Solution Thermodynamics: Theory

1
Solution Thermodynamics Theory

• Properties depend strongly on composition,
• T and P.
• Gas mixtures and liquid solutions
• New terms chemical potential, partial
  properties, fugacity, and excess property (e.g.
  excess Gibbs energy).
• Why we have to study Thermodynamics?.........
• The prediction of the equilibrium existing
  between phases, and to understand the process and
  to calculate equilibria.

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• What is the most important property ?

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• G.
• For pure component
• G G (T, P)
• For a homogeneous mixture e.g. 3 components
• G G (T, P, n1, n2, n3)

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Fundamental Property Relation

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Fundamental Property Relation

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• G plays a role of a generating function,
  providing the means of calculation of other
  thermodynamics properties by simple mathematical
  calculations and implicitly represents complete
  property information.

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The Chemical Potential

• Phase ? and ?.
• Transfer between phase

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The Chemical Potential

• Two phases are closed, and at equilibrium.

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• Chemical Potential (?)
• is an extensive property,
• provides a measure of the work of a system is
  capable when a change in mole numbers occurs e.g.
  chemical reaction or a transfer of mass.

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The Chemical Potential
? phases at equilibrium, and N is the number of
species.
Thus, multiple phases at the same T and P are in
equilibrium when the chemical potential of each
species is the same in all phases. What will be
happened?
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Partial Properties

• The partial molar property any extensive
  property of a solution changes with respect to
  the number of moles of any component i in the
  solution at constant T, P and composition of the
  others.

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Partial Properties
The number of moles of other species except i are
held constant
Always in terms Of ni never xi
Always hold the intensive properties P and T
constant
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Partial Properties
M molar property of solution
Thus, the partial molar Gibbs energy is the
chemical potential
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Partial Properties
Since ni xin
Similarly to this, one can write
Substitute these terms to Eq. 11.9, and then
rearrange
The only way that the left hand side of this
equation can be zero is for each term in brackets
to be zero.
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Partial Properties
Summability relations
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Partial Properties
Differentiate Eq. 11.11,
Compare with Eq. 11.10 yields Gibbs Duhem
equation

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Partial Properties
Gibbs/Duhem equation
If T and P constant
Solution property M Partial property
Pure-species property Mi
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Partial Properties
Binary system
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• For 2 components

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Example 11.3
The need arise in a laboratory for 2000 cm3 of an
antifreeze solution consisting of 30 mol
methanol in water. What volumes of pure methanol
and of pure water at 25 ?C must be mixed to form
the of antifreeze, also at 25 ?C ? Partial molar
volumes for methanol and water in a 30 mol
methanol solution and their pure-species molar
volume, both at 25 ?C , are Methanol (1) and
water (2)
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Solution
22
Solution
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We want to know the quantity of each component to
be mixed. V1 40.727 cm3 mol-1 and V2 18.068
cm3 mol-1.
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Relations among Partial Properties
Maxwell relation
We have two additional equations (Eq.11.17 is an
exact differential eq.)
One can write the RHS in the form of partial
molar, and change the composition from n to x.
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Relations among Partial Properties (contd.)

• Every equation that provides a linear relation
  among thermodynamic properties of a
  constant-composition solution has as its
  counterpart an equation connecting the
  corresponding partial properties of each species
  in the solution.

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Ideal-gas Mixture

• Dalton Law
• Every gas has the same V and T.
• Amagat Law
• Every gas has the same P and T.

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Ideal-gas Mixture

• Properties of each component species are
  independent of the presence of other species.
• A partial molar property (other than volume) of
  a constituent species in an ideal-gas mixture is
  equal to the corresponding molar property of the
  species as a pure ideal gas at the mixture
  temperature but at a pressure equal to its
  partial pressure in the mixture.

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Ideal-gas Mixture
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Ideal-gas Mixture
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Problem 11.1What is the change in entropy when
0.7 m3 of CO2 and 0.3 m3 of N2 each at 1 bar and
25 ?C blend to form a gas mixture at the same
condition? Assume ideal gases. Solution
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Homework11.12, 11.13 (Due date Feb. 29st,
2007)
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Difference between and
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Fugacity Fugacity Coefficient Pure Species
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Fugacity Fugacity Coefficient Pure Species
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Fugacity Fugacity Coefficient Pure Species
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Fugacity Fugacity Coefficient Pure Species
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Generalized Correlations for the Fugacity
Coefficient
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Generalized Correlations for the Fugacity
Coefficient
The average properties at the critical point and
the 2nd virial coefficient can be determined from
Equation 11.66-11.71
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Example 2 Problem 11.16
From the following compressibility factor data
for CO2 at 150ºC prepare plots of the fugacity
and fugacity coefficient of CO2 vs. P for
pressures up to 500 bar. Compare results with
those found from the generalized correlation
represented by Eq. (11.65)
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• At P 0, Z 1
• Calculate (Z-1)/P fi
• Plot (Z-1)/P vs. P
• ln ?i (fi fi-1)/2 (Pi- Pi-1)

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Example 2 Problem 11.16
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Agreement looks good upto about 200 bar.
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Example 3 Problem 11.17

• Good estimate of f and GR/RT of SO2 at
• 600 K and 300 bar.

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Example 3 Problem 11.17

• Good estimate of f and GR/RT of SO2 at 600 K and
  300 bar.
• For the given conditions, we see from Fig. 3.14
  that the Lee/Kesler correlation is appropriate.
• Pr P/Pc 300/78.84 3.805
• Tr T/Tc 600/430.8 1.393 ??
• Data from Table E.15 and E.16

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• Homework
• Problem 11.18 (select a or b)
• Problem 11.19 (select a or b)
• Due date March 3, 08.

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Vapor/Liquid Equilibrium for Pure Species
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Vapor/Liquid Equilibrium for Pure Species
For a pure species coexisting liquid and
vapor phases are in equilibrium when they have
the same temperature, pressure, and fugacity.
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Fugacity of a Pure Liquid

• fi of a compressed liquid is calculated in 2
  steps
• fi of saturated liquid and vapor
• Compressed liquid from Psat to P

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Example 4 Problem 11.21
From data in the steam table, determine a good
estimate for f / fsat for liquid water at 150 ?C
and 150 bar, where fsat is the fugacity of
saturated liquid at 150 ?C .
At 150 ?C, Psat 476.00 kPa 476 /100 4.76
bar P gt Psat water is a compressed liquid.
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Example 4 Problem 11.21 (cont.)
Vsat 1.091 cm3/g 1.091 cm3/g(18 g / g mol)
19.638 cm3/ g mol T 150 273.15 K
423.15 K
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• Homework
• Problem 11.22 ,
• Due date March 5, 2008.

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Fugacity and Fugacity Coefficient Species in
Solution

• fi of a solution is parallel to the pure
  solution.
• The ideal solution (analogous to the ideal gas)
• At equilibrium

Thus, multiple phases at the same T and P are in
equilibrium when the fugacity of each constituent
species is the same in all phases.
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Fugacity and Fugacity Coefficient Species in
Solution
A partial residual property,
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The Fundamental Residual-Property Relation
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Fugacity Coefficient from the Virial EOS

• For mixture
• e.g. binary mixture
• B y1y1B11 y1y2B12 y2y1B21
  y2y2B22
• B y12B11 2y1y2B12 y22B22
  (11.58)

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The Ideal Solution

• Serves as a standard to which real-solution
  behavior can be compared.

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The Ideal Solution The Lewis/Randall Rule

• Fugacity calculation of i in ideal solution.
• all the intermolecular forces are equal.

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The Ideal Solution

• ??????? Reference state ?????????????????????????
  ???? ???? gas ??????????????? ideal gas
  ??????????? ?? 2 ??????????? the Lewis/Randall
  rule ??? Henrys law

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Excess Properties
Fundamental of excess property relation
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The Excess Gibbs Energy and the Activity
Coefficient
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Activity Coefficient

• ??????? T, liquid phase composition P ??????? ?
  ??????? ???????????? f ??? deviation from
  ideal solution ideal solution ? 1.00 ?
  ?????????????

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The Excess Gibbs Energy and the Activity
Coefficient
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The Excess Gibbs Energy and the Activity
Coefficient
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Gibbs-Duhem Equation

• ??????????????? Thermodynamic consistency of
  thermodynamic data (??????????????????????????)
  ????????????? partial molar property ?????? j
  ?????????????????? i

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The Nature of the Excess Properties

• All MEs become 0 as either species approaches
  purity.
• Plot between GE vs. x1 is approximately
  parabolic in shape,
• Both HE and TSE exhibit individualistic
  composition
• dependencies
• When an excess property has a single sign (as
  does GE in
• all six cases, the extreme value of ME (maximum
  or minimum)
• Often occurs near the equimolar composition.

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Example 5 (Problem 11.25)

• For the system ethylene (1) / propylene (2) as a
  gas, estimate
• at t 150 ºC, P 30 bar, and y1 0.35.
• Through application of Eqs. (11.63)
• (b) Assuming that the mixture is an ideal
  solution.

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Example 5 (Problem 11.25)
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Example 5 (Problem 11.25)
Ethylene (1) ?1 0.087 Tc 282.3 K Pc
50.40 bar Vc 131. cm3 mol-1 Zc 0.281

Propylene (2) ?2 0.140 Tc 365.6 K Pc
46.65 bar Vc 188.4 cm3 mol-1 Zc
0.289
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• ij Tcij, K Pcij, bar Vcij, cm3mol-1 Zcij
  ?ij
• 282.3 50.4 131. 0.281 0.087
• 365.6 46.65 188.4 0.289 0.140
• 12 321.26 48.189 157.97 0.285 0.1135

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Ideal solution
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Review

• Mixtures
• What is the definition of partial molar
  property? Try saying it in words rather than
  equation.
• Why is the partial molar property not the same
  as the pure property? What can happen when we mix
  different species?
• How is excess property defined?
• Do ideal gases always form ideal mixtures when
  allowed to mixed?
• Pick a property, say V,. Review the ways we can
  calculate the partial molar volume. What about
  straight differentiation? What is the alternative
  way that only works for binary mixture? What
  about graphically?

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Review

• Mixtures
• What is Gibbs-Duhem equation? In what ways is it
  useful?
• For ideal solution, what is the molar volume?
• For ideal solution, what is the molar enthalpy?
• For ideal solution, what is the molar entropy?
• For ideal solution, what is the molar Gibbs free
  energy?
• What is the definition of infinite dilution
  property?

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• Homework
• Problem 11.30 (select one choice from a to g)
• Problem 11.35
• Problem 11.37 (select one choice from a to e)
• Due date March 6, 2008.