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