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1 Capillary Thermodynamics Without a Geometric Gibbs Convention F . C. G O O D R I C H

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

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


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.


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


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


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


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 :


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


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.


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.


Capillary Thermodynamics Without a Geometric ... - ACS Publications

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