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Sect' 7 Twocomponent Systems

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solution: homogeneous liquid or solid A-B phase ... (Stephen Hawking) 17. In terms of mole fractions instead of moles: - Divide (5) and (6) by nA nB: ... – PowerPoint PPT presentation

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Title: Sect' 7 Twocomponent Systems


1
Sect. 7 Two-component Systems
  • Components A and B (binary system)
  • Designations for single-phase A B systems

- mixture A-B gas or 2 immiscible condensed
phases
- solution homogeneous liquid or solid A-B phase
  • Conditions for mixture/solution ideality

- gases very closely ideal no intermolecular
interactions
- condensed strong A-A and B-B interactions when
pure
- for solution ideality A-B interaction must be
the mean of the A-A and B-B interactions
- A-B solutions are rarely ideal
2
Ideal Gas Mixtures
  • Daltons law of partial pressures (1807)

Each component of an ideal gas mixture occupies
the total volume as if it were alone
pAV nART pBV nBRT
  • pA, pB are the partial pressures of A and B
  • add (pA pB)V nRT p pA
    pB
  • divide individual eqns by the total

3
Entropy of Mixing
  • applies to gas mixtures, liquid solid
    solutions,ideal or nonideal use ideal gas to
    derive
  • ?smix entropy increase when xA moles of A and
    xB moles of B are mixed at constant T and p to
    form 1 mole of an ideal mixture

Dsmix
DsB
DsA
DsAB
4
  • Dalton route start with pure gases at pressure p

1. Reduce pressure of each to their partial
pressures in the mixture this incurs entropy
changes
2. Mix A and B at their partial pressures this
does not involve an entropy change (see Fig. 7.3)
For n moles total
  • applicable to solids and liquids as well as to
    gases

5
(No Transcript)
6
Examplemix 2 moles of soln 1 (xA 0.5) and 3
moles of soln 2 (xA0.2)
3.0
2.0
DSmix2
DSmix DSmix3 DSmix1 DSmix2
0.24R
7
insulated tanks
N2
He
2 moles He, 100oC, 1 atm
1 mole N2, 200oC, 0.5 atm
0.061m3 373 K moles 1 atm
0.078 m3 To 1 mole po
valve
Open valve tank contents mix what are DU, DH
DS as fns of po?
3CVmix(T Tref) 2CVHe(373 Tref) CVN2
(To Tref)
Tanks valve piping ? isolated system ? DU 0
T final temperature
Tref reference temperature (cancels out)
CVHe 3/2 R CVN2 5/2 R CVmix ? CVHe
?CVN2 1.83 R
8
Enthalpy Change 2 methods
?H1 3CPmix(TTref) 2CPHe(373Tref)
CPN2(ToTref)
?H2 ?U ?(pV) ?(pV) 3RT 2Rx373 RTo
Entropy change
Bring pure gases to final T and p
Mix at constant T and p
?S DSHe DSN2?Smix
9
specify po
10
Properties of liquid solid solutions
  • Temperature and composition are the only
    variables the effect of total pressure is
    neglected why?

T1300oC, p10.1 MPa
T2400oC, p210 MPa
Properties CP 3R 25 J/mole-oC
v 7.4x10-6 m3/mole a 6x10-5 oC-1
hA-h125x100 2500 J/mole
h2-hA 7.4x10-6(1-6x10-5x673)(10-0.1)x106 70
J/mole
The Dp contribution to Dh is 3 of the DT effect
11
Ideal solutions
  • Properties are the sum of pure-component values

. but
ni moles of component i
vi, hi, si molar properties of pure components
What is Dsmix for an ideal A/B solid?
12
Entropy of Mixing A/B solid
distribute NA atoms of A on NS lattice sites Atom
1 NS ways Atom 2 NS 1
ways . Atom NA NS - (NA 1) ways
But, this assumes atoms are distinguishable
is distinct from
A3
A1
A2
but is not distinct from
?Only 2 states
A
A
A

A
A
A
13
Remove distinguishibility by dividing by NA!
Now add B atom to NS-NA NB sites ? WB 1
Boltzmanns equation Smix Rln(WAWB)
Smix klnNS! ln(NS NA)! lnNA!
Stirlings approximation lnN! NlnN
lnWA -NAln(NA/NS) (NS NA)ln(NS-NA)/NS
xA NA/NS
xB NB/NS
Smix -R(xAlnxA xBlnxB)
Same as derived for ideal gas mixture!
14
Nonideal solutions
(?Z/?T)ni soln property (e.g., CP for Z H)
  • Physical meaning the change in property Z of a
    large quantity of solution when 1 mole of
    component i is added with T and mole numbers of
    all other components held constant

15
Proof of the equivalence of (1) and (2)
  • Integrate dZ at constant composition
    physically, this corresponds to adding components
    A and B at rates proportional to the composition
    of the final solution

After nA and nB moles are added
Which is the same as (1).
QED
Corollary from (4) (5)
_ _ nAdzAnBdzB0 (6)
16
I believe thermodynamics to be a science on a
par with relativity
  • (Stephen Hawking)

17
In terms of mole fractions instead of moles
_ - Solve (6a) for
dzB integrate from 0 to xA
(7)
  • _
    _
  • measuring as a function of composition gives

How to determine ?
18
_ Determination
of zA
  • partial molar properties cannot be measured
    directly only the molar properties are
    accessible
  • divide (4a) by dxA
  • multiply by 1-xA
  • add to (5a)

(8)
Graphical method measure slope at point P can
prove that intercepts
19
The excess property
  • excess property is an alternative to the
    partial molar property method as a measure of
    nonideality
  • zex excess z - can sometimes be measured
    directly when pure components A and B are mixed

- hex by heat released

- vex by volume change
  • Relation of excess property to partial molar
    properties
  • - equate (9) and (5a)

(10)
20
The chemical potential
  • mi is called the chemical potential of component
    i







21
The activity
  • The following is for nonaqueous solutions only -

ex liquid and solid alloys ceramics salts
gases
  • Relating ?i to solution composition is
    inconvenient because its range is -? lt ?i lt gi as
    0 lt xi lt 1.
  • A more convenient measure is the activity
  • range 0 lt ai lt 1 as 0 lt xi lt 1
  • gi molar free energy of pure i
  • the reason for excluding aqueous solutions from
    this treatment is that the species are ions for
    which there are no pure species.

22
Activity coefficient
?i ? ai / xi
  • direct measure of the deviation from ideality in
    solution
  • ?i ? 1 as xi ? 1 (pure component) Raoults Law
  • ?i ? const. as xi ? 0 (infinite dilution)
    Henrys Law
  • Shows positive and negative deviations from
    ideality

- ?i lt 1 means A and B attractstable solution
  • ?i gt 1 means A and B repel unstable solution
  • if gi is too positive, the solution separates
    into two immiscible solutions

23
Gibbs-Duhem equation
  • permits calculation of ?B if ?A is known as a
    function of composition
  • Start from (13a)
  • Eliminate mi dmi RTdlnai
  • Replace ai with gi ai xi gi
  • Result
  • Integrate at xA 0, gB 1, gA gA0

xAdlngA xBdlngB 0
24
Suppose lngA CxAxB2
Conditions to be satisfied
- As xA?1 ln?A 0 ?A ? 1 OK
- As xA?0 ln?A 0 ?A ? no!
Suppose lngB CxA2 DxA3, what is lngA?
Satisfies xB?1 limit as xB?0, lngB C - D
Substitute lngB into integrated G-D equation
25
Regular Solution Model
g xAgA xBgB gex gmix
gmix hmix TDsmix
hex
  • free energy change, pure components-to-solution

gsoln gpure comp. hex TDsmix
  • hex must ? 0 as either either xA ? 0 or xB ? 0

hex ?xAxB
Regular solution
? interaction energy ½(?AA ?BB) ?AB
26
Activity coefficients in the Regular-Solution
Model
?A g (1-xA)(dg/dxA)
?
3. Definition of activity ?A gA RTln(gAxA)
4. eliminate g(1 and 2) then mA (2 and 3)
27
Chemical potentials in ideal gas mixtures
  • Mix pure A and B at the partial pressures pA and
    pB
  • For this process ?s 0 (Dalton mixing) and ?h
    0

? D g 0, or gA and gB do not change.
(in mixture) ?A
gA depends on pA must relate to 1-atm value
  • in dgA -sAdT vAdpA,

dp, dT0
  • Integrate from 1 atm, where

, to pA
or
also
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