Gases and Liquids,

    5th ed, McGraw-Hill, New York, 2001 .

    35] P.M. Mathias , The second virial coefficient and the Redlich-Kwong equation, Ind. Eng. Chem. Res. 42 (2003) 7037–7044 P.M. Mathias, Response to “com-

    ments on ‘The second virial coefficient and the Redlich-Kwong equation’ “Ind. Eng. Chem. Res., 46 (2007), 6376–6378 .

    36] H.W. Woolley , The representation of gas properties in terms of molecular clus- ters, J. Chem. Phys. 21 (1953) 236–241 .

    37] C.R. Reid , J.M. Prausnitz , T.K. Sherwood , The Properties of Gases and Liquids,

    3rd ed, McGraw-Hill, New York, 1977 . 38] K.M. de Reuck , R.J.B. Craven , Methanol International Thermodynamic Tables of

    the Fluid State -12, Blackwell Scientific Publications, Oxford, 1993 . 39] H.G. Carlson , E.F. Westrum , Methanol: heat capacity, enthalpies of transition

    and melting, and thermodynamic properties from 5 - 300 °K, J. Phys. Chem. 54 (1971) 1464–1471 .

    40] K. Bennewitz , W. Rossner , Über die Molwärme von organischen Dämpfen, Z.

    Phys. Chem. 39B (1938) 126–144 . [41] J.C. Brown , A direct method for determining latent heat of evaporation, J.

    Chem. Soc., Trans. 83 (1903) 987–994 . 42] E.F. Fiock , D.C. Ginnings , W.B. Holton , Calorimetric determinations of thermal

    properties of methyl alcohol, ethyl alcohol and benzene, Bur. Stand. J. Res. 6 (1931) 881–900 .

    43] J. Konicek , Design and testing of a vaporization calorimeter. Enthalpies of

    vaporization of some alkyl cyanides, Acta Chem. Scand. 27 (1973) 1496–1502 .

    44] J.H. Mathews , The accurate measurement of heats of vaporization of liquids, J. Am. Chem. Soc. 48 (1923) 562–576 .

    45] K.G. McCurdy , K.J. Laidler , Heats of vaporization of a series of aliphatic alco-hols, Can. J. Chem. 41 (1963) 1867–1871 .

    46] D.M.T. Newsham , E.J. Mendez-Lecanda , Isobaric enthalpies of vaporization of

    water, methanol, ethanol, propan-2-ol, and their mixtures, J. Chem. Thermo- dyn. 14 (1982) 291–301 .

    [47] J. Polak , G.C. Benson , Enthalpies of vaporization of some aliphatic alcohols, J. Chem. Thermodyn. 3 (1971) 235–242 .

    48] M. Radosz , A. Lydersen , Heat of vaporization of aliphatic alcohols, Chem. Ing. Tech. 52 (1980) 756–757 .

    49] L.A . Staveley , A .K. Gupta , A semi-micro low-temperature calorimeter, and a comparison of some thermodynamic properties of methyl alcohol and methyl

    deuteroxide, Trans. Faraday Soc. 45 (1949) 50–61 .

    50] V. Svoboda , F. Vesely , R. Holub , J. Pick , Enthalpy data of liquids. II. The depen-dence of heats of vaporization of methanol, propanol, butanol, cyclohexane,

    cyclohexene, and benzene on temperature, Collect. Czech. Chem. Commun 38 (1973) 3539–3543 .

    [51] F. Vesely , L. Sváb , R. Provazník , V. Svoboda , Enthalpies of vaporization at highpressures for methanol, ethanol, propan-1-ol, propan-2-ol, hexane, and cyclo-

    hexane, J. Chem. Thermodyn. 20 (1988) 981–983 .

    52] I. Wadsö, Heats of vaporization for a number of organic compounds at 25 °C, Acta Chem. Scand. 20 (1966) 544–552 .

    53] T.K. Yerlett , C.J. Wormald , The enthalpy of methanol, J. Chem. Thermodyn. 18 (1986) 719–726 .

    54] M.-.H. Guermouche , J.-.M. Vergnaud , Détermination d’équations représentant la variation des grandeurs thermodynamiques de vaporisation d’hydrocarbures

    valables pour toute temperature, J. Chim. Phys 71 (1974) 1049–

    1052 . 55] K. Aim , M. Ciprian , Vapour pressures, refractive index at 20 °C, and vapour-liq-

    uid equilibrium at 101.325 kPa in the methyl tert-butyl ether-methanol system, J. Chem. Eng. Data 25 (1980) 100–103 .

    56] D. Ambrose , C.H. Sprake , Thermodynamic properties of organic compounds XXV. Vapour pressures and normal boiling temperatures of aliphatic alcohols,

    J. Chem. Thermodyn. 2 (1970) 631–645 .

    57] D. Ambrose , C.H. Sprake , R. Townshend , Thermodynamic properties of or- ganic oxygen compounds XXXVII. Vapour pressures of methanol, ethanol, pen-

    tan-1-ol, and octan-1-ol from the normal boiling temperature to the critical temperature, J. Chem. Thermodyn. 7 (1975) 185–190 .

    58] M. Antosik , Z. Fras , S.K. Malanowski , Vapour-liquid equilibrium in 2-ethoxyethanol + methanol at 313.15 to 333.15 K, J. Chem. Eng. Data 44 (1999) 368–372 .

    59] A. Apelblat , F. Kohler , Excess Gibbs energy of methanol + propionic acid andof methanol + butyric acid, J. Chem. Thermodyn. 8 (1976) 749–756 .

    60] A. Aucejo , S. Loras , R. Muñoz , Phase equilibria and multiple azeotropy in theassociating system methanol + diethylamine, J. Chem. Eng. Data 42 (1997) 1201–1207 .

    http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0001http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0001http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0001http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0002http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0002http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0002http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0002http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0003http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0003http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0003http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0003http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0003http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0004http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0004http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0004http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0005http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0005http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0005http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0006http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0006http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0006http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0007http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0007http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0008http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0008http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0008http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0009http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0009http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0009http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0009http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0010http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0010http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0011http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0011http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0012http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0012http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0012http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0013http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0013http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0014http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0014http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0014http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0015http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0015http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0015http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0016http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0016http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0016http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0016http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0017http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0017http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0017http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0018http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0018http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0018http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0018http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0019http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0019http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0019http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0019http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0020http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0020http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0020http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0021http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0021http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0022http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0022http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0022http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0022http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0023http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0023http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0023http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0023http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0024http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0024http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0024http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0024http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0024http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0025http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0025http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0025http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0025http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0025http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0026http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0026http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0026http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0026http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0027http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0027http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0027http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0028http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0028http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0029http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0029http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0029http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0030http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0030http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0031http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0031http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0032http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0032http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0032http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0033http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0033http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0034http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0034http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0034http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0034http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0035http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0035http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0036http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0036http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0037http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0037http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0037http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0037http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0038http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0038http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0038http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0039http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0039http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0039http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0040http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0040http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0040http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0041http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0041http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0042http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0042http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0042http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0042http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0043http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0043http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0044http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0044http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0045http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0045http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0045http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0046http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0046http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0046http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0047http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0047http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0047http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0048http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0048http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0048http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0049http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0049http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0049http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0050http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0050http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0050http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0050http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0050http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0051http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0051http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0051http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0051http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0051http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0052http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0052http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0053http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0053http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0053http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0054http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0054http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0054http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0055http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0055http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0055http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0056http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0056http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0056http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0057http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0057http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0057http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0057http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0058http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0058http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0058http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0058http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0059http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0059http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0059http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0060http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0060http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0060http://refhub.elsevier.com/S0378-3812(20)30399-X/sbref0060

  • G.H. Graaf and J.G.M. Winkelman Fluid Phase Equilibria 529 (2021) 112851

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    [

    61] D.P. Barton , V.R. Bethanabotla , S.C. Campbell , Binary total pressure measurements for methanol with 1-pentanol, 2-pentanol, 3-pen-

    tanol, 2-methyl-1-butanol, 2-methyl-2-butanol, 3-methyl-1-butanol, and 3-methyl-2-butanol at 313.15 K, J. Chem. Eng. Data 41 (1996) 1138–1140 .

    62] T. Boublík , K. Aim , Heats of vaporization of simple non-spherical molecule compounds, Collect. Czech. Chem. Commun. 37 (1972) 3513–3521 .

    63] M. Broul , K. Hlavaty , J. Linek , Liquid-vapour equilibrium in systems of elec-trolytic components, V., Collect. Czech. Chem. Commun. 34 (1969) 3428–3435 .

    64] I. Cervenková, T. Boublík , Vapour pressures, refractive indexes, and densities at

    20 °C, and vapour-liquid equilibrium at 101.325 kPa, in the tert-amyl methyl ether-methanol system, J. Chem. Eng. Data 29 (1984) 425–427 .

    65] P. Colmant , Etude expérimentale du système méthanol-sébacate de butyle à20 °, Bull. Soc. Chim. Belg 63 (1954) 5–39 .

    66] B. Coto , R. Wiesenberg , C. Pando , R.G. Rubio , J.A.R. Renuncio , Vapour-liquidequilibrium of the methanol-tert-butyl methyl ether (MTBE) system, Ber. Bun-

    sen-Ges. 100 (1996) 4 82–4 89 .

    67] D.F. Dever , A. Finch , E. Grunwald , The vapour pressure of methanol, J. Phys.Chem. 59 (1955) 668–669 .

    68] X. Esteve , S.K. Chaudari , A. Coronas , Vapour-liquid equilibria for methanol + tetraethylene glycol dimethyl ether, J. Chem. Eng. Data 40 (1995) 1252–1256 .

    69] R. Garriga , P. Sánchez , P. Pérez , M. Gracia , Vapour pressures at eight temper-

    atures between 278.15 K amd 323.15 K and excess molar enthalpies and vol-

    umes at T = 298.15 K of (n-propylether + methanol), J. Chem. Thermodyn. 29 (1997) 649–659 .

    70] H.F. Gibbard , J.L. Creek , Vapour pressure of methanol from 288.15 to 337.65 K,J. Chem. Eng. Data 19 (1974) 308–310 .

    [71] R.H. Harrison , B.E. Gammon , Private communication to the IUPAC Centre, 1989 presented in [38] .

    72] H. Holldorff, H. Knapp , Vapour pressures of n-butane, dimethylether,

    methylchloride, methanol and the vapour-liquid equilibrium of d