Solution Thermodynamics-chapter 12-PART II-modified (1)

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    Solution Thermodynamics:

    ApplicationsChapter 12-Part II

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    Find parameter values for the Margules equation that give the best fit of G E / T tothe data! and prepare a P x y diagram that compares the e"perimental points #ithcurves determined from the correlation

    $% &alculate E'(E )ME*TA+ values of activity coefficients $ and , and G E

    2211

    22

    22

    11

    11

    lnln/

    ;

    γ γ

    γ γ

    x x RT G P x

    P y

    P x

    P y

    E

    sat sat

    +=

    ==

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    In the previous class we showed

    21212121

    x A x A RT x x

    G E +=

    ])(2[ln

    ])(2[ln

    2211221

    2

    12

    1122112221

    x A A A x

    x A A A x

    −+=−+=

    γ

    γ

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    Now we have our analytical model

    2121

    683.0475.0 x x RT x x

    G E

    +=

    sat sat calc P x P x P 222111 γ γ +=

    +ets calculate ln $! ln , ! GE

    /" $" , T! and:

    calc

    sat calc

    P

    P x y 111

    1

    γ =

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    MS- S. T ( i0( icalc %, /n - 12345 6(a

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    Thermodynamic consistency

    • We need to check that the experimentallyo tained activity coe!!icients satis!y the"i s-#uhem e$uation%

    • I! the experimental data are inconsistentwith the "-# e$uation& they are not correct

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    Consistency test

    +−=

    =

    =

    +=

    1

    *2

    2

    1

    *1

    1

    2

    1

    1

    2

    1

    1

    1

    *

    *22

    *11

    *

    lnlnln

    )/(

    ln)/(

    )/(

    lnln

    dx

    d x

    dx

    d x

    dx

    RT Gd

    dx

    RT Gd

    dx

    RT Gd

    x x RT

    G

    E

    E

    E

    E

    γ γ γ γ

    δ δ

    γ γ

    γ γ

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    Consistency test

    0ln

    lnlnln0

    lnlnln

    )/(

    2

    1

    1

    *2

    21

    *1

    12

    1

    1

    *2

    21

    *1

    12

    1

    1

    =

    +−=

    +−=

    γ γ δ

    γ γ γ γ

    δ

    γ γ γ γ

    δ δ

    dxd

    xdx

    d x

    dxd

    xdx

    d x

    dx RT Gd E

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    The e"perimental data is not thermodynamically consistent

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    'n alternative consistency test()arker*s method

    • +it the model ", .T to make thecalculated pressures the closest possi le

    to the experimental data%

    • +or example& o tain '12 and '21 !or the

    /ar0ules e$uation to minimi e thecalculated pressures with respect to theexperimental values% see dashed lines in+i0ure 12%34

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    example

    • 5sin0 )arker*s method& !ind parameters!or the /ar0ules e$n that provide the est!it o! ", .T to the data& and prepare a Pxydia0ram that compares the experimentalpoints with curves determined !orm thecorrelation%

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    solution

    ])(2[ln])(2[ln

    2211221212

    1122112

    2

    21

    x A A A x x A A A x

    −+= −+=γ γ

    Minimi7e the sum of squares of the follo#ing function:

    2221211222

    121211211exp

    ]),,,(

    ),,,([

    sat

    sat

    i

    i

    P x x A A x

    P x x A A x P

    γ

    γ +−∑

    Starting #ith A$,-128! A,$-$! get A$,- 12985! A,$-12 38

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    Calculate the ./6 !or PMS- S. T ( i0( icalc %, /n - 12$;9 6(a