CHAPTER 1: CO 2 CAPTURE
3. Theory of Solution
3.5 Non-ideal solution
54 V_f = ª∂nfΩ
∂Pæ
i
!!!!!! 3145!
Based on Svf= − ®©™lø
©i´
j,kf and H_f= nf!+ TSvf, partial enthalpy, and entropy will be obtained:
9_∫!= 9_°∫=9_¡∫! 3146
≤_∫!= ≤_°∫=≤_¡∫! 3147!
The partial molar volume and internal energy of the solute are constant in the ideal-dilute range and are equal to the values at infinite dilution (M_¡¬, (M = V, H)!). When the pressure is equal to the standard pressure p∘, the ( M_¡¬) quantities are equal to the standard values (M_°¬).
•v∫ = ! •∗∫− °<*!'∫ 3148!
!
•v∫! = ! •v°∫− °!<*'∫!!!!!!!!! 3149!
! Positive deviation from Raoult’s Law
In this case, the vapor pressure of the mixture is always greater than that expected by Raoult’s Law. The intermolecular!forces!between!dissimilar!molecules!of!A!and!B!are less strong than!
those! in! pure! liquids.! Thus,! the! tendency! of! each! component! to! pass! to! the! vapor! phase!
increases.!
ƒ#> ƒ#∗$# or y# > y#∗$#
Molecules!in!this!case!are!breaking!away!more!easily!than!they!do!in!pure!one. Indeed, the resulting! solution! has! a!larger! enthalpy! of! the! solution! than! pure! components! of! the!
solution,!thus!the solution process needs heat to be absorbed to move forward!which!states!the!
process!is!endothermic.!If!the change of enthalpy in the solution process is large and positive, then the liquids may be immiscible.
!
! Negative deviation from Raoult’s Law
In!this!case,!the!intermolecular!forces!between!dissimilar!molecules!are greater than!those!in!
pure!liquids.!Thus,!the!tendency!of!each!component!to!pass!to!the!vapor!phase!is!low.!
ƒ#< ƒ#∗$#!!!!or!!!y#< y#∗$#!!
It states that the vapor pressures of the mixture are less than that would be expected for an ideal mixture via Raoult’s Law. In negative deviation from Raoult’s Law, the solution process is accompanying by releasing more energy in the form of heat which results in making the reaction process exothermic. It means more heat is given out when the new stronger bonds are made than was used in breaking the original weaker ones.
! Chemical potential of non-ideal solutions
If a mixture is not ideal, we can write a more accurate description of the chemical potentials of each component in condensed solutions after an activity coefficient γi is introduced within the logarithm. The chemical potential is then written as:
56
∞( = ∞(∘+ °D<*(…('() 3-50
Where p#∘ is the chemical potential of constituent i in the standards state. The activity coefficient of component i # is a dimensionless factor that takes into account the deviation from ideal behavior in the solution [170]. This coefficient considers the non-ideal characteristics of a mixture and it is between 0 and 1. For an ideal solution # approaches 1 and then the Raoult’s Law is accurate. If # > 1 or # < 1!then the substance i shows negative or positive deviation from ideality/Raoult’s Law respectively. By substituting the product #$# to the variable!4#which is known as the activity of component i the chemical potential is rewritten as:
!
∞( = ∞(∘+ °D<*(À()!! 3151
! Thermodynamic of mixing for non-ideal solution
Based on ∆Gìfä= G − G∗= ∑ xf f(nf− nf∗), and the definition of nf for the non-ideal solution the Gibbs free energy of mixing for non-ideal solutions is written as follows:
ü†;('! = °D / *(
!<*À(! 3152
all other thermodynamic mixing properties for non-ideal solutions are listed as follows ΔSìfä! = L∂ΔGìfä!
∂T O
j
!!= −RT / xf
f
ªδln!af!
∂T æ
j
− !R / xf
f
ln!af 3153
ΔHìfä! = ΔGìfä+!TΔSìfä! =!!− RTê/ xf
f
ªδln!af!
∂T æ
j
! 3154
ΔVìfä! = LδΔGìfä!
δP O
i,kl
!!= RT / xf
f
ªδln!γf!
∂P æ
i
! 3155!
! Excess functions
The deviation from ideal behavior is well understood by the thermodynamic excess property/function. It can provide information about the nature of intermolecular interactions, i.e.,
attraction or repulsion. Excess quantities are properties of mixtures that characterize their non- ideal behavior. If M represents the molar value of any extensive property, then an excess quantity
^Õ of a mixture is defined as the difference between the value of the extensive property of the real mixture and the value it would have if the solution is ideal [169] at the same temperature, pressure, and composition. Therefore,
CŒ = C − C(P!! 3156
for example:!dÕ = d − d#œ, similar definitions hold for other excess properties:
H– = H − Hf— 3157
S– = S − Sf— 3158
V– = V − Vf—! !
3159
Moreover, subtraction of Gf— = Hf—− TSf— from G!= H!− TS! gives G–= H–− TS–. The excess property bears a relationship to the property change of mixing.
∆M = M −/xi!Mi
i
3160
G–= G − Gf—= !G − (/ xf!Gf+ RT
f
/ xf!ln!xf!!
f !
3161
Thus, G–= ∆Gìfä− RT! / xf!ln!xf!!
f !
! 3162
S– = ∆Sìfä+ RT! / xf!ln!xf!!
f !
3163
The excess enthalpy and volume are both equal to the observed enthalpy and volume of mixing, because the ideal values are zero in each case thus,zÕ = ∆zí#Å andaÕ= ∆aí#Å. As for ideal solutions, the excess properties are equal to zero, thus,
∆Gìfä = RT! / xf!ln!xf!!
f !
3164
∆Sìfä = −RT! / xf!ln!xf!!
f !
3165
∆Vìfä = 0,!∆Hìfä=0!!
!
! 3166
58
Subtracting ∆M(id) = M(id) − ∑ xf f!Mf from ∆M = M − ∑ xf f!Mf gives us the following equation that represents the excess properties and states that the property changes of mixing are readily calculated one from the other.
M–= ∆M − ∆Mf—!!! 3167
3.5.5.1! Excess chemical potential
nf– = n÷− nf—!!! 3168
Based on!nf°(real) = nf°(ideal)!then:
nf– = (nf°− RTlnaf) − (nf°− RTlnxf) = RTlnγf!!! 3169 Gf–= G÷− Gf—!= RT / xflnaf
f
− RT / xflnxf
f
= ÿx / $#ú" #
#
! 3170
Similarly, other important relations include:
S!– = − ®©Ÿ©il⁄´
j= −R ∑ nf f®©¤k‹©il!´
j! 3171
H!– = Gf–− TSf– = −RTê!/ nf
f
Æ∂lnγf!
∂T ±
j
3172
∆V!– = − ª∂Gf–
∂Pæ
i
= −RTê/ nf
f
Æ∂lnγf!
∂P ±
i! 3173!
The nonzero values of the excess functions are due to the fact that interactions between particles of different components are different from the interactions between particles of the same component.
3.5.5.2!Partial molar volume
In a system that contains at least two substances, the total value of any extensive property of the system is the sum of the contribution of each substance to that property. The contribution of one
mole of a substance to the volume of a mixture is called the partial molar volume of that component. At constant T and P:
V = nèÆ∂V
∂nè± + nêÆ∂V
∂nê±
!
3174
zè,í = "èÆ›z
›"è±
[,\,Wfi
+ "êÆ›z
›"ê±
[,\,Wfl
3175
zè,í = Æ›z
›"è±
[,\,Wfi
!!! 3176!
zê,í = Æ›z
›"ê±
[,\,Wfl
!! 3177!!
The partial molar volume for the pure components is Vè,ì∗ =ìfl
‡fl!and are Vê,ì∗= ìfi
‡fi. The total molar volume and molar volume of components can be obtained from
zí = $è!zè,í+ $ê!zê,í = ·zè,í− zê,í‚$è!+ zê,í 3178 so,
zè,í = zí+ $êÆ›zí
›$è±
!
3179
zê,í = zí+ $èÆ›zí
›$è±
!
! 3180
∆zí#Å = zÕ = ∆zê+ $誛zÕ
›$èæ
!
=!∆zè− $ꪛzÕ
›$èæ
!
3181
The most common method of measuring partial molar volumes is to measure the dependence of the volume of a solution upon its composition. The observed volume can then be fitted to a function of the composition (usually using a computer), and the slope of this function can be determined at any composition of interest by differentiation.
3.5.5.3! Excess molar volume
Volume changes on mixing at constant pressure and temperature are an indicator of the non- idealities present in real mixtures. The change in volume on mixing two liquids, especially two
60
polar liquids, 1 and 2 can be attributed to several procedures including : (a) the breakdown of 1 - 1 and 2 - 2 intermolecular interactions which have a positive effect on the volume, (b) the formation of 1 - 2 intermolecular interactions which results in a volume contraction of the mixture, (c) packing effects caused by a difference in the size shape of the component species and which may have a positive or negative effect on the value, and (d) formation of new chemical species.
The excess molar volume upon mixing has been determined via two principal methods, directly and indirectly. The direct measurements involve mixing the liquids and observing the resultant volume change in dilatometers and the indirect measurements involve measuring the density of the pure liquid as well as the density of the mixture using densitometers or pycnometers. The dilatometer is filled with known masses of pure liquids, which are separated by mercury. The height of mercury in the calibrated graduated column is noted. The liquids are mixed by rotating the dilatometer and the volume change on mixing is indicated by the change in the height of the mercury in the calibrated capillary.
The indirect determination of excess volume for a binary mixture can be determined from density measurements of components in the pure and mixed state using the following equation:
zÕ =($è^è+ $ê^ê)
„èê − Æ$è^è
„è ± − Æ$ê^ê
„ê ± 3182
where x1 and x2 are the mole fractions, M1 and M2 are molar masses, ρ1, ρ2 and ρ are the densities where 1 and 2 refer to the component 1 and 2, respectively and ρ is the density of the mixture. A brief review of density measurement techniques has been discussed in the following chapter.