raoult's law and aerosol compositon
DESCRIPTION
Raoult's Law and Aerosol CompositonTRANSCRIPT
Raoult's Law
The vapor pressure and composition in equilibrium with a solution can yield valuable
information regarding the thermodynamic properties of the liquids involved. Raoult’s law
relates the vapor pressure of components to the composition of the solution. The law
assumes ideal behavior. It gives a simple picture of the situation just as the ideal gas law
does. The ideal gas law is very useful as a limiting law. As the interactive forces between
molecules and the volume of the molecules approache zero, so the behavior of gases
approach the behavior of the ideal gas. Raoult’s law is similar in that it assumes that the
physical properties of the components are identical. The more similar the components the
more their behavior approaches that described by Raoult’s law.
Using the example of a solution of two liquids, A and B, if no other gases are present the
total vapor pressure Ptot above the solution is equal to the sum of the vapor pressures of the
two components, PA and PB.
If the two components are very similar, or in the limiting case, differ only in isotopic
content, then the vapor pressure of each component will be equal to the vapor pressure of
the pure substance Po times the mole fraction in the solution. This is Raoult’s law.
Thus the total pressure above solution of A and B would be
Graphically this can be represented by a diagram in which the horizontal axis gives the
composition from one pure component to the other as shown below.
With actual mixtures the plots of vapor pressure vs. composition usually depart from the
straight lines shown above, some curve above, some below due to intermolecular forces.
Below is an example of a positive deviation from ideality. The thin lines represent the ideal
behavior.
As the curves approach the extremes of the pure components, molecules of the minor
component are surrounded by molecules of the major component. Thus the departures from
the ideal give information about the interaction of the components. The partial vapor
pressure above the solution is obviously related to the escaping tendency and thus the
chemical potential. At equilibrium the chemical potential of a component i (µi) is the same in
the solution and in the vapor phase.
Koo et al., 2003
Some thoughts & derivations….
Ptot = PA0xA + PB
0xB Raoult’s Law
cieq
= xi,kci*
where xi,k is mole fraction of species i in section k; ci* is effective saturation
concentration of species i
ct,i = cg,i + ca,i = αi ∆HCi Mass balance
ca,i = ct,i – xici* for i = 1,n
ci = m/V = PiMi/(RT) Ideal gas law
total secondary organic mass in aerosol = Ca = Σca,i = Σct,i – Σxici*
,
,0
0
a i
ii
a j
j j
c
Mx
cc
M M
=
+∑ definition of mole fraction, but what about water, inorganics?
, ,
,
, ,0 0
0 0
a i a i i
ii ia j j i j j a j
a j a jj i j j
j jj j
c c c
M MC HC c HC c
c cc c
M M M M
α α
∗
∗= ∆ − ⋅ = ∆ − =
+ +
∑∑ ∑ ∑ ∑
∑ ∑
,
,
,0
0
a i i i
i ia j j a j
a jj j
j j
c P M
MC HC c
cc
M M
α
∗
= ∆ − =
+
∑∑ ∑
∑
,
,
,0
0
a i i
ia j j a j
a jj j
j j
c P
C HC ccc
M M
α
∗
= ∆ − =
+
∑∑ ∑
∑
Define weighted average molecular weight of organic species condensed onto aerosol:
,
,
a j j
j
a k
k
c M
Mc
=∑
∑
Then can approximate mole fraction using this:
,
,
, 0,0
0
0
a i
a i
i
a j
j a j
j
ccMx
c c Mcc
MM M
≈ =
++∑ ∑
Leading to:
,
,
0,
0
a i i
ia j j a j
j j
a j
j
M c P
C HC cc M
cM
α
∗
= ∆ − =
+
∑∑ ∑
∑
Some limiting cases:
(A) all Pi* values equal 1 (atmosphere)
,
,
,
0
0
a i
ia j j a j
j ja j
j
c
C HC cc
c
M M
α= ∆ − =
+
∑∑ ∑
∑
,
, ,
, ,
0 0
0 0
11
a i
ij j a j a j
j j ja j a j
j j
c
HC c cc c
c c
M MM M
α
∆ = + = + + +
∑∑ ∑ ∑
∑ ∑
Thus,
,
,
0
0
11
j j
j
a a j
j
a j
j
HC
C c
cc
M M
α ∆
= = + +
∑∑
∑
(B) all Pi* values equal 0
,a j j a j
j j
C HC cα= ∆ =∑ ∑
(C) all Pi* values the same = P
*
0,
0
1
j j
j
a
a j
j
HC
C
P M
c Mc
M
α
∗
∆
= + +
∑
∑
Note that in these cases, Ca is in the denominator of the right hand side. If we rearrange and
solve for Ca (for case (C)):
0
0
0
1
11
a
o
j j
j
CM
Pc MM
HC c Mα
∗
= + + − ∆ ∑
Just for simplicity, assume M0 = M =M
0 0
0
1
11
1
a
j j
j
CMP
c c
HC cα
∗= + + − ∆
∑
(1) Should denominator include everything (not just organic) in the aerosol phase?
(2) Note the ci* is “effective” saturation concentration, which includes the effects of processes
like oligimer formation, acid-base equilibria, condensed phase oxidation, condensed
phase activity, …; and thus, in general ci* ≤ ci, and it could much less!
Let’s take case (C) and add water to the aerosol. cw is concentration of water in the aerosol
phase. Thus (with assumptions Pi*=P*
, and 0
M M M= = ),
,
,
,00 ,
01818
a i
a iii
a j wwa j
jj j
c
cMx
c c Mc cc c
M M
= =
+ ++ + ∑∑
,
,
0 ,18
a i
ia j j a j
wj ja j
j
P M c
C HC cc M
c c
α
∗
= ∆ − =
+ +
∑∑ ∑
∑
Leading to:
2
0 0
0
0
2 2
0 0
018 18
1
18
18
4 4
2 2
18
w wa a j j j j
j j
wj j
j
wj j
j
a
wj j
j
c M c MC C c P M HC HC c
l
c Mm c P M HC
c Mn HC c
m m nl m m nC
l
c Mc P M HC c
α α
α
α
α
∗
∗
∗
+ + + − ∆ − ∆ + =
=
= + + − ∆
=− ∆ +
− ± − − + −= =
− + + − ∆ + + =
∑ ∑
∑
∑
∑2
04
18 18
2
w wj j j j
j j
c M c MP M HC HC cα α∗
+ − ∆ + ∆ + ∑ ∑
2
- 1/2 co - 1/36 cw M - 1/2 Pstar M + 1/2 aHC + 1/36 (324 co + 36 co cw M
2 2 2
+ 648 co Pstar M + 648 co aHC + cw M + 36 cw M Pstar + 36 cw M aHC
2 2 2 1/2
+ 324 Pstar M - 648 Pstar M aHC + 324 aHC )
>>
2
0 0
2 2 2 2 20
2
324 36 648 648
136 36 324
2 36 2 2 36
648 324
o w o i i
ii i
w ia w w w i i
i
i i i i
i i
c c c M c P M c HCHC
c c M P MC c M c P M c M HC P M
P M HC HC
α
α
α
α α
∗
∗∗ ∗
∗
+ + + ∆ ∆ =− − − + + + + + ∆ + − ∆ + ∆
∑∑
∑
∑ ∑
( ) ( )2
0 0
0
2
2
18 36 181
2 2 36 2 3618
w w i ii i
ii w
a
i i
i
c Mc c Mc P M HCHCc c M P M
C
P M HC
αα
α
∗
∗
∗
+ + + + ∆ +∆ = − + + + − ∆
∑∑
∑
This equation is display graphically on the following plots. Parameters given in the plot titles.
Raoult's Law and Mass Balance Calculation
cw = co, Pstar=1e-4, M=200
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
aero
so
l p
has
e
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = 0, Pstar=1e-4, M=200
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
aero
so
l p
ha
se
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = 10 x co, Pstar=1e-4, M=200
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
ae
ros
ol p
hase
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = co, Pstar=1e-6, M=200
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
ae
ros
ol p
ha
se
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = co, Pstar=1e-2, M=200
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
ae
ros
ol p
hase
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = co, Pstar=1e-4, M=1000
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
aero
so
l p
has
e
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=
Raoult's Law and Mass Balance Calculation
cw = co, Pstar=1e-4, M=40
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
co
Fra
cti
on
in
ae
ros
ol p
hase
1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04
1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01
1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02
a*HC=