theory of solution thermodynamics
DESCRIPTION
Theory of Solution ThermodynamicsTRANSCRIPT
![Page 1: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/1.jpg)
188
Chapter 3: Solution Thermodynamics: Theory
3.0 The Notations for Solution Thermodynamics
M = the properties of the solution
(h, s, v)
iM = partial properties (e.g., ih , is )
of species i in the solution
iM = pure-species properties
(e.g., ig , is )
![Page 2: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/2.jpg)
189
3.1 Fundamental Property Relations
We can re-write the following
relationship (Eq. 2.13)
dg vdP sdT= − (2.13)
for a system with a total number of
moles of n, as follows
( ) ( ) ( )d ng nv dP ns dT= − (3.1)
Since ‘ng’ is f(P, T) (do you know
WHY?), we obtain the following
mathematical implication
( ) ( ) ( ), ,T n P n
ng ngd ng dP dT
P T∂ ∂⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ (3.2)
![Page 3: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/3.jpg)
190
By comparing Eq. 3.1 with Eq. 3.2,
we obtain the following relationships
( ),T n
ngnv
P∂⎛ ⎞
=⎜ ⎟∂⎝ ⎠ (3.3)
and ( ),P n
ngns
T∂⎛ ⎞
= −⎜ ⎟∂⎝ ⎠ (3.4)
where
n = total # of moles or # of moles
of all chemical species in the
solution
= 1 2 3 ... nn n n n+ + + +
(for n species)
Accordingly, ( )= 1 2 3f , , , , , ....ng T P n n n
![Page 4: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/4.jpg)
191
Thus, we can re-write Eq. 3.2 as
follows
( ) ( ) ( )
( ), ,
1 , , j
T n P n
n
ii i P T n
ng ngd ng dP dT
P T
ngdn
n=
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
∂⎛ ⎞+ ⎜ ⎟∂⎝ ⎠∑
(3.5)
where jn is all species EXCEPT species i
The term ( ), , ji P T n
ngn
∂⎛ ⎞⎜ ⎟∂⎝ ⎠
is defined as
“CHEMICAL POTENTIAL, μi”
( ), , j
ii P T n
ngn
μ∂⎛ ⎞
= ⎜ ⎟∂⎝ ⎠ (3.6)
![Page 5: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/5.jpg)
192
Combining Eqs. 3.2-3.6 together yields
( ) ( ) ( )1
n
i ii
d ng nv dP ns dT dnμ=
= − +∑ (3.7)
In the case that n = 1 i in x= , Eq. 3.7
can be re-written as follows
1
n
i ii
dg vdP sdT dxμ=
= − +∑ (3.8)
From Eq. 3.8, it implies that
( )= 1 2 3f , , , , , ....g T P x x x
Accordingly, it can be implied
mathematically that
,T x
g vP∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠
(3.9)
and ,P x
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.10)
![Page 6: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/6.jpg)
193
This confirms that Gibbs free
energy still serves as a generating
function for other thermodynamic
properties, even in the system where
compositions of all species vary with
time
![Page 7: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/7.jpg)
194
3.2 The Chemical Potential & Phase Equilibria
Consider a closed system where
2 phases co-exist and are in equilibrium
with each other
We can write Eq. 3.7 for each phase
(phases &α β ), as follows
( ) ( ) ( )1
n
i ii
d ng nv dP ns dT dnα α α α αμ=
= − +∑ (3.11)
( ) ( ) ( )1
n
i ii
d ng nv dP ns dT dnβ β β β βμ=
= − +∑ (3.12)
![Page 8: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/8.jpg)
195
A total property of the system are
obtained when the property of each
phase is combined together, as
shown in the following equation
( ) ( )nM nM nMα β= + (3.13)
where M = any system property
Hence, by combining Eq. 3.11 with
Eq. 3.12, we obtain
( ) ( ) ( )1 1
n n
i i i ii i
d ng nv dP ns dT dn dnα α β βμ μ= =
= − + +∑ ∑
(3.14) From Eq. 3.1, we knew that
( ) ( ) ( )d ng nv dP ns dT= − (3.1)
![Page 9: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/9.jpg)
196
Thus, Eq. 3.14 becomes
( ) ( )1 1
n n
i i i ii i
d ng d ng dn dnα α β βμ μ= =
= + +∑ ∑
1 1
0n n
i i i ii i
dn dnα α β βμ μ= =
+ =∑ ∑ (3.15)
Since this is a closed system, a
decrease in the number of moles of
Phase α will result in an increase in
the same number of moles of Phase
β , or vice versa, or it means that
i idn dnα β= − (3.16)
Combining Eq. 3.15 with Eq. 3.16
yields
![Page 10: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/10.jpg)
197
( )α α β αμ μ= =
+ − =∑ ∑1 1
0n n
i i i ii i
dn dn
( )α β αμ μ=
− =∑1
0n
i i ii
dn
and, eventually
i iα βμ μ= (3.17)
This principle can also be
extended to any phases, as follows
....i i i iα β χ πμ μ μ μ= = = =
“Multiple phases at the same T &
P are in equilibrium with one
another when the chemical
potential of each species is the
same for all phases”
![Page 11: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/11.jpg)
198
3.3 Partial Properties
We can extend the principle of
chemical potential (Eq. 3.6) to any other
thermodynamic properties, as follows
( ), , j
ii P T n
nMM
n∂⎛ ⎞
= ⎜ ⎟∂⎝ ⎠ (3.18)
where M = any system property
(e.g., h, s)
This is called “PARTIAL PROPERTY” or
“RESPONSE FUNCTION”, which represents
the change in the total property, nM, due
to the addition of a small (differential)
amount of (moles of) species i, at constant
T & P, to the solution
![Page 12: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/12.jpg)
199
Combining Eqs. 3.6 & 3.18 together
results in
( ), , j
i ii P T n
ngg
nμ
∂⎛ ⎞= = ⎜ ⎟∂⎝ ⎠
(3.19)
This implies that the ‘chemical
potential’ is, in fact, ‘partial Gibbs
free energy’
![Page 13: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/13.jpg)
200
3.3.1 Properties & Partial Properties
We can extend Eq. 3.5 to any other
thermodynamic properties (M) as
follows
( ) ( ) ( )
( ), ,
1 , , j
T n P n
n
ii i P T n
nM nMd nM dP dT
P T
nMdn
n=
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
∂⎛ ⎞+ ⎜ ⎟∂⎝ ⎠∑
(3.20)
and when combining Eq. 3.20 with Eq.
3.18, we obtain
( ) ( ) ( )
, ,
1
T n P n
n
i ii
nM nMd nM dP dT
P T
Mdn=
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
+∑
(3.21)
![Page 14: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/14.jpg)
201
Since n = total # of moles of all species
in the solution, it must be constant; thus,
( )
, ,
1
T n P n
n
i ii
M Md nM n dP n dTP T
Mdn=
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
+∑ (3.22)
For a special case, when n = 1, we
obtain the following relationships
, ,T n T x
M MP P
∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
and , ,P n P x
M MT T
∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
Thus, Eq. 3.22 can be re-written as
follows
( )
, ,
1
T x P x
n
i ii
M Md nM n dP n dTP T
Mdn=
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
+∑ (3.23)
![Page 15: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/15.jpg)
202
Consider the term 1
n
i ii
Mdn=∑ in Eq. 3.23
Since ii
nxn
= i in x n=
Thus,
( )i i i idn d x n x dn ndx= = +
Accordingly,
( )1 1
n n
i i i i ii i
Mdn M x dn ndx= =
= +∑ ∑ (3.24)
Consider the term ( )d nM in Eq. 3.23
( )d nM ndM Mdn= + (3.25)
Combining Eqs. 3.23-3.25 together
yields
![Page 16: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/16.jpg)
203
( )
, ,
1
T x P x
n
i i ii
M MndM Mdn n dP n dTP T
M x dn ndx=
∂ ∂⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
+ +∑
Rearranging the above equation
yields
, ,
0
i iT x P x
i i
M MdM dP dT Mdx nP T
M x M dn
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦+ − =⎡ ⎤⎣ ⎦
∑
∑
(3.26) Eq. 3.26 will be true only if
, ,
0i iT x P x
M MdM dP dT MdxP T
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞− − − =⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦∑
and
0i iM x M− =⎡ ⎤⎣ ⎦∑
![Page 17: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/17.jpg)
204
Hence,
, ,
i iT x P x
M MdM dP dT MdxP T
∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∑ (3.27)
i iM x M= ∑ (3.28)
Note that Eq. 3.27 is a special case of
Eq. 3.23 when 1n = and, thus, i in x=
Multiplying Eq. 3.28 with n yields
( )i inM x n M= ∑
i inM n M= ∑ (3.29)
Eq. 3.29 is called “summability relations”
Differentiating Eq. 3.28 gives
i i i idM x dM Mdx= +∑ ∑ (3.30)
![Page 18: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/18.jpg)
205
Combining Eq. 3.27 with Eq. 3.30 yields
, ,
i i i i
i iT x P x
x dM Mdx
M MdP dT MdxP T
+ =
∂ ∂⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
∑ ∑
∑
Rearranging the above equation gives
, ,
0i iT x P x
M MdP dT x dMP T
∂ ∂⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∑
(3.31)
Note that Eq. 3.31 is called a Gibbs/
Duhem equation In the case where T & P are constant, a
Gibbs/Duhem equation (Eq. 3.31) becomes
0i ix dM =∑ (3.32)
![Page 19: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/19.jpg)
206
3.3.2 Partial Properties in Binary Solutions
Consider a solution with 2 components
(in other words, a binary solution) Writing Eq. 3.28,
i iM x M= ∑ (3.28)
for a binary solution yields
1 1 2 2M x M x M= + (3.33)
Differentiating Eq. 3.33 results in
1 1 1 1 2 2 2 2dM x dM Mdx x dM M dx= + + +
(3.34)
![Page 20: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/20.jpg)
207
Writing the Gibbs/Duhem equation
for a system where T & P are constant
results in
0i ix dM =∑ (3.32)
Eq. 3.32 can be written for a binary
solution, as follows
1 1 2 2 0x dM x dM+ = (3.35)
Combining Eq. 3.34 with Eq. 3.35
yields
( )1 1 2 2 1 1 2 2dM Mdx M dx x dM x dM= + + +
1 1 2 2dM Mdx M dx= + (3.36)
0
![Page 21: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/21.jpg)
208
Since, for a binary solution, 1 2 1x x+ =
1 21x x= − ; hence, 1 2dx dx= − or 2 1dx dx= −
Eq. 3.36 can then be re-written as
follows:
( )
( )
1 1 2 1
1 1 2 1
1 2 1
dM Mdx M dxMdx M dxM M dx
= + −
= −
= −
1 21
dM M Mdx
= −
Rearranging the above equation
results in
1 21
dMM Mdx
= + (3.37)
![Page 22: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/22.jpg)
209
Rearranging Eq. 3.33 gives
2 2 1 1x M M x M= −
12 1
2 2
xMM Mx x
= − (3.38)
Substituting Eq. 3.38 into Eq. 3.37 and
rearranging the resulting equation yield
1
1 12 2 1
11 1
2 2 1
xM dMM Mx x dxx M dMM Mx x dx
⎛ ⎞= − +⎜ ⎟⎝ ⎠
+ = +
2 1 1 1
2 2 1
x M x M M dMx x dx+
= + (3.39)
![Page 23: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/23.jpg)
210
Multiplying Eq. 3.39 with x2 yields
2 1 1 1 21
dMx M x M M xdx
+ = +
( )2 1 1 21
dMx x M M xdx
+ = + (3.40)
Since 1 2 1x x+ = (for a binary system),
Eq. 3.40 can be written as follows:
1 21
dMM M xdx
= + (3.41)
By performing the same derivation
for 2M , we obtain the following equation
2 11
dMM M xdx
= − (3.42)
![Page 24: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/24.jpg)
211
Eqs. 3.41 & 3.42 illustrate that any
partial property ( iM) can be calculated
from the property of the solution (M),
when the composition of each species
is known, which can be illustrated
graphically as follows
Let’s consider a plot between M
(M can be either h, s, or v, or etc. –
any system property) and x1 (Figure
3.1)
![Page 25: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/25.jpg)
212
Figure 3.1: A Plot between M (any thermodynamic property) and x1
The slope of the line I1–I2 can be
calculated as follows
2 2
1 1 10M I M IdM
dx x x− −
= =−
(3.43)
and 1 21 2
1 1 0I IdM I I
dx−
= = −−
(3.44)
M
x1
I2
I1
![Page 26: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/26.jpg)
213
Rearranging Eq. 3.43 gives
2 11
dMI M xdx
= − (3.45)
and rearranging Eq. 3.44 yields
1 21
dMI Idx
= + (3.46)
Combining Eq. 3.46 with Eq. 3.45
and rearranging result in
( )
1 11 1
11
1
dM dMI M xdx dx
dMx Mdx
⎛ ⎞= + −⎜ ⎟
⎝ ⎠
= − +
but 1 2 1x x+ = 2 11x x= −
![Page 27: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/27.jpg)
214
Hence,
1 21
dMI M xdx
= + (3.47)
Comparing Eq. 3.47 with Eq. 3.41
and Eq. 3.45 with Eq. 3.42 yields
1 1I M= and 2 2I M=
![Page 28: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/28.jpg)
215
Let’s consider the value of 1
dMdx
when
x1 0 (or it is called “infinite dilution of
species 1”) from Figure 3.2
Figure 3.2: A Plot between M (any thermodynamic property) and x1 when x1 0
x1
M
M2
M1
1M∞
![Page 29: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/29.jpg)
216
Comparing Figure 3.2 with Figure
3.1 gives
1 1 1I M M∞= = and 2 2 2I M M= =
This indicates that when x1 0 (i.e.
pure x2), 2 2M M=
Let’s consider the value of 1
dMdx
when
x1 1 or x2 0 (i.e. infinite dilution of
species 2), in Figure 3.3
![Page 30: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/30.jpg)
217
Figure 3.3: A Plot between M (any thermodynamic property) and x1 when x1 1 (or x2 0)
Comparing Figure 3.3 with Figure 3.1
yields
1 1 1I M M= = and 2 2 2I M M∞= =
meaning that when x1 1 (i.e. pure
species 1): 1 1M M=
x1
M
M2
M1 2M∞
![Page 31: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/31.jpg)
218
Example The enthalpy of a binary
liquid system of species 1 & 2 at
constant T & P is presented by the
following equation
( )1 2 1 2 1 2400 600 40 20h x x x x x x= + + +
(3.48)
Determine the values of 1h , 2h , 1h∞ , &
2h∞
![Page 32: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/32.jpg)
219
Rearranging Eq. 3.48 to be a
function of x1 only (note that, for a
binary system, 2 11x x= − ) gives
31 1600 180 20h x x= − − (3.49)
Differentiating Eq. 3.49 with respect
to x1 yields
21
1
180 60dh xdx
= − − (3.50)
Writing Eqs. 3.41 & 3.42
1 21
dMM M xdx
= + (3.41)
2 11
dMM M xdx
= − (3.42)
for this case yields
![Page 33: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/33.jpg)
220
( )1 11
1 dhh h xdx
= + − (3.51)
2 11
dhh h xdx
= − (3.52)
Combining Eqs. 3.49 & 3.50 with
Eqs. 3.51 & 3.52 and rearranging give
( )( )
31 1 1
21 1
600 180 20
1 180 60
h x x
x x
⎡ ⎤= − −⎣ ⎦+ − − −
2 31 1 1420 60 40h x x= − + (3.53)
32 1600 40h x= + (3.54)
![Page 34: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/34.jpg)
221
From Figures 3.2 & 3.3, we knew
that, when x1 0:
2 2M M= and 1 1M M∞ = and when x1 1:
1 1M M= and 2 2M M∞ =
Hence, by substituting corresponding
numerical values into Eqs. 3.53 & 3.54, we
obtain:
( ) ( ) ( )2 31 1 1 1 420 60 1 40 1 400h h x= = = − + =
( ) ( )32 2 1 0 600 40 0 600h h x= = = + =
( ) ( ) ( )2 31 1 1 0 420 60 0 40 0 420h h x∞ = = = − + =
( )32 2 1( 1) 600 40 1 640h h x∞ = = = + =
![Page 35: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/35.jpg)
222
The values of 1h , 2h , 1h∞ , and 2h∞ can
be shown graphically as follows
0
100
200
300
400
500
600
700
0 0.2 0.4 0.6 0.8 1
x 1
h
h2
h1
2h∞
1h∞
![Page 36: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/36.jpg)
223
3.3.3 Relations among Partial Properties
Combining Eq. 3.7
( ) ( ) ( ) i id ng nv dP ns dT dnμ= − +∑ (3.7)
with Eq. 3.19
( ), , j
i ii P T n
ngg
nμ
∂⎛ ⎞= = ⎜ ⎟∂⎝ ⎠
(3.19)
results in
( ) ( ) ( ) i id ng nv dP ns dT g dn= − +∑ (3.55)
From an “exact differential expression”,
we obtain the following relationships
![Page 37: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/37.jpg)
224
( ) ( ), ,P n T n
nv nsT P
∂ ∂⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
, ,P n T n
v sT P∂ ∂⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
(3.56)
( ), , , j
i
P n i P T n
nsgT n
∂⎛ ⎞∂⎛ ⎞ = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ (3.57)
and ( ), , , j
i
T n i P T n
nvgP n
∂⎛ ⎞∂⎛ ⎞ =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ (3.58)
By employing Eq. 3.18
( ), , j
ii P T n
nMM
n∂⎛ ⎞
= ⎜ ⎟∂⎝ ⎠ (3.18)
we can re-write Eqs. 3.57 & 3.58 as follows
,
ii
P n
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.59)
,
ii
T n
g vP
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ (3.60)
![Page 38: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/38.jpg)
225
When comparing with Eqs. 3.10 & 3.9
,P n
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.10)
,T n
g vP∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠
(3.9)
we can conclude that
“Every equation that provides a linear
relation among thermodynamic
properties of a constant-composition
solution has its counterpart as an
equation connecting the corresponding
partial properties of each species in the
solution”
![Page 39: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/39.jpg)
226
Example In the case of a constant-
composition solution ( ). . 0ii e dn = , we
know that ( ),ig f T P=
Hence,
, ,
i ii
P n T n
g gdg dT dPT P
∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
Combining the above equation
with Eqs. 3.59 & 3.60 results in
i i idg sdT v dP= − + (3.61)
which is in accord with Eq. 2.13
dg sdT vdP= − + (2.13)
![Page 40: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/40.jpg)
227
3.4 Ideal-Gas Mixtures
If n moles of a mixture of ideal
gases contain in a container with a
volume of tV at the temperature of T,
the total pressure of the container
can be calculated using the
following equation:
tnRTPV
= (3.62)
![Page 41: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/41.jpg)
228
and if in moles of species i contain in the
same container, the partial pressure of
species i ( )iP at the same temperature,
T, is as follows:
ii t
nRTPV
= (3.63)
(3.63)/(3.62) gives
i
ti i
i
t
nRTP nV xnRTP n
V
= = =
or i iP x P=
![Page 42: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/42.jpg)
229
Partial volume of species i (ideal gas)
can be written in the form of equation
as follows:
, ,
, ,
, ,
j
j
j
igig
ii T P n
i
T P n
i T P n
nvvn
RTnP
n
RT nP n
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠
⎛ ⎞⎛ ⎞∂ ⎜ ⎟⎜ ⎟⎝ ⎠= ⎜ ⎟∂⎜ ⎟⎜ ⎟
⎝ ⎠
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠
Since i jn n n= +∑ , but jn is constant,
we obtain
( ), , , ,j j
i j
i iT P n T P n
n nnn n
⎛ ⎞∂ +⎛ ⎞∂= ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
∑
( ), , , ,
1j j
ji
i iT P n T P n
nnn n
⎛ ⎞∂⎛ ⎞∂= + =⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
∑ 0
![Page 43: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/43.jpg)
230
Thus,
ig igi i
RTv vP
= =
which illustrates that, in the case of an
ideal-gas mixture
ig igi iv v= (3.64)
The above relation can also be
extended to other thermodynamic
properties:
“A partial molar property of a constituent
species in an “ideal-gas” mixture is equal to
the corresponding molar property of the
species as a pure ideal gas at the same
temperature, but at a pressure equal to its
partial pressure in the mixture”
![Page 44: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/44.jpg)
231
which can be written in the form of
equation as follows:
( ) ( ), ,ig igi i iM T P M T P= (3.65)
3.4.1 For Enthalpy: ( ) ( )=, ,ig ig
i i ih T P h T P
However, we have known (from
Chapter 2) that an enthalpy of an ideal
gas is independent of P
Hence,
( ) ( ) ( ), , ,ig ig igi i i ih T P h T P h T P= =
meaning that
ig igi ih h= (3.66)
![Page 45: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/45.jpg)
232
Applying Eq. 3.28
i iM x M= ∑ (3.28)
to enthalpy, yields
ig ig igi i i ih x h x h= =∑ ∑ (3.67)
This indicates that, for an ideal-gas
solution, the summation of an enthalpy
of each “pure” species is equal to the
enthalpy of the solution
![Page 46: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/46.jpg)
233
3.4.2 For Entropy: Since an entropy of an ideal gas
depends on both T & P (see Chapter 2),
by considering Eq. 2.31 when T is
constant, we obtain
igi p
dT dPds c RT P
= −
ln
igi
dPds RP
Rd P
= −
= −
Integrating the above equation
from P = Pi to P = P yields
0
![Page 47: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/47.jpg)
234
ln
ln
i i
i
P Pig
iP P
P
P
ds Rd P
R d P
= −
= −
∫ ∫
∫
( ) ( ) [ ], , ln lnig igi i i is T P s T P R P P− = − −
( ) ( ), , lnig igi i i
i
Ps T P s T P RP
− = −
but ii
PxP
= i iP x P=
Accordingly,
( ) ( ) 1, , ln lnig igi i i
i i
Ps T P s T P R Rx P x
− = − = −
( ) ( ), , lnig igi i i is T P s T P R x− = (3.68)
![Page 48: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/48.jpg)
235
Applying Eq. 3.65 to entropy yields
( ) ( ), ,ig igi i is T P s T P= (3.69)
Combining Eq. 3.68 with Eq. 3.69 results in
( ) ( ), , lnig igi i is T P s T P R x− = (3.70)
Rearranging Eq. 3.70 gives
( ) ( ), , lnig igi i is T P s T P R x= − (3.71)
Multiplying Eq. 3.71 with xi gives
( ) ( ), , lnig igi i i i i ix s T P x s T P Rx x= − (3.72)
Applying Eq. 3.28 ( )i iM x M= ∑ to Eq. 3.72
yields
( ) ( ), , lnig igi i i i i ix s T P x s T P R x x= −∑ ∑ ∑
( ), lnig igi i i is x s T P R x x= −∑ ∑ (3.73)
![Page 49: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/49.jpg)
236
Rearranging Eq. 3.73 gives
( ) 1, lnig igi i i
i
s x s T P R xx
− =∑ ∑ (3.74)
Note that ( ),ig igi is x s T P−∑ = entropy
change of mixing for ideal gas
Since 1 ix > 1, ( ),ig igi is x s T P−∑ > 0
(positive, +, sign) in agreement
with the 2nd law of thermodynamics
(what does the 2nd law say?)
![Page 50: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/50.jpg)
237
Applying the “relations among
partial properties” to the Gibbs free
energy for an ideal gas
ig ig igg h Ts= − (3.75)
gives
ig ig igi i ig h Ts= − (3.76)
Combining Eq. 3.76 with Eqs. 3.66
& 3.71 yields
( )lnig ig igi i i ig h T s R x= − − (3.77)
Rearranging Eq. 3.77 results in
( ) lnig ig igi i i ig h Ts RT x= − +
lnig igi i i ig g RT xμ= = + (3.78)
![Page 51: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/51.jpg)
238
For an ideal gas when T is constant,
Eq. 2.13
dg vdP sdT= − (2.13)
becomes
igidg vdP
RT dPP
dPRTP
=
=
=
lnigidg RTd P= (3.79)
Integrating Eq. 3.79 yields
( )lnigi ig RT P T= + Γ (3.80)
where ( )i TΓ = integration constant
= f(T)
![Page 52: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/52.jpg)
239
Combining Eq. 3.78 with Eq. 3.80
results in
( )( ) ( )
lnln ln
ln ln
ig igi i i
i i
i i
g g RT xRT P T RT x
T RT x P
= +
= + Γ +⎡ ⎤⎣ ⎦= Γ + +
( ) lnigi i ig T RT x P= Γ + (3.81)
Applying Eq. 3.28 to Eq. 3.81 yields
( ) lnig igi i i i i ig x g x T RT x x P= = Γ +∑ ∑ ∑
(3.82)
![Page 53: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/53.jpg)
240
3.5 Fugacity and Fugacity Coefficient: Pure Species
Re-writing Eq. 3.80
( )lnigi ig RT P T= + Γ (3.80)
for pure & real fluids at constant T
gives
( )lni i ig RT f T= + Γ (3.83)
The property if in Eq. 3.83 is called
a “FUGACITY”
Accordingly, fugacity has the
same unit as pressure
![Page 54: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/54.jpg)
241
(3.83) – (3.80) yields
( ) ( )ln ln
ln
igi i i i i
i
g g RT f T RT P TfRTP
− = + Γ − + Γ⎡ ⎤⎣ ⎦
=
but ig Ri i ig g g− = (see Chapter 2); thus,
lnR ii
fg RTP
=
Note that ii
fP
φ= = FUGACITY
COEFFICIENT
Hence,
lnRi ig RT φ= (3.84)
![Page 55: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/55.jpg)
242
In the case of an ideal gas: if P=
1ii
fP
φ = = ( )ln 1 0Rig RT= =
Rearranging Eq. 3.84 gives
lnRi
igRT
φ = (3.85)
Combining Eq. 3.85 with Eq. 2.67
(see Chapter 2)
( )0
1PRg dPZ
RT P= −∫ (2.67)
yields
( )0
ln 1PR
ii i
g dPZRT P
φ = = −∫ (3.86)
![Page 56: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/56.jpg)
243
In the case of cubic EoS (e.g.,
Redlich-Kwong, Soave-Redlich-Kwong,
& Peng-Robinson EoS), the property Rg
RT
can be written for species i, as follows
( ) ( )ρ= − − − + −ln 1 1Ri
i ig b Z qI ZRT
(2.85a)
Combining Eq. 2.85a with Eq. 3.85
lnRi
igRT
φ = (3.85)
gives
( ) ( )ln ln 1 1R
ii i i
g b Z qI ZRT
φ ρ= = − − − + − (3.87)
![Page 57: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/57.jpg)
244
but, from Eq. 2.95 (See Chapter 2)
B bZ
ρ= (2.95)
or, for species i
i
i
B bZ
ρ= (2.95a)
Eq. 3.87 can, thus, be re-written as
follows:
( ) ( )ln ln 1R
ii i i i
g Z B qI ZRT
φ = = − − − + − (3.88)
![Page 58: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/58.jpg)
245
3.6 Vapour/Liquid Equilibrium (VLE) for Pure Species
When a system consists of liquid (l)
and vapour (v), we can write Eq. 3.83
for each phase, as follows:
( )lnv vi i ig RT f T= + Γ (3.89)
( )lnl li i ig RT f T= + Γ (3.90)
(3.89) – (3.90) yields
ln ln
ln
v l v li i i i
vil
i
g g RT f RT ffRTf
− = −
=
![Page 59: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/59.jpg)
246
At equilibrium,
v li ig g=
or 0v li ig g− =
Thus, this means that
ln 0v
il
i
ff=
or v l sati i if f f= = (3.92)
“For a pure species coexisting
liquid and vapour phases are in
equilibrium when they have the
same T, P, and FUGACITY”
![Page 60: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/60.jpg)
247
Dividing Eq. 3.92 with P results in
v l sat
i i if f fP P P= =
v l sati i iφ φ φ= = (3.93)
which implies that, when two phases
are in equilibrium with each other at
constant T & P, in addition to the fact
that the fugacity of each phase is
equal to each other, their fugacity
coefficients are also identical
![Page 61: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/61.jpg)
248
Example For H2O at a T of 300 oC and for
P up to 10,000 kPa (100 bar), calculate
values of if and iφ using data from steam
tables
Writing Eq. 3.83 for a constant-P system
at P = P gives
( )lni i ig RT f T= + Γ (3.83)
and at P = low P yields
( )* *lni i ig RT f T= + Γ (3.94)
(3.83) – (3.94) results in
**ln i
i ii
fg g RTf
− =
*
*ln i i i
i
f g gRTf−
= (3.95)
![Page 62: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/62.jpg)
249
Writing the following equation
i i ig h Ts= − (3.96)
for a low-P system gives
* * *i i ig h Ts= − (3.97)
(3.96) – (3.97) results in
( ) ( )* * *i i i i i ig g h h T s s− = − − − (3.98)
Dividing Eq. 3.98 with RT yields
( ) ( )* **i i i ii i
h h s sg gRT RT R
− −−= −
Substituting the above Eq. into Eq. 3.95
results in
( ) ( )* *
*ln i i i ii
i
h h s sfRT Rf− −
= − (3.99)
![Page 63: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/63.jpg)
250
Let low P = 6 kPa, and when T = 300 oC
(note that, at low P (& high T as T = 300 oC),
water vapour (H2O) can be assumed to be
an ideal gas; hence, * * 6 kPaif P= = )
h = 3,076.8 kJ/kg & s = 9.5162 kJ/kg-K (by reading from a superheated steam table)
At P = 4,000 kPa (4 MPa) & T = 300 oC
(superheated vapour), we obtain the
following values
h = 2,958.6 kJ/kg & s = 6.3531 kJ/kg-K Substituting corresponding numerical
values into Eq. 3.99 results in
![Page 64: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/64.jpg)
251
[ ]( )
[ ]( )
[ ]( )
*
kJ18.02 2,958.6 3,076.8 kmolln
kJ8.314 300 273 Kkmol-K
kJ18.02 6.3561 9.5162 kmol-K
kJ8.314 kmol-K
6.395
i
i
ff
−=⎛ ⎞ +⎜ ⎟⎝ ⎠
−−
⎛ ⎞⎜ ⎟⎝ ⎠
=
(note that we have to multiply the values of h
& s obtained from the steam table with MW of water
(= 18.02) to convert the units of h & s to be kJ/kmol
& kJ/kmol-K, respectively)
Thus,
* exp(6.395) 598.84i
i
ff= =
( )*598.84 598.84 6 kPa 3,593 kPai if f= = =
and
3,593 kPa 0.8984,000 kPa
ii
fP
φ = = =
![Page 65: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/65.jpg)
252
When P = 10,000 kPa & T = 300 oC
(the system (H2O) is now in the form of
“compressed liquid”), we obtain the
following
h = 1,347.3 kJ/kg & s = 3.2519 kJ/kg-K
Substituting corresponding numerical
values into Eq. 3.99 yields
[ ]( )
[ ]( )
[ ]( )
−=⎛ ⎞ +⎜ ⎟⎝ ⎠
−−
⎛ ⎞⎜ ⎟⎝ ⎠
=
*
kJ18.02 1,347.3 3076.8 kmolln
kJ8.314 300 273 Kkmol-K
kJ18.02 3.2519 9.5162 kmol-K
kJ8.314 kmol-K
7.057
i
i
ff
![Page 66: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/66.jpg)
253
Hence,
= =* exp(7.057) 1,160.96i
i
ff
( )*1,160.96 1,160.96 6 kPa 6,966 kPai if f= = =
and
6,966 kPa 0.69710,000 kPa
ii
fP
φ = = =
It can be observed, from this
Example, that when P is getting higher,
the discrepancy between the values of
if and P is getting larger Also, when P is getting higher, the
value of φi is lower and getting more
and more deviated from unity (i.e. 1)
![Page 67: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/67.jpg)
254
3.7 Fugacity and Fugacity Coefficient: Species in Solutions
Combining Eq. 3.78
lnig ig igi i i ig g RT xμ = = + (3.78)
with Eq. 3.80
( ) lniig
ig T RT P= Γ + (3.80)
gives the following equation:
( )( )ln lnig igi i i ig T RT P RT xμ = = Γ + +
Rearranging the above equation
gives
( ) ( )lnig igi i i ig T RT x Pμ = = Γ + (3.100)
which is the same as Eq. 3.81
![Page 68: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/68.jpg)
255
If this is a “real” solution (can be
either gaseous or liquid solution), Eq.
3.100 can be re-written as follows:
( ) ˆlni i i ig T RT fμ = = Γ + (3.101)
where
if = fugacity of species i in a solution
We have just learned that, if there
are 2 phases (e.g., phases α & β ) co-
existing in equilibrium at given T & P, it
results in the fact that
i if fα β=
![Page 69: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/69.jpg)
256
The above equation can also be
applied to any species in a solution
as follows
ˆ ˆi if fα β=
“Multiple phases at the same
T & P are in equilibrium when
the fugacity of each constituent
species is the same in all phases”
![Page 70: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/70.jpg)
257
We have learned, from Chapter 2,
that
R igM M M= − (2.55)
Hence, (3.101) – (3.100) results in
( )ˆln ln
R ig igi i i i i
i i
g g g
RT f RT x P
μ μ= − = −
= −
ˆ
lnR ii
i
fg RTx P
⎛ ⎞= ⎜ ⎟
⎝ ⎠ (3.102)
Let ˆ ˆ
ˆ i ii
i i
f fx P P
φ = =
Thus,
ˆlnRi ig RT φ= (3.103)
where iφ = fugacity coefficient of
species i in a solution
![Page 71: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/71.jpg)
258
We do the same as did previously
for iφ (see Page 242) and, thus,
obtain
( )0
ˆln 1P
i idPZP
φ = −∫ (3.104)
![Page 72: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/72.jpg)
259
3.8 Generalised Correlations for the Fugacity Coefficient
Combining Eq. 3.86
( )0
ln 1P
i idPZP
φ = −∫ (3.86)
with the following equations
rc
PPP
=
c rP P P= (3.105)
( )c r
c r
dP d P PP dP
=
= (3.106)
results in
![Page 73: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/73.jpg)
260
( )0
ln 1rP
c ri i
c r
P dPZP P
φ = −∫
( )0
ln 1rP
ri i
r
dPZP
φ = −∫ (3.107)
We can write Pitzer et al theorem
of corresponding states
( ) ( )ω= +0 1Z Z Z (1.21)
for species i as follows
( ) ( )ω= +0 1i i iZ Z Z (3.108)
![Page 74: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/74.jpg)
261
Subtracting both sides of Eq. 3.108
with 1 results in
0 11 1i i iZ Z Zω− = − + (3.109)
Combining Eq. 3.107 with Eq. 3.109
gives
( )
( )
( )
φ
ω
φ ω
= −
= − +
= − +
∫
∫
∫ ∫
0
0 1
0
0 1
0 0
ln 1
1
ln 1
r
r
r r
Pr
i ir
Pr
i ir
P Pr r
i i ir r
dPZP
dPZ ZP
dP dPZ ZP P
![Page 75: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/75.jpg)
262
Let
( )0 0
0
ln 1rP
ri i
r
dPZP
φ = −∫
and 1 1
0
lnrP
ri i
r
dPZP
φ = ∫
Thus,
0 1ln ln lni i iφ φ ω φ= +
Rearranging the above equation
yields
( )
( )( )
0 1
0 1
ln ln ln
ln
i i i
i i
ω
ω
φ φ φ
φ φ
= +
⎡ ⎤=⎣ ⎦
( )( )0 1i i i
ωφ φ φ= (3.110)
![Page 76: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/76.jpg)
263
Note that the values of 0iφ and 1
iφ in table
format (Tables A.3.1 and A.3.2) are
included at the end of this Chapter (on
Pages 284-285 and 286-287, respectively)
Example Determine the values of
fugacity coefficient ( iφ ) and fugacity ( if)
of R-134a at 300 oC, and (a) 80 bar and
(b) 284 bar
For R-134a:
374.26 KcT =
40.59 barcP =
0.326ω =
![Page 77: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/77.jpg)
264
(a)
80 bar 1.9740.59 barr
c
PPP
= = =
[ ]300 273 K1.53
374.26 Krc
TTT
+= = =
By performing double interpolations
of ( )φ 0i and ( )φ 1
i appeared in Tables A.3.1
and A.3.2, we obtain
( )φ =0 0.8495i
( )φ =1 1.1828i
Hence,
( )( )0.3260.8495 1.18280.897
iφ =
=
and ( )( )φ= = =0.897 80 bar 71.8 bari if P
![Page 78: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/78.jpg)
265
(b) 284 bar 7.0
40.59 barrc
PPP
= = =
[ ]300 273 K1.53
374.26 Krc
TTT
+= = =
Thus, also by reading from Tables A.3.1
and A.3.2 and performing double
interpolations, we obtain
( )φ =0 0.6866i
( )φ =1 1.5817i Hence,
( )( )0.3260.6866 1.58170.797
iφ =
=
and
( )( )0.797 284 bar226.3 bar
i if Pφ= =
=
![Page 79: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/79.jpg)
266
3.9 The Ideal Solution In the case that we are dealing
with an ideal liquid solution, Eq. 3.78
lnig igi i ig g RT x= + (3.78)
can be re-written for an ideal solution
as follows:
lnidi i ig g RT x= + (3.111)
Differentiating Eq. 3.111 with respect
to T, while P and composition are kept
constant, yields
, ,,
lnidi i i
P x P xP x
g g RT xT T T
⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
![Page 80: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/80.jpg)
267
Combining the above equation with Eq.
3.59
,
ii
P x
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.59)
results in
,
lnid ii i
P x
gs R xT
∂⎛ ⎞− = +⎜ ⎟∂⎝ ⎠
and ,
lnid ii i
P x
gs R xT
∂⎛ ⎞= − −⎜ ⎟∂⎝ ⎠ (3.112)
Combining Eq. 3.10
,P x
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.10)
re-written for species i
,
ii
P x
g sT
∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (3.10a)
with Eq. 3.112 gives
![Page 81: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/81.jpg)
268
( ) lnln
idi i i
i i
s s R xs R x
= − − −
= − (3.113)
Differentiating Eq. 3.111 with respect
to P, while T and composition are kept
constant, yields
, ,,
lnidi i i
T x T xT x
g g RT xP P P
⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (3.114)
Combining Eq. 3.114 with Eq. 3.60
,
ii
T x
g vP
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ (3.60)
results in
,
0id ii
T x
gvP
∂⎛ ⎞= +⎜ ⎟∂⎝ ⎠
Note that ln iRT x is a constant
![Page 82: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/82.jpg)
269
By combining the above equation
with Eq. 3.9 re-written for species i
,
ii
T x
g vP
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ (3.9a)
we obtain
idi iv v= (3.115)
When we applied the following
equation
g h Ts= −
to an ideal solution, we obtain
id id idi i ig h Ts= −
or
id id idi i ih g Ts= + (3.116)
![Page 83: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/83.jpg)
270
Combining Eq. 3.116 with Eqs. 3.111 &
3.113 gives
[ ] [ ]= + + −
= + =
ln lnidi i i i i
i i i
h g RT x T s R xg Ts h
which indicates that
idi ih h= (3.117)
Applying Eq. 3.28
i ii
M x M= ∑ (3.28)
to an ideal solution gives id id
i ii
M x M= ∑
![Page 84: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/84.jpg)
271
Thus,
ln
ln
id idi i i i i i
i i i i
g x g x g x RT x
x g RT x x
= = +
= +∑ ∑ ∑∑ ∑
(3.118)
lnid idi i i i i is x s x s R x x= = −∑ ∑ ∑ (3.119)
id idi i i iv x v x v= =∑ ∑ (3.120)
id idi i i ih x h x h= =∑ ∑ (3.121)
![Page 85: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/85.jpg)
272
3.10 The Lewis/Randall Rule
When applying Eq. 3.101
( ) ˆlni i i ig T RT fμ = = Γ + (3.101)
to an ideal solution, we obtain
( ) ˆlnid id idi i i ig T RT fμ = = Γ + (3.122)
Combining Eq. 3.83
( ) lni i ig T RT f= Γ + (3.83)
with Eq. 3.111
lnidi i ig g RT x= + (3.111)
yields
![Page 86: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/86.jpg)
273
( ) ( )ˆln ln lnidi i i i iT RT f T RT f RT xΓ + = Γ + +⎡ ⎤⎣ ⎦
Rearranging the above equation
gives
( )
ˆln ln lnln
idi i i
i i
RT f RT f RT xRT x f
= +
=
idi i if x f= (3.123)
Eq. 3.123 = “Lewis/Randall Rule”
“Fugacity of each species in an
ideal solution is proportional to its
mole fraction, where fugacity of
pure species i is a proportionality
constant”
![Page 87: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/87.jpg)
274
Dividing Eq. 3.123 with ix P yields
id
i i i i
i i
f x f fx P x P P
= =
We have defined (see Page 257) that
ˆ
ˆ ii
i
fx P
φ =
and when applying the above definition to
an ideal solution, we obtain
ˆ
ˆid
id i ii
i
f fx P P
φ = =
![Page 88: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/88.jpg)
275
Since
ii
fP
φ =
(see Page 241)
idi iφ φ= (3.124)
(Note: for IDEAL SOLUTION only)
![Page 89: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/89.jpg)
276
3.11 Excess Properties
As well as a residual property
R igM M M= − (2.55)
an excess property indicates how far
a real solution deviates from an ideal
solution, which can be written in the
form of equation as follows
E idM M M= − (3.125)
(3.125) – (2.55) results in
( ) ( )( )
E R id ig
id ig
M M M M M M
M M
− = − − −
= − − (3.126)
![Page 90: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/90.jpg)
277
Since an ideal-gas mixture is an
ideal-gas solution, when we replace ig
in Eq. 3.118
= +∑ ∑ lnidi i i ig x g RT x x (3.118)
with igig , the resulting equation is as
follows:
= +∑ ∑ lnig igi i i ig x g RT x x (3.127)
(by applying Eq. 3.28 to Eq. 3.78)
Eqs. 3.119-3.121 can also be re-
written as a function of igiM in the same
manner as Eq. 3.127 (please try to do it
yourself)
![Page 91: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/91.jpg)
278
Thus, (3.118) – (3.127) yields − = − =∑ ∑ ∑id ig ig R
i i i i i ig g x g x g x g
When performing the same derivation
for Eqs. 3.119-3.121, we obtain the following
general equation for any property id ig ig R
i i i i i iM M x M x M x M− = − =∑ ∑ ∑
(3.128)
When combining Eq. 3.128 with
Eq. 3.126, we obtain
( )E R Ri iM M x M− = − ∑
E R Ri iM M x M= −∑ (3.129)
![Page 92: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/92.jpg)
279
Note that Eq. 3.125 can be applied
to any partial property, as follows:
E idM M M= − (3.130)
![Page 93: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/93.jpg)
280
3.12 The Excess Gibbs Energy and the Activity Coefficient
We can write Eq. 3.101
( ) ˆlni i i ig T RT fμ = = Γ + (3.101)
for an ideal solution, as follows
( ) ˆlnid idi i ig T RT f= Γ + (3.131)
Combining Eq. 3.131 with Eq. 3.123
(Lewis/Radall Rule)
idi i if x f= (3.123)
gives
( ) lnidi i i ig T RT x f= Γ + (3.132)
![Page 94: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/94.jpg)
281
(3.101) – (3.132) results in
ˆ
lnid E ii i i
i i
fg g g RTx f
− = = (3.133)
Let ii
i i
fx f
γ = = activity coefficient of
species i in a solution
Thus,
lnEi ig RT γ= (3.134)
and
lnEi
igRT
γ= (3.135)
![Page 95: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/95.jpg)
282
When applying Eq. 3.28 to Eq. 3.135,
we obtain
lnEEi
i i igg x x
RT RTγ= =∑ ∑ (3.136)
The Nature of Excess Properties
(See Figure 3.4)
• Strongly depend on T
• Weakly depend on P, especially
at moderate T
• Depend on composition, but
the dependence varies from
solution to solution
![Page 96: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/96.jpg)
283
Figure 3.4: Examples of the Nature of Excess Properties
(from Introduction to Chemical Engineering Thermodynamics: 7th ed, by Smith, Van Ness, and Abbott, 2005)
![Page 97: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/97.jpg)
284
Table A.3.1: Values of φ0 as a function of Tr and Pr (from Introduction to Chemical Engineering Thermodynamics:
7th ed, by Smith, Van Ness, and Abbott, 2005)
![Page 98: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/98.jpg)
285
Table A.3.1: Values of φ0 as a function of Tr and Pr (cont.) (from Introduction to Chemical Engineering Thermodynamics:
7th ed, by Smith, Van Ness, and Abbott, 2005)
![Page 99: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/99.jpg)
286
Table A.3.2: Values of φ1 as a function of Tr and Pr (from Introduction to Chemical Engineering Thermodynamics:
7th ed, by Smith, Van Ness, and Abbott, 2005)
![Page 100: Theory of Solution Thermodynamics](https://reader031.vdocuments.site/reader031/viewer/2022012303/563db7cc550346aa9a8e0a99/html5/thumbnails/100.jpg)
287
Table A.3.2: Values of φ1 as a function of Tr and Pr (cont.) (from Introduction to Chemical Engineering Thermodynamics:
7th ed, by Smith, Van Ness, and Abbott, 2005)