solution thermodynamics-chapter 12-part ii-modified (1)
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8/20/2019 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
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