1 Capillary Thermodynamics Without a Geometric Gibbs Convention F . C. G O O D R I C H
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Institute of C o l l o i d and Surface Science, Clarkson College of Technology, Potsdam, N. Y .
It is shown in a number geometric
conventions
of capillary braic
thermodynamics
formalism
surfaces."
reliable
in which
The resulting
fore explicitly algebraic
of important employed
special
by Gibbs
are
replaceable
no mention
capillary
cases that
in his by
is made of
excess quantities
defined and may be interrelated
operations. tool than
the
surfaces,
the method
ment of
Gibbs.
by
To the extent that algebra intuitive
manipulation
is preferable
i b b s ' t h e r m o d y n a m i c analysis of
of
to the original
fluid
the
treatment an
alge-
"dividing are
there-
standard is a more geometric develop-
systems c o n t a i n i n g a p l a n e
interface ( 2 ) is c h a r a c t e r i z e d b y the c o n s t r u c t i o n t h r o u g h the i n t e r f a c i a l r e g i o n of i m a g i n a r y m a t h e m a t i c a l surfaces w h i c h are
supposed
to locate the extent of the separate phases i n a n e q u i v a l e n t m o d e l system i n w h i c h b y c o n v e n t i o n e v e r y t h i n g to one side of the m a t h e m a t i c a l sur face is h o m o g e n e o u s phase a w h i l e e v e r y t h i n g to the other side is h o m o geneous phase p. G i b b s w e l l u n d e r s t o o d the arbitrariness of this d i v i s i o n , a n d i n d e e d i n his w o r k m a d e use of this arbitrariness to d r a w different m a t h e m a t i c a l surfaces t h r o u g h the i n t e r f a c i a l r e g i o n , e a c h s u c h m a t h e m a t i c a l surface b e i n g i n v o l v e d w i t h a different set of
thermodynamic
quantities associated w i t h the c a p i l l a r y layer. W h a t e v e r m a y b e s a i d for the m a t h e m a t i c a l u t i l i t y of this p r o c e d u r e , its p e d a g o g i c results h a v e b e e n disastrous, a n d I k n o w of no other w i d e l y u s e d area of t h e r m o d y n a m i c s w h e r e so m u c h c o n f u s i o n exists. E v e n to this d a y the l i t e r a t u r e is f u l l of errors a n d i m p r e c i s e l y defined c a p i l l a r y q u a n t i t i e s , a l l of t h e m g o i n g b a c k u l t i m a t e l y to a too l i t e r a l i n t e r p r e t a t i o n of the G i b b s d i v i d i n g surfaces.
Guggenheim (3)
a n d D e f a y (1)
have
p a r t l y a l l e v i a t e d this s i t u a t i o n b y e m p h a s i z i n g that e x p e r i m e n t a l l y m e a 1
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
2
ADSORPTION
F R O M
AQUEOUS
SOLUTION
surable q u a n t i t i e s m u s t i n the t h e r m o d y n a m i c equations b e i n v a r i a n t to the l o c a t i o n o f a d i v i d i n g surface, b u t as has b e e n s h o w n i n a s p e c i a l case b y H a n s e n ( 4 ) , i t is b o t h possible a n d p r e f e r a b l e to h a n d l e the t h e r m o d y n a m i c s w i t h o u t ever i n t r o d u c i n g m a t h e m a t i c a l surfaces at a l l , a n d this w i l l b e the a p p r o a c h a d o p t e d here. One Component Systems A s a first e x a m p l e , consider a p u r e l i q u i d i n e q u i l i b r i u m w i t h its v a p o r . B e c a u s e I w i s h to focus a t t e n t i o n o n the l i q u i d / g a s i n t e r f a c e to the e x c l u s i o n of a d s o r p t i o n effects at s o l i d b o u n d a r i e s , I s h a l l suppose
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the c o n t a i n i n g vessel to b e c h e m i c a l l y inert. T h e G i b b s - D u h e m e q u a t i o n for the system is t h e n S d T - VdP + Ady + n d ^ = 0,
(1)
i n w h i c h the symbols h a v e t h e i r c o n v e n t i o n a l m e a n i n g s a n d w h e r e the extensive q u a n t i t i e s S, V, A , a n d n refer to the entire t w o phase system e n c l o s e d b y the c o n t a i n i n g vessel a n d not to some p a r t of i t i s o l a t e d artificially f r o m the rest b y m a t h e m a t i c a l b o u n d a r i e s . It is not u n c o m m o n i n t h e l i t e r a t u r e to find d e r i v e d f r o m (1) equations of the t y p e (2)
(dy/dT) =-S/A Pll
w h i c h are c o m p l e t e l y f a l l a c i o u s , for a one c o m p o n e n t , t w o phase system has f r o m the phase r u l e o n l y one degree of f r e e d o m , so that it is i m p o s sible to alter the t e m p e r a t u r e of the system w i t h o u t s i m u l t a n e o u s l y alter i n g the pressure a n d the c h e m i c a l p o t e n t i a l . T h i s is to say that E q u a t i o n 2 is false because the intensive v a r i a b l e s T, ?, y, a n d /u, are not i n d e p e n d e n t l y variable. T h e latter fact m a y b e e m p h a s i z e d b y c o n s i d e r i n g s m a l l samples of m o l a r content n
a
and n
d r a w n f r o m the b u l k phases i n regions far f r o m
e
the interface. T h e size a n d shape of these samples n e e d h a v e n o r e l a t i o n s h i p to the g e o m e t r y of the i n t e r f a c e — a n y i r r e g u l a r l y s h a p e d s p e c i m e n of b u l k phase w i l l do. F o r e a c h of these b u l k phase samples a a n d /? w e h a v e G i b b s - D u h e m equations S dT - V dP a
a
+ n«d/x = 0
S^dT - V*dP + n*dfjL = 0 or u p o n d i v i d i n g t h r o u g h e a c h b y n . n a
S«dT - V dP a
&
respectively
+ dfi = 0
S*dT - V * d P + d/x = 0 i n w h i c h as u s u a l the b a r r e d s y m b o l s are the m o l a r entropies a n d v o l u m e s of the respective phases.
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
1.
Capillary
GOODRICH
3
Thermodynamics
E q u a t i o n s 1 a n d 3 c o m p r i s e a set of three relationships b e t w e e n f o u r differentials, d T , d P , d y , a n d d/A. T h e r e thus r e m a i n s a single degree of f r e e d o m i n the system i n agreement w i t h the phase r u l e . I n order to e l i m i n a t e u n w a n t e d v a r i a b l e s , m u l t i p l y E q u a t i o n s 3 b y a l g e b r a i c m u l t i p l i e r s x a n d y r e s p e c t i v e l y a n d subtract f r o m E q u a t i o n 1. (S - xS« - y S 0 ) d T -
( V - xV« - y V 0 ) d P + A d y + (n - x - y ) d
/ A
= 0 (4)
E q u a t i o n 4 is v a l i d for a r b i t r a r y c h o i c e of x a n d y, a n d w e s h a l l o b t a i n different s p e c i a l equations for different choices of t h e m u l t i p l i e r s . E a c h s u c h c h o i c e corresponds to a different G i b b s c o n v e n t i o n .
T o understand
this fact, note for e x a m p l e that a q u a n t i t y S — x S " - ^ y S
is the t o t a l
0
e n t r o p y of the heterogeneous system m i n u s a n a m o u n t x S of the e n t r o p y Downloaded by 80.82.77.83 on May 5, 2018 | https://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0079.ch001
a
of the b u l k a phase m i n u s a n a m o u n t y S of the b u l k p phase.
Depending
0
therefore o n h o w w e choose x a n d y, w e s h a l l be d e f i n i n g excess entropies w h i c h c o m p a r e the r e a l , t w o phase system c o n t a i n i n g a n interface j v i t h fictitious
systems c o m p r i s i n g t w o b u l k phases of e n t r o p y x S
the absence of a n i n t e r f a c i a l r e g i o n .
a
and y S
3
in
T h e m u l t i p l i e r s thus p e r f o r m the
same f u n c t i o n as a G i b b s d i v i d i n g surface, b u t because w e are u s i n g algebraic methods
i n s t e a d of geometric
intuition, w e shall be able
to
make our thermodynamics more explicit. T o o b t a i n a n i n t e r p r e t a t i o n of the e x p e r i m e n t a l l y m e a s u r a b l e q u a n t i t y d y / d T , choose x a n d y i n E q u a t i o n 4 to m a k e the coefficients of d P a n d d/x v a n i s h . x + y = n xV
(5)
+ yW = V
a
T h e s e equations are solvable e x p l i c i t l y , nW-V = -;y = y/3 — ya w h e n c e u p o n s u b s t i t u t i o n into E q u a t i o n X
=
=r
r d y / d T = -S = - A - i S ' |_
V - nV ^ y/3 _ y a 4 a n d rearrangement, a
(6)
y/s — y v —V - "1 ™—JS« - S* . y/3 _ y a y/3 _ y a J n
n
a
(7)
T h e i n t r o d u c t i o n of the s u p e r s c r i p t ( n , V ) i m p l i e s the a d o p t i o n of a G i b b s c o n v e n t i o n a l g e b r a i c a l l y expressed b y E q u a t i o n s 5, w h i c h state that w e are c o m p a r i n g the r e a l system w i t h a f i c t i t i o u s one c o n s i s t i n g of t w o b u l k phases i n contact i n the absence of a n i n t e r f a c i a l r e g i o n , a n d w i t h the a d d e d specification that the n u m b e r s of moles n a n d the t o t a l v o l u m e V of the The quantity S Defay (1))
( r ?
fictitious '
y )
system s h a l l b e the same as i n the r e a l one.
( w h i c h is c a l l e d S
A
CT
by Guggenheim (3)
a n d sf
by
is the e n t r o p y change p e r u n i t area of interface c r e a t e d w h e n
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
4
ADSORPTION
F R O M
AQUEOUS
SOLUTION
a t a l l , slender, c l o s e d vessel of fixed v o l u m e V c o n t a i n i n g n moles of a substance i n g a s - l i q u i d e q u i l i b r i u m is l a i d i s o t h e r m a l l y o n its side, t h e r e b y e n l a r g i n g the i n t e r f a c i a l area.
N e i t h e r the t o t a l v o l u m e n o r the t o t a l
moles w i t h i n the vessel change, a n d the e n t r o p y c h a n g e is e x c l u s i v e l y d u e to the c r e a t i o n of n e w interface f r o m m a t e r i a l p r e v i o u s l y present i n the b u l k phases. Confidence
i n the correctness
of E q u a t i o n 7 is i n c r e a s e d b y
an
h e u r i s t i c a r g u m e n t w h i c h w i l l not b e r e p r o d u c e d here, b u t w h i c h shows that x a n d y i n E q u a t i o n s 6 are g o o d estimates of the t o t a l n u m b e r s of moles of l i q u i d a n d of v a p o r i n the r e a l system. It f o l l o w s t h a t x S " a n d yS
3
are c o r r e s p o n d i n g l y g o o d estimates of the t o t a l entropies to b e as
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s i g n e d to the l i q u i d a n d v a p o r phases i n the r e a l system, so t h a t S — x S
a
— x S is the a m o u n t b y w h i c h the e n t r o p y of the r e a l system exceeds the 3
c o m b i n e d entropies of its l i q u i d a n d v a p o r phases.
T h e phrase " g o o d
estimate" here has m o r e of a l i t e r a r y t h a n a m a t h e m a t i c a l i n t e r p r e t a t i o n ; f o r the diffuseness of the i n t e r f a c i a l r e g i o n p r o h i b i t s a n y m a t h e m a t i c a l l y rigorous d e f i n i t i o n of the v o l u m e s of the respective phases, a n d this fact w a s the w h o l e m o t i v a t i o n of G i b b s ' use of a d i v i d i n g surface. Two
Component Systems T u r n i n g n o w to a d s o r p t i o n e q u i l i b r i u m , let us a p p l y a l g e b r a i c m e t h
ods to a t w o c o m p o n e n t
1,2 phase system.
F r o m the phase r u l e there
w i l l b e t w o degrees of f r e e d o m , b u t w e s h a l l r e d u c e this to one b y m a i n t a i n i n g the t e m p e r a t u r e constant. T h e n for the t o t a l system there exists a G i b b s - D u h e m equation - V d P + A d y + riid/x! + n dp 2
2
(8)
= 0
a n d for samples of a r b i t r a r y shape a n d size d r a w n f r o m the interiors of the b u l k phases - V « d P + n ^ d / x i + n dfx a
2
-V(*dP
2
+ n^d/xx + n/d^
D i v i d i n g through Equations 9 by V
a
and V
3
= 0
^
= 0. respectively we obtain equa
tions w h i c h are i n d e p e n d e n t of the size of the samples c h o s e n :
i n w h i c h the cs
- d P + cfd^
+ c dfi
-dP + c ^
+ c/d^^O
a
2
2
= 0 (
1
0
)
are the b u l k phase concentrations i n moles p e r liter of
the several components i n the i n d i c a t e d phases. A s i n o u r p r e v i o u s w o r k , E q u a t i o n s 8 a n d 10 c o m p r i s e three relations b e t w e e n f o u r differentials, so that i n agreement w i t h the phase r u l e a single r e l a t i o n b e t w e e n t w o differentials is i m p l i e d . T o o b t a i n s u c h a
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
1.
Capillary
GOODRICH
5
Thermodynamics
r e l a t i o n , m u l t i p l y E q u a t i o n s 10 b y m u l t i p l i e r s x a n d y a n d s u b t r a c t f r o m E q u a t i o n 8. - ( V - x - y ) d P + A d y + ( n - x « - y c ^ d ^ + ( n - xc x
C l
a
2
2
- yc/)d^
2
= 0 (11)
W e h a v e c o n c e p t u a l l y a r r i v e d at t h e same p o i n t as w e d i d i n t h e a r g u m e n t l e a d i n g t o E q u a t i o n 4; f o r E q u a t i o n 11 is v a l i d f o r a r b i t r a r y c h o i c e of m u l t i p l i e r s x a n d y, a n d e a c h s u c h c h o i c e corresponds t o a different G i b b s convention.
T h e c h o i c e w h i c h leads t o t h e G i b b s a d s o r p t i o n
e q u a t i o n is that w h i c h makes t h e coefficients of d P a n d d/xi ( c o n v e n t i o n a l l y defined to b e t h e s o l v e n t ) v a n i s h :
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x + y= V x c
i
a
(12)
+ y c i = fti 3
w h i c h is t o say
x
=-^r^r;
y =
C
i
._
C
i
,
•
< > 13
S u b s t i t u t i n g i n t o E q u a t i o n 11 a n d r e a r r a n g i n g ,
(
9
r
/
S
B
)
r
= -
r
!
< . , ,
-A
-
1
[
%
_ 5 ^
v
- ^
t
f
]
i
M
)
T w o things are t o b e n o t e d a b o u t E q u a t i o n 14. T h e first is t h a t t h e pressure is n o t h e l d constant. I e m p h a s i z e this p o i n t because t h e state m e n t is f r e q u e n t l y seen i n t h e l i t e r a t u r e that t h e G i b b s a d s o r p t i o n e q u a t i o n is v a l i d o n l y u n d e r c o n d i t i o n s of constant t e m p e r a t u r e a n d pressure, a r e s t r i c t i o n w h i c h f o r a t w o phase, t w o c o m p o n e n t system reduces t h e n u m b e r of degrees of f r e e d o m to zero. T h e second t h i n g to note is that the G i b b s c o n v e n t i o n i m p l i e d b y E q u a t i o n 14 is c o n t a i n e d i n E q u a t i o n s 12, w h i c h state t h a t w e are c o m p a r i n g t h e r e a l system w i t h a
fictitious
one h a v i n g t h e same v o l u m e a n d t h e same n u m b e r s of moles o f solvent as the r e a l system. F i n a l l y a n h e u r i s t i c a r g u m e n t v a l i d f o r most c o n d i t i o n s of e x p e r i m e n t a l interest shows that t h e q u a n t i t y i n brackets o n t h e r i g h t h a n d side of E q u a t i o n 14 is t h e difference b e t w e e n t h e t o t a l moles n
2
of solute a c t u a l l y present i n t h e system m i n u s t h e t o t a l moles of solute present i n t h e b u l k l i q u i d a n d v a p o r phases. A n o t h e r excess q u a n t i t y l Y
2 )
is defined b y i n t e r c h a n g i n g i n d i c e s 1
a n d 2 i n E q u a t i o n s 12, 13, a n d 14> a n d f o r this q u a n t i t y t h e G i b b s c o n v e n t i o n is different, f o r w e are n o w c o m p a r i n g t h e r e a l system w i t h a fictitious
one defined to h a v e t h e same t o t a l moles n of solute as has t h e 2
r e a l one. D e s p i t e t h e fact that l Y
2 )
andr
2
( 1 )
are defined f o r different
G i b b s conventions, t h e y are a l g e b r a i c a l l y r e l a t e d , as t h e reader m a y
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
6
ADSORPTION
F R O M
AQUEOUS
SOLUTION
r e a d i l y c o n f i r m for himself. D i r e c t l y f r o m t h e i r a l g e b r a i c definitions one c a n s h o w that
(c «-c fl)r 1
1
2
( 1 )
+ (c
a
2
-c/)lY
2 )
= 0.
(15)
T h i s e q u a t i o n assumes a m o r e f a m i l i a r f o r m i f w e are w i l l i n g to ignore the gas phase (/?)
concentrations i n c o m p a r i s o n w i t h those of the l i q u i d
phase, for t h e n a p p r o x i m a t e l y
c r 1
a
2
+ c r!
( 1 )
2
a
(2)
= o,
or w h a t is t h e same t h i n g XilV^ + X a l V ^ O , Downloaded by 80.82.77.83 on May 5, 2018 | https://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0079.ch001
in which Xi and x
2
(16)
are the m o l e fractions of the l i q u i d phase. E q u a t i o n
16 is f a m i l i a r , b e i n g q u o t e d , for example, b y D e f a y
(1).
A s a final example, let us d e r i v e another set of excess quantities f r o m E q u a t i o n 11 b y c h o o s i n g the G i b b s c o n v e n t i o n x + y= V x ( « + c « ) + y ( * + cf) C l
a
C l
= (ti! + n ) .
(17)
2
U p o n s o l v i n g these equations for x a n d y a n d i n t r o d u c i n g the results i n t o E q u a t i o n 11 w e have a n a d s o r p t i o n e q u a t i o n dy +
+ r
2
( n )
d^ = 0
(18)
in which = A"
1
V(
a
Cl
(fix + n ) - V(c^
+
c/)
(Cj" +
+
Co*)
2
|\
+ c)
(ci« + c « ) -
-
2
(C/
ci
a
(n + n )
-
a
2
2
C °) x
2
(c^ + c / )
C
l
J
( 1 9 )
a n d a s i m i l a r e q u a t i o n for r d e r i v e d f r o m E q u a t i o n 19 b y i n t e r c h a n g i n g i n d i c e s 1,2. W h i l e the a p p e a r a n c e of E q u a t i o n 19 is f o r m i d a b l e , the quantities i Y 7 0 a n d r 2 ( r j ) are v e r y p r e c i s e l y defined a n d m a y be s h o w n d i r e c t l y f r o m t h e i r a l g e b r a i c definitions to satisfy the l i n e a r r e l a t i o n 2
(w)
iy
w)
+r
2
(w)
=o
(20)
w h i c h is to be c o m p a r e d w i t h E q u a t i o n 15. T h e G i b b s c o n v e n t i o n ( E q u a t i o n 17) states that w e c o m p a r e the r e a l system w i t h a fictitious one h a v i n g the same t o t a l v o l u m e a n d t o t a l n u m bers of moles of a l l constituents as the r e a l system. U n d e r this c o n v e n t i o n the t o t a l moles of i n d i v i d u a l components 1 a n d 2 w i l l differ b e t w e e n the r e a l a n d the fictitious systems, b u t because the t o t a l of a l l moles of b o t h components is the same, the surface excesses of each m u s t s u m to zero, a n d this is the m e a n i n g of E q u a t i o n 20.
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
1.
Capillary
GOODRICH
7
Thermodynamics
T h i s c o n v e n t i o n has the a d v a n t a g e t h a t b o t h of the consituents of the m i x t u r e are defined a c c o r d i n g to the same c o n v e n t i o n , whereas the quantities l Y
2 )
and r
2
( 1 )
are separately defined b y different conventions.
It w o u l d b e a m a t t e r of some d i f f i c u l t y to p r o v e b y geometric m a n i p u l a t i o n of d i v i d i n g surfaces that, for e x a m p l e , r
2
and r
( 1 )
2
( n )
are l i n e a r l y r e l a t e d .
F r o m t h e i r a l g e b r a i c definitions, h o w e v e r , i t is easy to s h o w that
(21) exactly. If w e neglect the concentrations i n the gas (/?)
phase, this s i m
plifies to w
r< > = Downloaded by 80.82.77.83 on May 5, 2018 | https://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0079.ch001
2
X l
r
( 1 ) 2
(22)
,
w h i c h is the r e l a t i o n most c o m m o n l y f o u n d i n the l i t e r a t u r e ( 5 , 6 ) . W h i l e I h a v e g i v e n here o n l y a f e w examples, i t turns out that the entirety of c a p i l l a r y t h e r m o d y n a m i c s m a y b e f o u n d e d u p o n m e t h o d s , r e s u l t i n g i n a great i m p r o v e m e n t
algebraic
i n the ease w i t h
which
the c a p i l l a r y excess q u a n t i t i e s m a y b e defined a n d m a n i p u l a t e d . Acknowledgment I a m i n d e b t e d to G . S c h a y for s h o w i n g m e a c o p y of his a r t i c l e i n a d v a n c e of p u b l i c a t i o n . Literature
Cited
(1) Defay, R., Priogogine, I., Bellemans, A., Everett, D. H., "Surface Tension and Adsorption," Wiley and Sons, Inc., New York (1966). (2) Gibbs, J. W., "Collected Works," Vol. I, Yale University Press, New Haven (1948). (3) Guggenheim, E. A., "Thermodynamics," Interscience, New York (1949). (4) Hansen, R. S., J. Phys. Chem. 66, 410 (1962). (5) Kipling, J. J., "Adsorption from Solutions of Non-Electrolytes," Academic Press, New York (1965). (6) Schay, G., "Surface and Colloid Science," Interscience, New York (in press). RECEIVED October 26, 1967.
Weber and Matijevi; Adsorption From Aqueous Solution Advances in Chemistry; American Chemical Society: Washington, DC, 1968.
