Research: Science and Education
The Gibbs Energy Basis and Construction of Boiling Point Diagrams in Binary Systems Norman O. Smith1 Department of Chemistry, Fordham University, Bronx, NY 10458
An earlier paper (1) illustrated how excess Gibbs energies of the components in binary systems can be used to construct melting point diagrams. The present paper is an extension of the treatment to boiling point diagrams. It will be limited to completely miscible liquid pairs. Current textbooks include discussion of such systems but rarely, if ever, cover the underlying causes of the various types of behavior in terms of intermolecular forces, and do not do so quantitatively. This article attempts to do this and shows how, from quantitative data, one can calculate the coexisting liquid and vapor compositions in boiling point diagrams with acceptable accuracy. Because T–x diagrams (isobars) are generally more practical than vapor pressure diagrams (isotherms) it is desirable and instructive to show how one can be obtained quantitatively from information normally used to construct the other, such as activity coefficients. The method, although somewhat laborious, is well worth the effort and permits study of the factors determining the results, as will be shown. It can be adapted later to the computer for routine, timesaving use and makes a good preliminary study for that purpose. Method
Table 1. Properties of Componentsa ∆vH o/kJ mol᎑1
∆vSo/J K᎑1 mol᎑1
329.28
29.087
88.34
Chloroform
334.88
29.37
87.7
Methanol
337.9
35.27
104.4
Cyclopentane
322.41
27.296
84.66
Cyclohexane
353.88
30.083
85.01
Benzene
353.25
30.765
87.09
Component
Tb°/K
Acetone
a
Data from ref 2.
liquid solutions, all that is required is an expression of intermolecular forces as they affect the excess Gibbs energies or the activity coefficients. For this purpose the quantities g0 and g1 have been chosen, which are parameters in the following relations:
There are three types of boiling behavior for miscible, volatile liquid pairs:
G A E = RT ln γ A =
I. those in which the boiling points of the solutions lie between those of the components,
(g0
− g 1 ) x B2 + 2 g 1 x B3
(1)
G B E = RT ln γ B = (g 0 + 2 g 1 ) x A2 − 2 g 1x A3
II. those in which they pass through a maximum as the composition changes, and III. those in which they pass through a minimum.
These will be denoted as Types I, II, and III, respectively, by analogy with the corresponding melting point types (1). As the calculated diagrams are to be for 1 atm, the following data will be needed for the pure components: the normal boiling points (T/K), the enthalpies of vaporization (∆vH °) and/or the entropies of vaporization (∆vS °) at T °. These data are provided in Table 1 for six liquids. For the
(2)
The form of eq 2 is dictated by that of eq 1 according to the Gibbs–Duhem relation, xAdGAE + xBdGBE = 0 at constant T and P. γA and γB are the activity coefficients; xA, xB are the mole fractions; GAE, GBE are the molar excess Gibbs energies of A and B in the liquid solutions; and R and T are the gas constant and temperature. (More extended expressions with additional parameters can be used for more complicated relationships if necessary.) The standard states for A and B are the respective pure liquids at T. The parameters given in Table 2 were used in all subsequent calculations.
Table 2. Types and Excess Gibbs Energy Parameters for Systems Discussed System
g0/J mol᎑1
g1/J mol᎑1
1118.15
᎑115.65
318.15
ref 3, p 206
᎑1857
᎑371
323.15
ref 3, p 206
III
1391
494
323.15
ref 3, p 227
III
990
0
352
ref 4, p 286
Type
Component A
Component B
Cyclopentane
Benzene
I
Acetone
Chloroform
II
Acetone
Methanol
Benzene
Cyclohexane
T/K
a
Source of Data
a
For which parameters given.
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Research: Science and Education
For the vapor phases, however, ideal behavior will be assumed, so that GAE = GBE = 0, and thus the activity coefficients taken to be unity. This assumption may be questioned, but is commonly made. In ref 1, Table 1, where g1 for both phases is zero, there is a progression from Type II through Type I to Type III melting behavior, for a given hypothetical pair of components, as g0l – g0s is varied from large positive through zero to large negative values. Analogous behavior is seen in liquid-vapor systems, where g0l – g0s is replaced by g0v – g0l, except that g0v is taken to be zero or close to it. Thus one can expect a Type II boiling point diagram when g0l is strongly negative, Type I when it is small (negative or positive), and Type III when it is strongly positive. These conclusions must be applied with caution, however, because close proximity of the boiling points to each other can produce effects that override them (see below). Finally, attention may be drawn to the fact that Type II melting point diagrams, where g0l – g0s is large and positive, are rare, whereas Type II boiling point diagrams, where g0v – g0l is large and positive, are common. Type II melting occurs, then, when g0l is large and positive but g0s is large and negative, an almost contradictory situation considering the close proximity of the molecules in condensed phases. In Type II boiling, even with g0v near zero, g0l can be sufficiently large and negative to make g0v – g0l large and positive a common occurrence. When g0l is excessively large and positive, partial miscibility in the liquid will occur. For the construction of boiling point diagrams two equations are set up at a given temperature T, one for each component, describing the changes in Gibbs energy for the transfer of one mole of component through an isothermal, isobaric cycle at 1 atm as follows: (i) 1 mole of pure A(l ) is transferred to the equilibrium liquid solution (xAl = x), then (ii) transferred to the equilibrium vapor (xAv = y), then (iii) to pure A(v), and finally (iv) the vapor is condensed to pure A(l ) to complete the cycle. The Gibbs energies associated with these steps total zero, and are given, respectively, by
systems and is equivalent to assuming that the entropy of mixing is ideal. The solution of eqs 3 and 4 for x and y at various assigned Ts permits the construction of the boiling point diagram for the entire system. If an azeotrope is present there will be two pairs of roots for each T for several temperatures chosen. Details of solving eqs 3 and 4 will be given below. Before attempting to solve eqs 3 and 4, however, it is desirable and helpful to establish whether the system has an azeotrope and, if so, where it lies. Both its temperature (Tm) and composition (xm) can be found by setting x = y = xm and T = Tm in eqs 3 and 4. This causes two terms in each equation to cancel, giving respectively, 2
(g0
− g 1 ) (1 − x m ) + 2 g 1 (1 − x m )
(g0
(
)
with the help of eq 1 and the Gibbs–Helmholtz relation, (∂∆vGAo/∂T )p = ᎑∆vSAo. Analogously for B,
(
Journal of Chemical Education
(4)
)
•
)
(6)
These are two simultaneous equations in two unknowns, Tm and xm, which can be solved as follows. First Tm is expressed explicitly as a function of xm from eq 5, namely Tm =
1 2 g 1x m3 − ( g 0 + 5 g 1 ) x m2 ∆ v S A° + 2 ( g 0 + 2 g1 ) xm − ( g 0 + g1 )
(7)
+ ∆ v S A° TA°
and from eq 6, namely 2 g 1x m3 − ( g 0 + 2 g 1 ) x m2 + ∆ v S B° TB° ∆ v SB°
(8)
Equating the right sides of eqs 7 and 8 and collecting terms gives
(
)
F ( x m ) = 2 g 1 ∆ v S B° − ∆ v S A° x m3 2 + ( g 0 + 2 g 1 ) ∆ v S A° − ( g 0 + 5 g 1 ) ∆ v SB° x m
(
In the last terms of eqs 3 and 4 it is assumed in the integration of the Gibbs–Helmholtz equation that ∆vS° is independent of T over the short temperature range of integration. It will also be assumed that, for a given concentration, the GAE and GBEvalues, determined experimentally at neighboring temperatures (given in Table 2), are valid at all temperatures of interest. This is an acceptable assumption for many 420
(
+ 2 ( g 0 + 2 g 1 ) ∆ v S B° x m + ∆ v S A° ∆ v S B° TA° − TB°
RT ln (1 − x ) + ( g 0 + 2g 1 ) x 2 − 2 g 1 x 3 + 0 1 + ∆ v S B° T − TB° = f B = 0 + RT ln 1 − y
(5)
+ 2 g 1 ) x m2 − 2 g 1 x m3 + ∆ v S B° Tm − TB° = 0
3
(3)
)
and
RT ln x + ( g 0 − g 1 ) (1 − x ) + 2 g 1 (1 − x ) + 0 1 = fA = 0 + RT ln + ∆ v S° A T − TA° y
(
+ ∆ v S A° Tm − TA° = 0
Tm = 2
3
− ( g 0 + g 1 ) ∆ v S B° = 0
)
(9)
This cubic equation in xm can be solved by estimating a value for xm (xm0), calculating F(xm0) and its first derivative, F´(xm0), and evaluating ∆xm0 = F(xm0)/F´(xm0). The value of xm0 – ∆xm0 or xm1, is the first improved approximation of xm. Continued repetition of this process causes xm to converge on the final solution. Tm may readily be found from this by using eq 8 (or eq 7). Its magnitude tells whether it is a maximum or a minimum.
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If the system has no azeotrope, this will be evident either by difficulty in getting xm to converge, or if it does, the value is unacceptable, outside the range from 0 to 1. Having determined the composition and temperature of the azeotrope (if any) the rest of the diagram can now be found by solving eqs 3 and 4 for x and y at various assigned values of T, starting with temperatures near TA° or TB° and working toward Tm. The procedure, illustrated in ref 1, is one of successive approximation, analogous to that shown above, but with two unknowns, not one. After estimating a pair of roots (x0, y0) the following equations are solved for ∆x and ∆y:
∂ f ∂ f B ∆ x + B ∆ y = f B ∂ x ∂ y
The new roots (x´, y´) are determined iteratively as x´ = x0 – ∆x and y´ = y0 – ∆y. This operation is continued until x and y converge. Repetition of this process at other temperatures yields the entire diagram. The results so calculated for each of the four systems chosen for illustration are given in Table 3 and shown graphically in Figure 1. Results and Discussion A useful rule that emerges from application of eq 9 is the following: for systems showing a maximum boiling azeotrope, the mole fraction at the azeotrope of the lower boiling component will be lower than that of the higher boiling component; for systems showing a minimum boiling azeotrope, the mole fraction at the azeotrope of the lower boiling component will be higher than that of the higher boiling component. The ideal boiling point diagram for each system can be readily obtained by letting g0 = g1 = 0 in eqs 3 and 4 to give, respectively,
(
)
(10)
1 − x ∆ S ° ln = v B TB° − T RT 1 − y
(
)
(11)
from which x and y can be found for various assigned values of T. The results are included in Table 3 and Figure 1. No azeotrope is possible in ideal systems, although x approaches y at every temperature as TA° approaches TB°, and the liquidus and vaporus become coincident in the limit. This condition is approximated in the system benzene–cyclohexane, Figure 1(c). The genesis and development of an azeotrope with increase in nonideality (as measured by the g parameters) can be illustrated by the benzene–cyclohexane system where, conveniently, g1 = 0. For g0 = 990 J/mol, xm = 0.524 (Table 3). By substituting xm = 1, into eq 9 and solving for g0, one finds it to be 53.6 J/mol. For g0 = 53.6, no azeotrope is possible, www.JCE.DivCHED.org
Calculated
T/K
xA
Ideal
γB
γA
yA
xA
yA
(a) Cyclopentane (A)–Benzene (B) [Type I] 353.23 0
0
(1.461)
1.000 0
345.0
0.123
0.320
1.350
1.005 0.189 0.369
0
335.0
0.362
0.623
1.182
1.049 0.483 0.708
327.0
0.707
0.846
1.039
1.220 0.787 0.908
322.41 1.000
1.000
1.000
(1.526) 1.000 1.000
(b) Acetone (A)–Chloroform (B) [Type II]
∂ f ∂ f A ∆ x + A ∆ y = f A ∂ x ∂ y
x ∆ S ° ln = v A TA° − T RT y
Table 3. Calculated Data for Construction of 1-Atmosphere Boiling Point Diagrams and Activity Coefficientsa in Typical Binary Systems
•
334.88 0
0
(0.436)
1.000 0
0
336.0
0.077
0.049
0.502
0.994 —
—
337.5
0.200
0.161
0.609
0.964 —
—
338.3b
0.361b 0.361b 0.742
0.893 —
—
338.0
0.480
0.523
0.828
0.825 —
—
337.0
0.568
0.643
0.882
0.770 —
—
335.0
0.687
0.780
0.939
0.693 —
—
333.0
0.803
0.884
0.977
0.618 0.302 0.340
331.0
0.907
0.954
0.995
0.554 0.672 0.710
329.28 1.000
1.000
1.000
(0.501) 1.000 1.000
(c) Benzene (A)–Cyclohexane (B) [Type III] 353.88 0
0
(1.403)
1.000 0
0
352.6
0.106
0.136
1.310
1.004 —
—
353.7
—
—
351.5
0.247
0.284
353.6
—
—
350.7b
0.524b 0.524b 1.080
353.4
—
—
351.5
0.808
0.777
1.013
1.247 —
—
352.6
0.943
0.926
1.003
1.351 —
—
353.25 1.000
1.000
1.000
(1.403) 1.000 1.000
—
—
1.211 —
—
—
0.232 0.235
1.021 — 1.097 — —
—
0.484 0.488 —
0.756 0.759
(d) Acetone (A)–Methanol (B) [Type III] 337.9
0
0
(2.017)
1.000 0
336.0
0.051
0.114
1.850
1.002 0.225 0.278
0
333.0
0.170
0.294
1.553
1.024 0.572 0.645
330.0
0.414
0.509
1.208
328.6b
0.774b 0.774b 1.022
1.433 —
—
329.0
0.927
0.917
1.002
1.597 —
—
329.28 1.000
1.000
1.000
(1.678) 1.000 1.000
1.134 0.917 0.939
aThe activity coefficients, as such, were not used to construct Figures 1(a)–(d): they were, in effect, replaced by the g parameters, and are included here for purposes of discussion (see text). bAzeotrope.
Table 4. Ranges of TB° Needed To Yield Azeotrope with TA° Fixed System
Range of TB°/K
Cyclopentane (A)–Benzene (B)
310.6–335.3
353.2
Acetone (A)–Chloroform (B)
308.1–354.4
334.9
Benzene (A)–Cyclohexane (B)
341.9–364.9
353.9
Acetone (A)–Methanol (B)
307.9–357.6
337.9
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Actual TB°/K
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421
Research: Science and Education A 355
Temperature / K
350 345
V
340 335
L
330 325 320 0.0 A
0.5
1.0 B
Mole Fraction B
B 340
Temperature / K
338
V 336 334 332
L 330 328 0.0 A
0.5
1.0 B
Mole Fraction B
C
Temperature / K
354
V
353
352
L
351
0.0 A
0.5
1.0 B
Mole Fraction B
D
336
Temperature / K
Acknowledgment The author is grateful to Norman Bray of Lehman College, The City University of New York, Bronx, NY 10468 for his assistance in preparing the diagrams and in transcribing the manuscript onto a disk for publication.
338
V
334
Note
332
330
1. Present address to which all correspondence should be sent: 811 East Central Rd. Apt. 112, Arlington Heights, IL 60005.
L
Literature Cited
328 0.0 A
0.5
1.0 B
Mole Fraction B
Figure 1. Calculated boiling point diagrams at 1 atmosphere for typical binary miscible liquid systems: (a) the system cyclopentane (A)-benzene (B) (Type I), (b) the system acetone (A)-chloroform (B) (Type II), (c) the system benzene (A)-cyclohexane (B) (Type III), and (d) the system acetone (A)-methanol (B) (Type III). Solid line (––––): calculated actual; dashed line (———): calculated ideal.
422
but as g0 increases above 53.6 the azeotrope emerges from xm = 1 and T = TA° to smaller xm and Tm values, reaching xm = 0.524, Tm = 350.7 K when g0 has become 990 J/mol. Eq 7 or 8 can be used to find Tm for each value of xm. It is to be noted that, because of the proximity of TA° and TB°, very little nonideality (g0 = 53.6) is needed to give an azeotrope. The dominating influence here of the proximity of the component boiling points over the g values is thus apparent. If, on the other hand, a system (for which g1 = 0) with g0 less than 0 instead of greater than 0, but with TA° still close to but less than TB° had been chosen for illustration instead of benzene–cyclohexane, a different but analogous situation would have resulted involving a maximum boiling azeotrope. There would have been a negative threshold value of g0 between zero and which there would be no azeotrope but for more negative values than which a maximum boiling azeotrope would appear, having emerged from the xm = 0 side of the diagram at TB°. The importance of TA° – TB° in determining the presence of an azeotrope has been mentioned above. By imagining TA° to be held constant while TB° is changed in eq 9, but keeping ∆vSA°, ∆vSB° and the g parameters constant, the range of values within which xm will lie between 0 and 1, that is, within which an azeotrope will be formed, can be easily calculated. Table 4 gives the results for the systems given in Table 3. In only the first system, the one with no azeotrope in reality, does TB° lie outside the calculated range, as expected. It can be seen from the foregoing discussion that Type I behavior can be shown by systems with γA and γB less than 1 or greater than 1 provided that TA° and TB° are sufficiently separated. In all nonideal systems, azeotropes can result when TA° is sufficiently close to TB°. Finally, in molecular terms, the magnitude of the activity coefficients reflects the relative intermolecular attractive forces: with γA and γB less than 1, the attraction of A for B molecules dominates those of A for A and B for B molecules, whereas with γA and γB greater than 1, the attraction of A for A and B for B causes “antisocial” behavior that, if excessive, can produce partial liquid miscibility and two liquid phases.
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1. Smith, N. O. J. Chem. Educ. 1997, 74, 1080–1084. 2. Stull, D. R.; Westrum, E. F., Jr.; Sinke, G. C. Thermodynamics of Organic Compounds; John Wiley and Son: New York, 1969. 3. Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall, Inc.: New Jersey, 1969. 4. Lewis, G. N.; Randall, M.; revised by Pitzer, K.; Brewer, L.; Thermodynamics; McGraw-Hill: New York, 1961.
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