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MSE 303
Thermodynamics & Equilibrium Processes
Solution Theory (Gaskell Chapter 9)
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Terminology
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Liquid A
T
Vapor Pressure
Pressure gauge
PA0
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At equilibrium
Rates of Evaporation & Condensation for Single Component
0)( AAc kpr
0)( AAe kpr
Eqn 9.1
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Rates of Evaporation & Condensation for Single Component
)()( BcBe rr
0')()( BBcBe pkrr
E
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Rates of evaporation & condensation for a solution
If the mole fraction of A in the solution is XA and the atomic
diameters of A and B are similar, then assuming that the
composition of the surface of the liquid is the same as that
of the bulk liquid, the fraction of the surface sites occupied
by A atoms is XA.
Liquid A B
PA PB
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As A can only evaporate from surface sites occupied by A atoms, the rate of evaporation of A scales by a factor XA
Also, since at equilibrium, the rates of evaporation and condensation are equal to one another, the equilibrium vapor pressure of A exerted by the A-B solution is decreased from pA0 to pA.
AAAe kpXr )(
BBBe pkXr'
)(
Eqn 9.3
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Eq 9.5
Eq 9.6
Raoults Law
0AAA pXp
0BBB pXp
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So far we made the assumption that
Deviation from Raoults Law
are independent
A B B
A A B
B
B A B
B
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')(Aer
AAAe kpXr'
)( Eqn. 9.7 (See Eqn 9.3)
B
B A B
B
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0)( AAe kpr AAAe kpXr
')(
')(Aer
AAA XkpHenrys Law:
,
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Liquid A B
PA PB
')(Aer
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)('
)( AeAe rr
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Activity
0 ofactivity i
ii f
fai
.
the fugacity of a real gas is an effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations.
At constant T,
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Activity for ideal solutions
0i
ii p
pa Eqn 9.12
ii Xa
which is an alternative expression of Raoults law
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Basically, the introduction of activity normalizes the vapor pressure-composition relationship with respect to the saturated vapor pressure exerted in the standard state
0i
ii p
pa iii Xkpiii Xka
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Gibbs-Duhem Equation
Let Q be a thermodynamic properties
At constant T and P, the variation in Q with the composition of the solution
Define:
Then:
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kth component entire solution
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How is this useful?
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Gibbs Free Energy formation of a Solution AAG
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Example: The vapor pressures of ethanol and methanol are 44.5 mm and 88.7 mm Hg respectively. An ideal solution is formed at the same temperature by mixing 60 g of ethanol with 40 g of methanol. Calculate the total vapor pressure for solution and the mole fraction of methanol in the vapor.
Mol. mass of ethyl alcohol = C2H2OH = 46 No. of moles of ethyl alcohol = 60/46 = 1.304 Mol. mass of methyl alcohol = CH3OH = 32 No. of moles of methyl alcohol = 40/32 = 1.25 Then, XA, mole fraction of ethyl alcohol = 1.304/(1.304+1.25) = 0.5107 XB, mole fraction of methyl alcohol = 1.25/(1.304+1.25) = 0.4893 Partial pressure of ethyl alcohol = XA. pA0 = 0.5107 44.5 = 22.73 mm Hg Partial pressure of methyl alcohol = XB.pA0 =0.4893 88.7 = 43.73 m Hg Total vapour pressure of solution = 22.73 + 43.40 = 66.13 mm Hg Mole fraction of methyl alcohol in the vapour = Partial pressure of CH3OH/Total vapour pressure = 43.40/66.13 = 0.6563
Solution:
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Change in Gibbs Free Energy Due to the Formation of a Solution
dpPRTdG i
i
ppRTG
0
ln
And Recall 0i
ii p
pa
iiii aRTpureGsolutioninGG ln)()(
The difference between the two Gs (solution vs pure) is the change in the Gibbs free energy accompanying the introduction of 1 mole of component i into the solution
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Is there any change of volume in mixing?
For binary A-B solution,
Hence,
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The change in the volume in the formation of an ideal solution is zero
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Heat of formation of ideal solutions
Heat of formation of ideal solutions i
Mi XRTG lnFor an ideal solution
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Entropy of formation of ideal solutions
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With Sterlings approximation
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Substitute for nA and nB
Term inside brackets is always negative So Sconf is always positive during the formation of a solution
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Non-ideal Solutions
Non-ideal Solutions
0i
ii p
pa
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Statistical Model for Regular Solutions
Z
Then,
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Statistical Model for Regular Solutions
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Statistical Model for Regular Solutions
=
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Then,
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Of course
then
38 This is also equal to GXS, the excess Gibbs free energy of the solution
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From the Gibbs-Duhem equation,
Because
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On the other hand, according to the definition of activity,
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Example
Solution
So,
similarly
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From Table A-4, Gaskell