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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

Kuang-Hui Li

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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

Kuang-Hui Li

Kuang-Hui Li

Kuang-Hui Li

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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,

Kuang-Hui Li

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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