chemical engineering thermodynamics lecturer: zhenxi jiang (ph.d. u.k.)

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Chemical Engineering Chemical Engineering ThermodynamicsThermodynamics

Lecturer: Zhenxi Jiang (Ph.D. U.K.)Lecturer: Zhenxi Jiang (Ph.D. U.K.)

School of Chemical EngineeringSchool of Chemical Engineering

Zhengzhou UniversityZhengzhou University

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Chapter 12Chapter 12Solution Thermodynamics: ApplicationSolution Thermodynamics: Application

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All of the fundamental equations and All of the fundamental equations and necessary definitions of solution necessary definitions of solution thermodynamics are given in the thermodynamics are given in the preceding chapter. In this chapter, we preceding chapter. In this chapter, we examine what can be learned from examine what can be learned from experiment. Considered first are experiment. Considered first are measurements of vapor/liquid measurements of vapor/liquid equilibrium (VLE) data, from which equilibrium (VLE) data, from which activity coefficient correlations are activity coefficient correlations are derived.derived.

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Second, we treat mixing Second, we treat mixing experiments, which provide data for experiments, which provide data for property changes of mixing. In property changes of mixing. In particular, practical applications of particular, practical applications of the enthalpy change of mixing, called the enthalpy change of mixing, called the heat of mixing, are presented in the heat of mixing, are presented in detail in Sec. 12.4.detail in Sec. 12.4.

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12.1 Liquid phase property from VLE data12.1 Liquid phase property from VLE data

Figure 12.1 shows a vessel in which a Figure 12.1 shows a vessel in which a vapor mixture and a liquid solution vapor mixture and a liquid solution coexist in vapor/liquid equilibrium. coexist in vapor/liquid equilibrium.

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The temperature T and P are uniform The temperature T and P are uniform throughout the vessel, and can be throughout the vessel, and can be measured with appropriate measured with appropriate instruments. Vapor and liquid instruments. Vapor and liquid samples may be withdrawn for samples may be withdrawn for analysis, and this provides analysis, and this provides experimental values for mole experimental values for mole fractions in the vapor {fractions in the vapor {yyii} and mole } and mole fractions in the liquid {fractions in the liquid {xxii}.}.

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FugacityFugacity

For species For species I I in the vapor mixture, Eq. in the vapor mixture, Eq. (11.52) is written:(11.52) is written:

The criterion of vapor/liquid The criterion of vapor/liquid equilibrium, as given by Eq. (11.48), equilibrium, as given by Eq. (11.48), is that . Therefore,is that . Therefore,

ˆ ˆv vi i if y P

ˆ ˆl vi if f

ˆ ˆl vi i if y P

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Although values for vapor-phase Although values for vapor-phase fugacity coefficient are easily fugacity coefficient are easily calculated (Secs. 11.6 and 11.7), calculated (Secs. 11.6 and 11.7), VLE measurements are very often VLE measurements are very often made at pressure low enough (P ≤ 1 made at pressure low enough (P ≤ 1 bar) that the vapor phase may be bar) that the vapor phase may be assumed an ideal gas. In this case assumed an ideal gas. In this case = 1, and the two preceding = 1, and the two preceding equations reduce to:equations reduce to:

vi

vi

ˆ ˆl vi i if f y P

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Thus, the fugacity of species Thus, the fugacity of species ii (in (in both the liquid and vapor phases) is both the liquid and vapor phases) is equal to the partial pressure of equal to the partial pressure of species species ii in the vapor phase. Its in the vapor phase. Its value increases from zero at infinite value increases from zero at infinite dilution to Pdilution to Pii

satsat for pure species for pure species ii. this . this is illustrated by the data of Table is illustrated by the data of Table 12.1 for the methyl ethyl 12.1 for the methyl ethyl ketone(1)/toluene(2) system at 50ketone(1)/toluene(2) system at 50 . ℃. ℃

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The first three columns list a set of experimental P-x1-y1 data and columns 4 and 5 show:The first three columns list a set of experimental P-x1-y1 data and columns 4 and 5 show:

and and 1 1f y P 2 2f y P

1111


The fugacities are plotted in Fig 12.2 The fugacities are plotted in Fig 12.2 as solid lines. The straight dashed as solid lines. The straight dashed lines represent Eq. (11.83), the lines represent Eq. (11.83), the Lewis/Randall rule, which expresses Lewis/Randall rule, which expresses the composition dependence of the the composition dependence of the constituent fugacities in an ideal constituent fugacities in an ideal solution:solution:

(11.83)(11.83)

1ˆ id

i if x f

1212


1313


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Although derived from a particular Although derived from a particular set of data, Fig. 12.2 illustrates the set of data, Fig. 12.2 illustrates the general nature of the fugacities of general nature of the fugacities of components 1 and 2 vs. components 1 and 2 vs. xx11 relationships for a binary liquid relationships for a binary liquid solution at constant solution at constant TT. The . The equilibrium pressure equilibrium pressure PP varies with varies with composition, but its influence on the composition, but its influence on the liquid phase values of and is liquid phase values of and is negligible. negligible. Thus a plot at constant T and P would look the same, as indicated Thus a plot at constant T and P would look the same, as indicated in Fig. 12.3 foe species I (I = 1, 2) in a binary solution at constant in Fig. 12.3 foe species I (I = 1, 2) in a binary solution at constant T and P.T and P.

1f 2f

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Activity CoefficientActivity Coefficient

The lower dashed line in Fig. 12.3, The lower dashed line in Fig. 12.3, representing the Lewis/Randall rule, representing the Lewis/Randall rule, is characteristic of ideal-solution is characteristic of ideal-solution behavior. It provides the simplest behavior. It provides the simplest possible model for the composition possible model for the composition dependence of , representing a dependence of , representing a standard to which actual behavior standard to which actual behavior may be compared.may be compared.

1f

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The activity coefficient as defined by The activity coefficient as defined by Eq. (11.90) formalizes this Eq. (11.90) formalizes this comparison:comparison:

Thus the activity coefficient of a Thus the activity coefficient of a species in solution is the ratio of its species in solution is the ratio of its actual fugacity to the value given by actual fugacity to the value given by the Lewis/Randall rule at the same the Lewis/Randall rule at the same TT, , PP, and composition., and composition.

ˆ ˆ

ˆi i

i idi i i

f f

x f f

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For the calculation of experimental For the calculation of experimental values, both and are values, both and are eliminated in favor of measurable eliminated in favor of measurable quantities:quantities:

(12.1)(12.1)

This is a restatement of Eq. (10.5), modified Raoult’s This is a restatement of Eq. (10.5), modified Raoult’s law, and is adequate for present purposes, allowing law, and is adequate for present purposes, allowing easy calculation of activity coefficients from easy calculation of activity coefficients from experimental low pressure VLE data. Values from this experimental low pressure VLE data. Values from this equation appears in the last two columns of Table 12.1.equation appears in the last two columns of Table 12.1.

ifid

if

( 1, 2, , )i ii sat

i i i i

y P y Pi N

x f x P

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The solid lines in both Figs 12.2 and The solid lines in both Figs 12.2 and 12.3, representing experimental 12.3, representing experimental values of , become tangent to the values of , become tangent to the Lewis/Randall rule lines at Lewis/Randall rule lines at xxii = 1. This = 1. This is a consequence of the Gibbs/Duhem is a consequence of the Gibbs/Duhem equation. Thus, the ratioequation. Thus, the ratio

is indeterminate in this limit, is indeterminate in this limit, and application of 1’Hopital’s rule and application of 1’Hopital’s rule yields:yields:

(12.2)(12.2)

ii xf

i0

0

ˆ ˆlim

i

i

i i

xi i x

f df

x dx

Ή

if

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Equation (12.2) defines Henry’s Equation (12.2) defines Henry’s constant constant HHii as the limiting slope of as the limiting slope of the curve at the curve at xxii = 0. as shown = 0. as shown by Fig. 12.3, this is the slope of a line by Fig. 12.3, this is the slope of a line drawn tangent to the curve at drawn tangent to the curve at xxii = 0.= 0.

ii xf ˆ

2020



The equation of this tangent line The equation of this tangent line expresses Henry’s law:expresses Henry’s law:

(12.3)(12.3)

Applicable in the limit as xi -> 0, it is also of Applicable in the limit as xi -> 0, it is also of approximate validity for small values of xi. approximate validity for small values of xi. Henry’s law as given by Eq. (10.4) follows Henry’s law as given by Eq. (10.4) follows immediately from this equation when , immediately from this equation when , i.e., when has its ideal gas value.i.e., when has its ideal gas value.

if

iˆi if x Ή

ˆi if y P

2121



Henry’s law is related to the Henry’s law is related to the Lewis/Randall rule through the Lewis/Randall rule through the Gibbs/Duhem equation. Gibbs/Duhem equation.

Writing Eq. (11.14) for a binary Writing Eq. (11.14) for a binary solution and replacing by solution and replacing by gives:gives:

xx11 d dμμ11 + + xx22 d dμμ22 = = 0 (const 0 (const T, P)T, P)

Differentiation of Eq. (11.46) at constant Differentiation of Eq. (11.46) at constant T T and and P P yields: yields: dμdμi i = RT dln = RT dln ;; whence,whence,

ˆiM ˆ

iG iμ

if

2222



xx11dln + dln + xx22dln = dln = 0 (const 0 (const T, P)T, P)

Upon division by dUpon division by dxx11 this becomes: this becomes:

(12.4)(12.4)

This is a special form of the This is a special form of the Gibbs/Duhem equation.Gibbs/Duhem equation.

1f 2f

1 21 2

1 2

ˆ ˆln ln0 (const , )

d f d fx x T P

dx dx

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Activity CoefficientActivity CoefficientThroughThrough more operations the Eq. more operations the Eq. (12.5) is obtained.(12.5) is obtained.

(12.5)(12.5)

This equation is the exact expression This equation is the exact expression of the Lewis/Randall rule as applied of the Lewis/Randall rule as applied to real solutions.to real solutions.

1

11

1 1

ˆ

x

dff

dx

2424



Henry’s law applies to a species as it Henry’s law applies to a species as it approaches infinite dilute in a binary approaches infinite dilute in a binary solution, and the Gibbs/Duhem solution, and the Gibbs/Duhem equation insures validity of the equation insures validity of the Lewis/Randall rule for the other Lewis/Randall rule for the other species as it approaches purity.species as it approaches purity.

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12.1 Liquid phase property from VLE data12.1 Liquid phase property from VLE dataThe fugacity shown by Fig. 12.3 is for a species The fugacity shown by Fig. 12.3 is for a species with positive deviations from ideality in the with positive deviations from ideality in the sense of the Lewis/Randall rule. Negative sense of the Lewis/Randall rule. Negative deviations are less common, but are also deviations are less common, but are also observed; the curve then lies below the observed; the curve then lies below the Lewis/Randall line. In Fig. 12.4 the fugacity of Lewis/Randall line. In Fig. 12.4 the fugacity of acetone is shown as a function of composition acetone is shown as a function of composition for two different binary liquid solutions at 50 for two different binary liquid solutions at 50 . ℃. ℃When the second species is methanol, acetone When the second species is methanol, acetone exhibits positive deviations from ideality. When exhibits positive deviations from ideality. When the second species is chloroform, the deviations the second species is chloroform, the deviations are negative. The fugacity of pure acetone is of are negative. The fugacity of pure acetone is of course the same regardless of the identity of the course the same regardless of the identity of the second species. However, Henry’s constants, second species. However, Henry’s constants, represented by slopes of the two dotted lines, represented by slopes of the two dotted lines, are very different for the two cases.are very different for the two cases.

ii xf ˆ

2727


Excess Gibbs EnergyExcess Gibbs Energy

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In Table 12.2 the first three columns In Table 12.2 the first three columns repeat the P-repeat the P-xx11--yy11 data of Table 12.1 data of Table 12.1 for system methyl ethyl ketone(1)/ for system methyl ethyl ketone(1)/ toluene(2). toluene(2).

These data points are also shown as These data points are also shown as circles on Fig. 12.5(a). Values of lncircles on Fig. 12.5(a). Values of lnγγ11 and lnand lnγγ22 are listed in columns 4 and 5, are listed in columns 4 and 5, and are shown by the open squares and are shown by the open squares and triangles of Fig. 12.5(b). and triangles of Fig. 12.5(b).

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They are combined for a binary system in They are combined for a binary system in accord with Eq. (11.99):accord with Eq. (11.99):

(12.6)(12.6)

The values ofThe values of G G EE/ RT / RT so calculated are so calculated are then divided by xthen divided by x11 x x22 to provide values of to provide values of

G G EE//xx11xx22RTRT; the two sets of numbers are ; the two sets of numbers are listed in columns 6 and 7 of Table 12.2, listed in columns 6 and 7 of Table 12.2, and appear as solid circles on Fig. 12.5(b). and appear as solid circles on Fig. 12.5(b).

1 1 2 2ln lnEG

x xRT

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The four thermodynamic functions, The four thermodynamic functions, lnlnγγ1,1, ln lnγγ22 , , G G EE/ RT , and G / RT , and G EE//xx11xx22RT, RT, are are properties of the liquid phase. Figure properties of the liquid phase. Figure 12.5(b) shows how their experimental 12.5(b) shows how their experimental values very with composition for a values very with composition for a particular binary system at a particular binary system at a specified temperature. This figure is specified temperature. This figure is characteristic of systems for which:characteristic of systems for which:

1 and ln 0 ( 1, 2)i i i

3131


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In such cases the liquid phase shows positive deviations from Raoult’s law In such cases the liquid phase shows positive deviations from Raoult’s law behavior.behavior.

Because the activity coefficient of a Because the activity coefficient of a species in solution becomes unity as species in solution becomes unity as the species becomes pure, each the species becomes pure, each tends to zero as tends to zero as xxii → → 1. At the 1. At the other limit, where other limit, where

xxii → → 0 and species 0 and species ii becomes becomes infinitely dilute, approaches a infinitely dilute, approaches a finite limit, namely, . finite limit, namely, .

ln ( 1,2)i i

ln iln i

3333



The Gibbs/Duhem equation, written The Gibbs/Duhem equation, written for a binary system, is finally divided for a binary system, is finally divided to give:to give:

(12.7)(12.7)

AndAnd

(12.8)(12.8)

1 21 2

1 1

ln ln0 (const , )

d dx x T P

dx dx

1

1 2

( / )ln

Ed G RT

dx

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Data ReductionData Reduction

Of the sets of points shown in Fig. Of the sets of points shown in Fig. 12.5(b), those for 12.5(b), those for G G EE//xx11xx22RT RT most most closely confirm to a simple closely confirm to a simple mathematical relation. Thus a mathematical relation. Thus a straight line provides a reasonable straight line provides a reasonable approximation to this set of points, approximation to this set of points, and mathematical expression is and mathematical expression is given to this linear relation by the given to this linear relation by the equation:equation:

(12.9a)(12.9a)

21 1 12 21 2

A AEG

x xx x RT

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where where AA2121 andand A A1212 are constants in are constants in any particular application. any particular application. Alternatively,Alternatively,

(12.9b)(12.9b)

Expressions for and are Expressions for and are derived from Eq. (12.9b) by derived from Eq. (12.9b) by application of Eq. (11.96).application of Eq. (11.96).

1ln 2ln 21 1 12 2 1 2(A A )

EGx x x x

RT

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Further reduction leads to:Further reduction leads to:

(12.10a)(12.10a)

(12.10b)(12.10b)

21 2 12 21 12 1ln [A 2(A A ) ]x x

22 1 21 12 21 2ln [A 2(A A ) ]x x

3737



These are the Margules equations, These are the Margules equations, and they represent a commonly used and they represent a commonly used empirical model of solution behavior. empirical model of solution behavior. For the limiting conditions of infinite For the limiting conditions of infinite dilution, they becomedilution, they become

andand 1 12 1ln 0A x 2 21 2ln 0A x

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For the methyl ethyl ketone/toluene For the methyl ethyl ketone/toluene system considered here, the curves of system considered here, the curves of Fig. Fig. 12.5(b) 12.5(b) for for G G EE/ RT, / RT, and and represent Eqs. (12.9b) and (12.10) represent Eqs. (12.9b) and (12.10) with:with:

AA1212 = 0.372 and A = 0.372 and A2121 = 0.198 = 0.198

These are values of the intercepts at These are values of the intercepts at xx1= 0 and 1= 0 and xx1 = 1 of the straight line 1 = 1 of the straight line drawn to represent the drawn to represent the data points. data points.

1ln 2ln

1 2/EG x x RT

3939



A set of VLE data has here been A set of VLE data has here been reduced to a simple mathematical reduced to a simple mathematical equation for the dimensionless excess equation for the dimensionless excess Gibbs energy:Gibbs energy:

This equation concisely stores the This equation concisely stores the information of the data set. information of the data set.

1 2 1 20.198 0.372EnG

x x x xRT

4040



For binary system:For binary system:

1 1 1 2 2 2sat satP x P x P

1 1 11

1 1 1 2 2 2

sat

sat sat

x Py

x P x P

4141



4242



A second set of data, for A second set of data, for chloroform(1)/ 1,4-dioxane(2) at chloroform(1)/ 1,4-dioxane(2) at 50°C, is given in Table 12.3, along 50°C, is given in Table 12.3, along with values of pertinent with values of pertinent thermodynamic functions. Figures thermodynamic functions. Figures 12.6(a) and 12.6(b) display as points 12.6(a) and 12.6(b) display as points all of the experimental values. This all of the experimental values. This system shows negative deviations system shows negative deviations from Raoult's-law behavior.from Raoult's-law behavior.

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Although the correlations provided by Although the correlations provided by the Margules equations for the two the Margules equations for the two sets of VLE data presented here are sets of VLE data presented here are satisfactory, they are not perfect. satisfactory, they are not perfect.

Finding the correlation that best Finding the correlation that best represents the data is a trial represents the data is a trial procedure. procedure.

4545


Thermodynamic ConsistencyThermodynamic Consistency

4646



4747



4848


This end of Section 1 of Chapter 12This end of Section 1 of Chapter 12

Questions?Questions?

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AssignmentAssignment

5050


THANKSTHANKS

chemical engineering thermodynamics lecturer: zhenxi jiang (ph.d. u.k.)

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