thermo ii lecture 2 solutionthermo

8/9/2019 Thermo II Lecture 2 SolutionThermo

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Solution Thermodynamics

CH2351 Chemical Engineering Thermodynamics II

Unit – I, II

www.msubbu.in

Dr. M. Subramanian

Associate Professor

Department of Chemical Engineering

Sri Sivasubramaniya Nadar College of Engineering

alava!!am " #$% &&$' anchipuram (Dist)

Tamil Nadu' *ndia

msubbu.in+AT,gmail.com

Jan-2012

w w w .m s u b b u .i n


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Contents

- N*T */ P01PE0T*ES 12 S13T*1NS" Partial molar properties' ideal and non4ideal solutions' standard

states definition and choice' 5ibbs4Duhem e6uation' e7cessproperties of mi7tures.

- N*T **/ P8ASE E9*3*:0*A" Criteria for e6uilibrium bet;een phases in multi component non4

reacting systems in terms of chemical potential and fugacity

Jan-2012 M Subramanian



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Typical Chemical Plant




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Multicomponent Systems " :asic 0elations

- Single component system/

" *ntensive properties/ depends on Pressure' Temperature" E7tensive properties/ depends on Pressure' Temperature' and

amount

- Multicomponent system/" *ntensive properties/ depends on Pressure' Temperature' and

composition

" E7tensive properties/ depends on Pressure' Temperature'amount of each component




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Composition

Mole fraction

For binary solution

In dealing with dilute solutions it is convenient to speak of thecomponent present in the largest amount as the solvent, while thediluted component is called the solute.




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1ther Measures of Composition

- Mass fraction " preferable ;here the definition of molecular;eight is ambiguous (eg. Polymer molecules)

- Molarity " moles per litre of solution

- Molality " moles per !ilogram of solvent. The molality is usuallypreferred' since it does not depend on temperature or pressure';hereas any concentration unit is so dependent.

- =olume fraction

- Mole ratio or volume ratio (for binary systems)




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Properties of Solutions

- The properties of solutions are' in general' not additiveproperties of the pure components.

- The actual contribution to any e7tensive property is designatedas its partial property. The term partial property is used todesignate the property of a component ;hen it is in admi7ture

- :ecause most chemical' biological' and geological processesoccur at constant temperature and pressure' it is convenient toprovide a special name for the partial derivatives of all

thermodynamic properties ;ith respect to mole number atconstant pressure and temperature. They are called partialmolar properties




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Molar volumes:

Water: 18 mL/molEthanol: 58 mL/mol

Partial molar volumes(at 50 mole% of

Ethanol):

Water: 16.9 mL/molEthanol: 57.4 mL/mol

Ethanol4>ater System at


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1 liter of ethanol and 1 liter of water are mixed at constant temperature and

pressure. What is the expected volume of the resultant mixture ?



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Partial Molar Properties

- The partial molar property of a given component in solution isdefined as the differential change in that property ;ith respect to

a differential change in the amount of a given component underconditions of constant pressure and temperature' and constantnumber of moles of all components other than the one underconsideration.

;here M is any thermodynamic property.

- The concept of partial molar quantity can be applied to any

extensive state function.Jan-2012 M Subramanian



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Partial Molar =olume

- Benzene-Toluene: :en?ene and toluene form an ideal solution.The volume of & mole pure ben?ene is @@. mlB the volume of &mole pure toluene is &$#. ml. @@. ml ben?ene mi7ed ;ith

&$#. ml toluene results in @@. ml &$#. ml' or &.% ml ofsolution. (ideal solution)

- Ethanol-Water:

" The volume of & mole pure ethanol is @.$ ml and the volumeof & mole pure ;ater is &@.$ ml. 8o;ever' & mole ;atermi7ed ;ith & mole ethanol does not result in @.$ ml &@.$ml' or F#.$ ml' but rather F.% ml.

" >hen the mole fraction is $.' the partial molal volume of

ethanol is F. ml and the partial molal volume of ;ater is. ml. (non-ideal solution)




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2undamental E6uations of Solution

Thermodynamics


w w w .m s u

b b u .i n


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w w w .m s u

b b u .i n


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w w w .m s u

b b u .i n


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5ibbs4Duhem E6uation

- This e6uation is very useful in deriving certain relationshipsbet;een the partial molar 6uantity for a solute and that for the

solvent.




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Determination of artial !olar roperties of Binary "olutions




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Partial Molar 9uantities " Physical *nterpretation

- The partial molar volume of component i in a system is e6ual tothe infinitesimal increase or decrease in the volume' divided bythe infinitesimal number of moles of the substance ;hich isadded' ;hile maintaining T' P and 6uantities of all othercomponents constant.

- Another ;a to visuali?e this is to thin! of the chan e in volume

on adding a small amount of componentito an infinite totalvolume of the system.

- Note/ partial molar 6uantities can be positive or negativeG




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Example ro#lem




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7& 8 (HImol)84ideal(HImol)

8&bar(HImol)

8


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550

600

650

700

( J / m o l )

H-mix

H-ideal

H1bar

H2bar

350

400

450

0 0.2 0.4 0.6 0.8 1

x1 (-)

H




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b u .i n


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Partial Molar Properties from E7perimental Data

- Partial molar volume/

" Density data (ρ vs. 7&)

- Partial molar enthalpy/

" Enthalpy data (8 vs. 7&)B can be directly used

" 8eat of mi7ing (also called as enthalpy change on mi7ing)data (∆8mi7 vs. 7&)

- 1btained using differential scanning calorimetry

- 0eported normally as HImol of soluteB to be converted to HImol ofsolution


w w w .m s u b

b u .i n


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Density data for Water ($) - !ethanol (%) system at %&'$ *

7&

ρ (!gIm%) avgM> = =ideal

∆=mi7

m% I!mol m% I!mol m% I!mol

$ F@#.@# %


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0.025

0.03

0.035

0.04

0.045

e ( m 3 / k m

o l )

V

V-ideal

0.01

0.015

0.02

0 0.2 0.4 0.6 0.8 1

x1

V o

l u




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-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

0 0.2 0.4 0.6 0.8 1

V m i x

-1.20E-03

-1.00E-03

-8.00E-04

x1




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;tJ8

8(!HImol)

∆8mi7

(!HImol)∆8

mi7Model

(!HImol)

$


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2.00

6.00

10.00

( k J / m o l )

-10.00

-6.00

-2.00 . . . . . .

x1

H




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0edlich4ister Model

- Also !no;n as 5uggenheim4Scatchard E6uation

- 2its ;ell the data of ∆Mmi7 vs. 7&




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-10

-8

-6

-4

-2

0

0.00 0.20 0.40 0.60 0.80 1.00

H m i x

-16

-14

-12

x1

Data

RK Model fit

ao 4.


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Calculate the partial molar volume of Ethanol and Water as afunction of composition.



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5ibbs

- Hosiah >illard 5ibbs (&@% " &$)

- 5ibbs greatly e7tended the field of thermodynamics' ;hich

originally comprised only the relations bet;een heat andmechanical ;or!. 5ibbs ;as instrumental in broadening the fieldto embrace transformations of energy bet;een all the forms in

' ' '

electrical' chemical' or radiant.

- 8e is considered to be the founder of chemicalthermodynamics.

- 8e is an American theoretical physicist' chemist' andmathematician. 8e devised much of the theoretical foundationfor chemical thermodynamics as ;ell as physical chemistry.




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2undamental E6uation for Closed System

- The basic relation connecting the 5ibbs energy to thetemperature and pressure in any closed system/

" applied to a single4phase fluid in a closed system ;herein nochemical reactions occur.



2 d t l E ti f 1 S t


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2undamental E6uation for 1pen System

- Consider a single4phase' open system/

- Definition of chemical potential/

∑

∂

∂+

∂

∂+

∂

∂=

i

i

nT PinPnT

dn

n

nGdT

T

nGdP

P

nGnGd

j,,,,

)()()()(

(The partial derivative of 5 ;ith respect to the mole number ni at constantT and P and mole numbers n K L n K )

- The fundamental property relation for single4phase fluid systemsof constant or variable composition/

jnT Pi

i n

nG

,,

)(

∂

∂≡

µ

∑+−=i

iidndT nS dPnV nGd µ )()()(




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When n = 1,

∑+−=i

iidxSdT VdPdG µ ,...),...,,,,( 21 i x x xT PGG =

xPT

GS

,

∂

∂=

xT P

GV

,

∂

∂=

The Gibbs energy is expressed as a

function of its canonical variables.

Solution properties, M

Partial properties,Pure-species properties, M i

i M

ii G≡ µ




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Chemical potential and phase e6uilibria

- Consider a closed system consisting of t;o phases ine6uilibrium/

∑+−= ii dndT nS dPnV nGd α α α α α µ )()()( ∑+−= ii dndT nS dPnV nGd β β β β β µ )()()(

β α )()( nM nM nM +=

∑∑ ++−=i

ii

i

ii dndndT nS dPnV nGd β β α α µ µ )()()(

β α

µ µ ii =

β α ii dndn −=

Mass balance:

Multiple phases at the same T and P are in equilibriumwhen chemical potential of each species is the same in all phases.Jan-2012 M Subramanian

Since the two-phase system isclosed,



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Partial Molar Energy Properties and Chemical


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Partial Molar Energy Properties and Chemical

Potentials

The partial molar Gibbs free energy ischemical potential; however, the other partial

molar energy properties such as that ofinternal energy, enthalpy, and Helmholtz freeenergy are not chemical potentials: becausechemical otentials are derivatives withrespect to the mole numbers with the natural

independent variables held constant.


w w w .m s u b b

u .i n


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w w w .m s u b b

u .i n

= i i f i h T d P


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=ariation of µ ;ith T and P

- =ariation of µ ;ith P/ partial molar volume

- =ariation of µ ;ith T/ partial molar entropy' can bee7pressed in terms of partial molar enthalpy


w w w .m s u b b

u .i n


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Entropy Change


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Entropy Change

The second law of thermodynamics

in which the equality refers to a system undergoing a reversible change and the

For an isolated system

.

For systems that are not isolated it will be convenient to use the criteria ofreversibility and irreversibility such as in the following equation:



Mi7ing at Constant T and P


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Mi7ing at Constant T and P

To carry out the mixing process in a reversible manner, the external pressure P’ onthe right piston is kept infinitesimally less than the pressure of B in the mixture; andthe external pressure P’’ on the left piston is kept infinitesimally less than thepressure of A in the mixture.




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A h i i i i h l d h i i id l


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As the mixing process is isothermal, and the mixture is an ideal gas,



Entropy Change of Mi7ing


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- Consider the process' ;here nA moles of ideal gas A are confinedin a bulb of volume =A at a pressure P and temperature T. Thisbulb is separated by a valve or stopcoc! from bulb : of volume=: that contains n: moles of ideal gas : at the same pressure Pand temperature T. >hen the stopcoc! is opened' the gas

molecules mi7 spontaneously and irreversibly' and an increase inentropy ∆Smi7 occurs.

- e en ropy c ange can e ca cu a e y recogn ? ng a e

gas molecules do not interact' since the gases are ideal. ∆Smi7 isthen simply the sum of ∆SA' the entropy change for thee7pansion of gas A from =A to (=A =:) and ∆S:' the entropychange for the e7pansion of gas : from =: to (=A =:). That is'

∆Smix = ∆SA + ∆SB




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For the isothermal process involving ideal gases, ∆H is zero.Therefore,



Partial Molar Entropy of Component i in an ideal gas mi7ture


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5ibbs free energy change of mi7ing


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5ibbs free energy change of mi7ing


Chemical potential of component i in an ideal gas mi7ture


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i

ig

i

ig

i

ig

i y RT GG ln+=≡ µ

P RT T G iigi ln)( +Γ =

P y RT T iig

i ln)( +Γ =



0esidual Property


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2ugacity of a component i


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Chemical potential of component i in a solution

in terms of fugacity


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2ugacity and fugacity coefficient/ species


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in solution- 2or species i in a mi7ture of real gases or in a solution

of li6uids/

iii f RT T ˆln)( +Γ ≡ µ

Fu acit of s ecies i in solution

- Multiple phases at the same T and P are in e6uilibrium;hen the fugacity of each constituent species is thesame in all phases/

(replacing the particle pressure)

π β α

iii f f f ˆ...ˆˆ ===



ig R−=The residual property:


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igii

Ri M M M −=The partial residual property:

ig

ii

R

i GGG −=

iii f RT T ˆln)( +Γ ≡ µ

P y RT T iiig

i ln)( +Γ = µ

P y

f

RT i

iig

ii

ˆ

ln=− µ µ

i

R

i RT G φ ̂ln=

i

nT Pi

i Gn

nG

j

=

∂

∂≡

,,

)( µ

P y

f

i

ii

ˆˆ ≡φ

The fugacity coefficient of species i

in solution

For ideal gas,0=

R

iG

1ˆ

ˆ ==

P y

f

i

iiφ P y f ii =

ˆ


w w w .m s u b b u

.i n

The e7cess 5ibbs energy and the activity


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coefficient- The e7cess 5ibbs energy is of particular interest/

id E GGG −≡

iii f RT T G ˆln)( +Γ =

iii

id

i f x RT T G ln)( +Γ =

ii

i E i

f x f RT Gˆ

ln=

ii

ii

f x

f ̂≡γ

i

E

i RT G γ ln=The activity coefficient of species

iin solution.A factor introduced into Raoult’s law to account for

liquid-phase non-idealities.For ideal solution,

c.f.

i

R

i RT G φ ̂ln= 1,0 == i E

iG γ


w w w .m s u b b u

.i n


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1ˆ

ˆ ==P

f iiφ P y f ii =

ˆ ii

x

f ̂≡γ

i

E

i RT G γ ln=

0= R

iG1,0 == i

E

iG γ

For ideal solution,

w w w .m s u b b u

.i n

2ugacity of a pure li6uid


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- The fugacity of pure species i as a compressed li6uid/

sat

i

isat

ii f

f RT GG ln=− )( processisothermaldPV GG

P

P i

sat

ii sat i

∫=−

=P

isat

i

sat dPV

f 1ln

i i

Since Vi is a weak function of P

RT

PPV

f

f sat il

i

sat

i

i )(ln −=

sat

i

sat

i

sat

i P f φ = RT

PPV P f

sat

i

l

isat

i

sat

ii

)(exp

−= φ



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Infinite dilution of girls in boys



thermo ii lecture 2 solutionthermo

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