mse 303 note7_solution theory

1

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

4

At equilibrium

Rates of Evaporation & Condensation for

Single Component

0

)( AAc kpr

0

)( AAe kpr

Eqn 9.1

5

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

Raoult’s Law

0

AAA pXp

0

BBB pXp

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So far we made the assumption that

Deviation from Raoult’s Law

are independent

A B

B

A A B

B

B A B

B

10

'

)( Aer

AAAe kpXr '

)( Eqn. 9.7

(See Eqn 9.3)

B

B A B

B

11

0

)( AAe kpr AAAe kpXr '

)(

'

)( Aer

AAA Xkp Henry’s Law:

,

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Liquid A B

PA

PB

'

)( Aer

13

)(

'

)( AeAe rr

14

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

0

i

ii

p

pa Eqn 9.12

ii Xa

which is an alternative

expression of Raoult’s 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

0

i

ii

p

pa

iii Xkp iii 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 Gibb’s Free Energy Due to

the Formation of a Solution

dpP

RTdG

i

i

p

pRTG

0

ln

And Recall 0

i

ii

p

pa

iiii aRTpureGsolutioninGG ln)()(

The difference between the two G’s (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

M

i XRTG lnFor an ideal solution

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Entropy of formation of ideal solutions

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With Sterling’s 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

0

i

ii

p

pa

32

Statistical Model for Regular Solutions

Z

Then,

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Statistical Model for Regular Solutions

34

35

Statistical Model for Regular Solutions

=

36

Then,

37

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

40

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