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A determination of ionic diffusion coefficients

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A determination of ionic diffusion coefficients

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Content A DETERMINATION OF IONIC DIFFUSION COEFFICIENTS
A D i s s e r t a t i o n
P r e s e n t e d t o
t h e F a c u lt y o f th e D epartm en t o f C h e m istry
U n i v e r s i t y o f S o u th e rn C a l i f o r n i a
In P a r t i a l F u l f i l l m e n t
o f t h e R e q u ire m e n ts f o r th e D egree
D o c to r o f P h ilo s o p h y
by
J u l i a n M. N ie ls e n
May 1951
U M I Number: DP21752
All rights reserved
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PH. 0 0- 'SI
This dissertation, written by
..................J u l i a n N i e l s e s .......
under the guidance of h.X.P. Faculty Committee
on Studies, and approved by all its members, has
been presented to and accepted by the Council
on Graduate Study and Research, in partial ful
fillment of requirements for the degree of
D O C T O R O F P H I L O S O P H Y
H* Deuel.*.... J r * .............
Dean
.H^...W...._Patmore
Secretary
Date.. ....................
Com m ittee on Studies
_ Chairman
.............................. ..................................
i
TABLE OF CONTENTS
CHAPTER PAGE
I . INTRODUCTION ....................................... .............................. 1
I I . THEORY . . . . . ..................................................................... 5
LIMITING LAWS FOR DIFFUSION ................................ 5
The "Thermodynamic" L im i ti n g Law
' f o r S e l f D i f f u s i o n ............................................... 7
The "C onductance" L im i ti n g Law
f o r S e l f D i f f u s i o n .....................................................11
The " R e la x a tio n " L im itin g Law
f o r S e l f D i f f u s i o n .....................................................12
THEORY OF THE DIAPHRAGM METHOD................................ 14
I I I . EXPERIMENTAL METHODS .......................................................... 21
Use o f th e Diaphragm C e l l ......................................21
P r e p a r a t i o n o f S o l u t i o n s ..................................... 22
P r e p a r a t i o n o f T r a c e r s . . . . . . . . . 22
C o u n tin g th e R a d io a c tiv e T r a c e r s . . . . 26
IV . THE EFFECT OF STIRRING............................................................31
N o n -m ec h an ic al S t i r r i n g .................................... 32
M e c h a n ic a l S t i r r i n g . . . . . . . . . . 35
. V. CALIBRATION OF THE DIAPHRAGM C E L L S ............................ 42
V I. SURFACE EFFECTS ............................................................49
V I I . RESULTS AND SUMMARY................................................................ 57
V I I I . DISCUSSION................................ 64
BIBLIOGRAPHY ................................................................................................ 67
ii
LIST OF TABLES
TABLE PAGE
I . S e l f a b s o r p t i o n c u rv e o f s u l f u r 3 3 i n
b a riu m s u l f a t e ................................................... 29
I I . Com parison o f e f f e c t o f r a t e of s t i r r i n g
on th e a l t e r n a t e r o t a t i o n and m ag n e tic
s t i r r i n g m e t h o d s ................................................... 41
I I I . D i f f u s i o n c e l l c a l i b r a t i o n by
c o n d u c ta n c e m easurem ents ...................................... 4 7
IV. C om parison o f e f f e c t o f a d s o r p t i o n on
g l a s s and s t e e l diaphragm on
d i f f u s i o n c o e f f i c i e n t s . . . . ......................... 54
V. S e l f d i f f u s i o n c o e f f i c i e n t s o f Ha*" and Cl"
i n aqueous sodium c h l o r i d e s o l u t i o n s
a t 25°C............................................................................ 58
V I. S e l f _ d if f u s io n c o e f f i c i e n t s o f Na* and
S O / i n aqueou s sodium s u l f a t e
s o l u t i o n s a t 25°C. ................................... . 59
iii
LIST OF FIGURES
FIGURE PAGE
1 . Diaphragm d i f f u s i o n c e l l . . . . . . . . . . 20
2. S e l f A b s o r p tio n f a c t o r s o f S 3 3 in BaS 0 4 . . . 30
3. Dependence o f s e l f d i f f u s i o n c o e f f i c i e n t s
o f H a ^ i n 0 ^ 3 8 0 4 on c e l l c o n s t a n t i n
u n s t i r r e d r u n s .....................................................................34
4 . S e c t i o n a l c o n d u c ta n c e and d i f f u s i o n c e l l s . . 44
5. T r a n s f e r e n c e a p p a r a t u s ..................................................... 45
6 . C ond uctance c e l l c o n s t a n t s o f HaCI
s o l u t i o n s i n diaphragm c e l l ..................................51
7. C onductance c e l l c o n s t a n t s o f ^ 3 8 0 4
s o l u t i o n s i n diaphragm c e l l . .......................... . 52
8 . S e l f d i f f u s i o n c o e f f i c i e n t s o f Na+
a nd Cl~ a t 2 5 ° C . ............................... 60
9 . S e l f d i f f u s i o n c o e f f i c i e n t s o f Ha4 "
and SOf a t 25°C .................................................................61
INTRODUCTION i
I
I
D i f f u s i o n i n s o l u t i o n h as been s u b j e c t t o s t u d y J
s i n c e 1850 when Graham^ began h i s e x p e r im e n ta l w ork. I
F ic k ^ began th e t h e o r e t i c a l work s h o r t l y t h e r e a f t e r by j
p o i n t i n g o u t t h e a n a lo g y betw een d i f f u s i o n and h e a t c o n - ;
l
d u c ti o n and t h i s l e d t o t h e la w s b e a r i n g h i s name. The :
f i r s t s t u d i e s w ere l a r g e l y e m p i r i c a l b u t found e a r l y ;
l
a p p l i c a t i o n in th e d e t e r m i n a t i o n o f io n s i z e by th e app
l i c a t i o n o f S to k e s ' law .
The d i f f u s i o n c o e f f i c i e n t i s th e p r o d u c t o f th e
m o b i l i t y tim e s th e d r i v i n g f o r c e . N e rn s t^ c o n s i d e r e d '
t h e d r i v i n g f o r c e t o be th e o sm o tic p r e s s u r e g r a d i e n t
and i n 1888 r e l a t e d t h e d i f f u s i o n c o e f f i c i e n t o f an e l
e c t r o l y t e t o t h e e l e c t r i c m o b i l i t i e s o f i t s c o n s t i t u e n t
i o n s a s d e te r m in e d from c o n d u c ta n c e m ea su re m e n ts. As
e x p e r im e n ta l work became a v a i l a b l e i t was found t h a t
t h i s law gave a v a lu e o f th e d i f f u s i o n c o e f f i c i e n t v a l
i d o n ly a t i n f i n i t e d i l u t i o n .
In 1931 H a r tle y ^ s u g g e s te d t h a t t h e d r i v i n g f o r c e
f o r d i f f u s i o n was th e g r a d i e n t o f th e c h e m ic a l p o t e n t i a l .
( l ) T. Graham, P h i l . T r a n s . , 1 4 0 . 1 , 805 (1 8 5 0 ).
(3 ) A. F ic k , Pogg. A nn., 9 4 , 59 ( l 8 5 5 ) .
(3 ) W. N e r n s t , Z. p h y s i k . Chem., 2, 613 (1 8 8 8 ).
(4 ) G. S. H a r t l e y , P h i l . M ag., 1 3 , 473 (1 9 3 1 ).
3 !
O n sag er and F uoss^ a p p l i e d th e i o n i c a t t r a c t i o n th e o r y
t o t h e d i f f u s i o n p ro b lem , o b t a i n i n g th e r e s u l t t h a t th e
i
« <
d i f f u s i o n m o b i l i t i e s f o r th e io n s a r e s l i g h t l y g r e a t e r j
i
th a n th e e l e c t r i c m o b i l i t i e s , and i n c r e a s e s l i g h t l y ;
i
o v e r th e i n f i n i t e d i l u t i o n v a lu e a s th e c o n c e n t r a t i o n I
j
i n c r e a s e s . However, t h i s law s h o u ld be v a l i d o n ly a t 1
i
sm a ll c o n c e n t r a t i o n s , where t h e i n t e r i o n i c a t t r a c t i o n
I
t h e o r y i s v a l i d . The c h e m ic a l p o t e n t i a l f a c t o r d e c r e a s
es w i t h i n c r e a s e in c o n c e n t r a t i o n . T h is o f f s e t s th e
i n c r e a s e i n m o b i l i t y and c a u s e s th e d i f f u s i o n c o e f f i c
i e n t to d e c r e a s e . Thus th e O n sa g e r-F u o ss t h e o r y r e l a t e d
th e d i f f u s i o n c o e f f i c i e n t s t o io n m o b i l i t i e s and a c t i v i t y
I
c o e f f i c i e n t s . T h is t h e o r y has b een p a r t i a l l y c o n firm e d
by Harned and French® .
The b e h a v io r o f s o l u t i o n s o f e l e c t r o l y t e s has
b een m easured u n d e r t h e i n f l u e n c e o f e x t e r n a l f o r c e s
su c h as m e c h a n ic a l s h e a r and e l e c t r i c a l f o r c e s t o g iv e
p r e c i s e v i s c o s i t y and c o n d u c ta n c e d a t a and y e t th e s tu d y
o f e l e c t r o l y t e s o l u t i o n s w ith no e x t e r n a l f o r c e s b u t
o n ly i n t e r n a l th e r m a l f o r c e s i s o n ly now y i e l d i n g e x a c t
r e s u l t s . R e ce n t mean d i f f u s i o n c o e f f i c i e n t d e te r m in a
(5) L. O nsager and R. M. F u o s s , J . P h y s. Chem .,
3 6 . 3689 (1 9 3 2 ). |
( 6 ) H. S. H arned and D. M. F re n c h , Ann. N.Y. Acad.
S c i . , 46 , 26? (1 9 4 5 ). j
t i o n s o f H arned and c o -w o r k e r s ’ 7'® and S to k e s 9 have p r o
v i d e d p r e c i s e d a t a on some 1 0 e l e c t r o l y t e s o v e r a w ide
ra n g e o f c o n c e n t r a t i o n s . These d a t a g iv e th e m o b i l i t i e s
o f io n s in s o l u t i o n u n d e r t h e c o n d i t i o n s t h a t o p p o s i t e l y
c h a rg e d io n s move t o g e t h e r a s an e l e c t r i c a l l y n e u t r a l
p a r t i c l e .
The s e l f d i f f u s i o n m o b i l i t i e s o f th e i n d i v i d u a l
io n s a s t h e y move a lo n e w ith o u t e l e c t r o n e u t r a l i t y r e s t r
i c t i o n s a r e o f more fu n d a m e n ta l i n t e r e s t i n t h a t th e y
r e l a t e t o t h e p r o p e r t i e s o f i n d i v i d u a l io n s r a t h e r th a n
t o t h e a v e ra g e p r o p e r t y o f two o r more i o n s . T h is i n
v e s t i g a t i o n o f a s i n g l e io n s p e c i e s i n s o l u t i o n i s made
p o s s i b l e th r o u g h th e u s e o f i s o t o p i c t r a c e r s . Ah io n
s p e c i e s may be ta g g e d by su c h t r a c e r s and t h e i r m otio n
f o llo w e d u n i n f lu e n c e d by e x t e r n a l f o r c e s o r c o n c e n t r a t i o n
g r a d i e n t s . The m o b i l i t y th u s m easured i s a r e s u l t o f
th e rm a l m o tio n a s m o d if ie d by th e i n t e r i o n i c a t t r a c t i o n
o f th e c h a rg e d i o n s .
P r i o r to 1948 when t h i s i n v e s t i g a t i o n was begu n
o n ly th e s e l f d i f f u s i o n o f heavy w a te r i n t o l i g h t w a te r
(7) H. S. H arned and R. L. N u t t a l l , J . Am. Chem. S o c .,
6 9 . 736 (1 9 4 7 ).
( 8 ) H. S. Harned and A. L. L evy, i b i d . , .71, 3781
( 1 9 4 9 ) .
(9) *R. H. S t o k e s , i b i d . , 72, 763, 2243 (1 9 5 0 ).
4
a t s e v e r a l temperatures***®, Na* and C l” i n aqueous sod
ium c h l o r i d e s o l u t i o n s a t 35°C. ■ * ■ ■ * " and Na^ in aqueous
sodium i o d i d e s o l u t i o n s a t 2 5 ° C . ^ had b e en r e p o r t e d .
I n c r e a s e d i n t e r e s t i s shown by t h e f a c t t h a t s i n c e t h a t
tim e s e v e r a l more s t u d i e s have b e en made. Na+ i n aq
ueo u s sodium c h l o r i d e s o l u t i o n s a t 250C.'*‘3 , Na- * " and 1 “
i n sodium i o d i d e s o l u t i o n s a t 250C .^ 4 and Ag- * " i n s i l v e r
n i t r a t e s o l u t i o n s a t 2 5 °C .15
The p u rp o s e o f t h i s r e s e a r c h was t o d e v e lo p and
a p p ly t h e d iaphragm c e l l method f o r d i f f u s i o n m easure
ment to s i n g l e io n d i f f u s i o n w ork. The s i n g l e io n d i f f
u s i o n c o e f f i c i e n t s o b ta in e d w ere d e s i r e d a s a g e n e r a l
c o n t r i b u t i o n t o th e n a t u r e o f e l e c t r o l y t e s o l u t i o n s and
t o s e r v e t o t e s t t h e o r i e s as th e y a r e now b e i n g d e v e l
oped.
(10) W. J . C. O rr and J . A. V. B u t l e r , J . Chem. S o c .,
1273 (1 9 3 5 ).
(11) L. P. J e h l e , Ph. D. D i s s e r t a t i o n , U n iv . o f
C a l i f . , B e r k e le y , 1938.
(12) A. W. Adamson, J . Chem. P h y s ., 1 5 , 762 (1 9 4 7 ).
(1 3 ) A. W. Adamson, J . W. Cobble and J . M. N i e l s e n ,
i b i d . . 1 7 , 740 (1 9 4 9 ).
(1 4 ) J . H. Wang and J . W. Kennedy, J . Am. Chem. S o c .,
7 2 , 2080 ( 1 9 5 0 ).
(1 5 ) S. 6 . W hiteway, D. F. MacLennan and C. C. C off
i n , J . Chem. P h y s ., 18, 229 ( 1 9 5 0 ).
THEORY
LIMITING LAWS FOR DIFFUSION
A c o m p le te t h e o r y o f s e l f d i f f u s i o n h a s n o t b e en
d e v e lo p e d . However, s e v e r a l l i m i t i n g e q u a t io n s have been
d e r i v e d from t h e O n s a g e r-F u o ss t h e o r y o f e l e c t r o l y t e
d i f f u s i o n ^ . S in c e s u f f i c i e n t d a t a to a d e q u a te ly t e s t j
t h e s e l i m i t i n g v a lu e s have n o t b e en a v a i l a b l e , work on
t h e t h e o r y h a s la g g e d . The l i m i t i n g law s a r e e x p r e s s e d
i n t h e form o f e q u a t io n s r e l a t i n g th e s e l f d i f f u s i o n
c o e f f i c i e n t o f t h e io n t o th e c o n c e n t r a t i o n . These eq
u a t i o n s s h o u ld become e x a c t o n ly a s i n f i n i t e d i l u t i o n i s
i
a p p ro a c h e d , i . e . , th e y g iv e a l i m i t i n g law f o r t h e v a r -
i 1
' i a t i o n in d i f f u s i o n c o e f f i c i e n t s w ith c o n c e n t r a t i o n . j
« I
j The O n sa g e r-F u o ss t h e o r y o f mean e l e c t r o l y t e
! I
i d i f f u s i o n c o n s i d e r s t h r e e f a c t o r s i n th e n e t m otion o f
io n s a c r o s s a c o n c e n t r a t i o n g r a d i e n t . The f i r s t i s th e
Htim e o f r e l a x a t i o n e f f e c t . ” I f t h e m o tio n i s su c h t h a t
1
j a f o r c e c a u s e s th e io n to move away from i t s a tm o s p h e re ,
t h e a tm o sp h e re i s deform ed and t h e io n w i l l s u f f e r an
e l e c t r i c a l d ra g t e n d i n g t o re d u c e i t s v e l o c i t y . T h is
d r a g may be r e l a t e d to t h e tim e o f r e l a x a t i o n and c a l c u
l a t i o n s made f o r c o r r e c t i o n s . I n i o n i c d i f f u s i o n b o th j
th e p o s i t i v e and n e g a t i v e io n s a r e r e q u i r e d by e l e c t r i c a l
6 ;
n e u t r a l i t y to move w ith t h e same a v e r a g e v e l o c i t y . Under
t h e s e c o n d itio n s , th e io n a tm o sp h e re s a r e s y m m e tric a l and 'j
t h e tim e o f r e l a x a t i o n e f f e c t v a n i s h e s .
The se co n d f a c t o r a r i s e s from th e r e s i s t a n c e t o !
m o tio n o f t h e s o l v e n t e n t r a i n e d hy th e io n a tm o sp h e re ;
t h i s i s t r e a t e d a s an a d d i t i v e c o r r e c t i o n t o t h e io n
v e l o c i t i e s and- i s c a l l e d th e ‘’e l e c t r o p h o r e t i c " e f f e c t .
Net d i f f u s i o n i s a c o n seq u en ce o f a c o n c e n t r a t i o n 1
g r a d i e n t and t h i s i n t u r n in v o lv e s an a c t i v i t y c o e f f i c
i e n t g r a d i e n t . T h is g r a d i e n t r e p r e s e n t s t h e th erm o - \
dynamic c o r r e c t i o n t o th e d r i v i n g f o r c e on th e io n and
a p p e a rs as a t h i r d f a c t o r in th e form o f a te rm ,
( 1 + c d in y / d c ) , w here i s th e mean a c t i v i t y c o e f f i c
i e n t and £ i s t h e c o n c e n t r a t i o n i n m oles p e r l i t e r .
In th e c a s e o f s e l f d i f f u s i o n t h e r e a r e no b u lk
c o n c e n t r a t i o n g r a d i e n t s , t h e r e f o r e , t h e r e w i l l be no
s o l v e n t c o u n te r f lo w and th e e l e c t r o p h o r e t i c te rm p l a y s
no p a r t . T h is l e a v e s t h e tim e o f r e l a x a t i o n and t h e a c t
i v i t y c o e f f i c i e n t e f f e c t s t o b e c o n s i d e r e d . T here i s
much d is a g r e e m e n t on t h e s e two f a c t o r s . H u izen g a,
G r ie g e r and Wall'*'® and G o s tin g and H arned1 ^ h o ld t h a t
(16) J . R. H u iz en g a , P. F. G r ie g e r and F. T. W all,
J . Am. Chem. S o c ., 72., 4328 ( 1 9 5 0 ). >
(1 7 ) L. J . G o s tin g and H. S. H arned, i b i d , 73, I
159 (1 9 5 1 ). !
7
t h e a b s e n c e of b u lk c o n c e n t r a t i o n g r a d i e n t s i n s e l f d i f f
u s io n means t h a t t h e r e can be no a c t i v i t y c o e f f i c i e n t
i
i
g r a d i e n t s and hence no a c t i v i t y te rm . G o s tin g and Harned)
compute t h e v a r i a t i o n o f th e d i f f u s i o n c o e f f i c i e n t w ith !
c o n c e n t r a t i o n a s d e p en d e n t e n t i r e l y upon a c a l c u l a t e d j
tim e o f r e l a x a t i o n e f f e c t . The l i m i t i n g s lo p e e q u a t io n j
1 fz 1
o f Adamson, Cobble and N ie ls e n ^ and f h i te w a y , MacLennan
and C o f f i n 1 5 i n c l u d e s no tim e o f r e l a x a t i o n e f f e c t on th e
! c o n c e n t r a t i o n d e p e n d e n c e , w hich i s g iv e n i n s t e a d by th e
j a c t i v i t y c o e f f i c i e n t te rm . T hese d i f f e r e n t p o i n t s o f
! view w i l l be d i s c u s s e d in t h e t h e o r y t o f o l lo w .
i
The wThermodynamic L im i ti n g Law f o r S e l f D i f f u s
i o n . F ic k * s f i r s t law** s t a t e s t h a t in a column o f u n i t
c r o s s s e c t i o n a l a r e a t h e q u a n t i t y , J[, in e q u i v a l e n t s p e r
i cm2 p e r s e c . o f s o l u t e d i f f u s i n g th r o u g h a g iv e n p e rp e n
d i c u l a r p l a n e i s p r o p o r t i o n a l t o th e g r a d i e n t o f t h e con-
J c e n t r a t i o n , V c, where: £ i s in e q u i v a l e n t s p e r cra^
JjL * - D j V c 5* CVi ( l )
, w h e re , Di , i s t h e p r o p o r t i o n a l i t y f a c t o r c a l l e d t h e d i f f -
j u s i o n c o e f f i c i e n t f o r th e io n . has th e d im e n sio n s o f
I and i s e x p r e s s e d in cm2 p e r s e c . ( o r more c o n v en -
i
I i e n t l y , i n cm2 p e r d a y ) . The q u a n t i t y , v-^, i s th e v e l -
I
o c i t y o f th e p a r t i c l e i n cm. p e r s e c . T h is v e l o c i t y may
i
I be e x p r e s s e d a s th e p ro d u c t o f t h e d i f f u s i o n m o b i l i t y i
and t h e c h e m ic a l p o t e n t i a l ,
v i = - c J i V ^ i ( 3 )
o^i i s th e m o b i l i t y o f t h e i ^ 11 io n i n th e s o l u t i o n . T his
m o b i l i t y i s n o t t h e same a s th e c o n d u c ta n c e m o b i l i t y ,
f o r t h e l a t t e r i s s u b j e c t t o a d ra g due t o t h e o p p o sin g I
m o tio n o f t h e io n a tm o sp h e re 1 ®. At th e l i m i t i n g c a s e o f j
i n f i n i t e d i l u t i o n th e two m o b i l i t i e s a r e e q u a l . In th e !
I
O n sa g e r-F u o ss t r e a t m e n t o f d i f f u s i o n th e io n a tm o sp h e re |
e f f e c t on was e v a l u a t e d i n term s o f t h e c o n d u c ta n c e j
m o b i l i t i e s a t i n f i n i t e d i l u t i o n and i t was a l s o i n d i c a t e d
t h a t would b e a f u n c t i o n o f t h e v i s c o s i t y o f th e s o l
u t i o n . The a n a la g o u s d e r i v a t i o n f o r t h e c a s e o f s e l f
d i f f u s i o n h as n o t b e e n t r e a t e d b u t s h o u ld be s i m i l a r .
V a rio u s p u b l i c a t i o n s 1 ®,1 4 ,1 5 ,1 ? have a p p e a re d 1
u s i n g f o r cj^ e i t h e r th e l i m i t i n g c o n d u c ta n c e m o b i l i t y a t
i n f i n i t e d i l u t i o n o r th e c o n d u c ta n c e m o b i l i t i e s a t th e
c o n c e n t r a t i o n o f t h e s o l u t i o n . N e i th e r i s p r o p e r b u t
may b e c o n s i d e r e d a s a p p r o x im a tio n s . At i n f i n i t e d i l u
t i o n th e d i f f u s i o n m o b i l i t y e q u a ls th e c o n d u c ta n c e mob
i l i t y and may be w r i t t e n
^ ( 3 )
1 i * J T T ( 3 )
w here i s th e l i m i t i n g d i f f u s i o n m o b i l i t y , i s th e
(18) R. H. S t o k e s , p r i v a t e com m un icatio n.
9
l i m i t i n g e q u i v a l e n t c o n d u c ta n c e , | z ^ \ i s th e a b s o l u t e
v a lu e o f t h e v a le n c e o f th e io n and F i s th e F a ra d a y in
coulom bs p e r e q u i v a l e n t .
The q u a n t i t y , i s t h e g r a d i e n t o f th e chem
i c a l p o t e n t i a l and, e x p r e s s e d in v o l t s
~ yj.c (4) ;
w i t h R, th e g as c o n s t a n t , 8 .3 1 6 v o l t coulom bs p e r
j
d e g re e p e r m ole, T, th e a b s o l u t e t e m p e r a t u r e , and !
t h e i o n i c a c t i v i t y c o e f f i c i e n t on t h e c o n c e n t r a t i o n
s c a l e , c_. E q u a tio n (1) now becomes
DiVc « c ^ I ^ V 'l n y xc (5) ;
l
o r
» i = " i f 1 0 + ° ( 6 ) :
A g e n e r a l p r e c e p t h a s b e e n t h a t i n a con densed system
m o tio n i s due t o t h e p r o d u c t o f a f o r c e and a m o b i l i t y .
Thus t h i s e q u a t io n i s o b t a i n e d by fo rm al a n a lo g y w ith
t h e t h e o r y f o r i o n i c d i f f u s i o n and th e f i n a l t e s t o f
i t s v a l i d i t y m ust w a i t u n t i l a p r o p e r t r e a tm e n t o f
i n t e r i o n i c a t t r a c t i o n e f f e c t s on th e Brownian m otion o f
io n s p r o v i d e s a fu n d a m e n ta l b a s i s f o r t h e t r e a tm e n t o f
i o n i c s e l f d i f f u s i o n . U ■
At i n f i n i t e d i l u t i o n t h e b r a c k e t t e d th erm o dyn a-
1 0 i
mic term i n e q u a tio n ( 6 ) d i s a p p e a r s , = co? and u s i n g ;
i
e q u a t io n (3) I
D -j (cm ^/day) = 0 .0 2 3 0 3 - i ^ ( 7 ) !
T h is e q u a tio n i s t h e N e rn s t l i m i t i n g e q u a t i o n 3 f o r th e I
i
s p e c i a l c a s e o f s e l f d i f f u s i o n . To o b t a i n an a p p ro x - !
I
|
irnate l i m i t i n g s lo p e i t i s p o s s i b l e to l e t = u>l and i
!
u s e t h e Debye-Hlickel t h e o r e t i c a l v a lu e f o r £ i
- l o g = 0 .5 0 9 z f Vu ( 8 )
w here u i s t h e i o n i c s t r e n g t h . S u b s t i t u t i o n i n t o equa
t i o n ( 6 ) g iv e s t h e l i m i t i n g s lo p e e q u a t io n
T h is i s t h e l i m i t i n g s lo p e e q u a t i o n o f Adamson,
C obble and N i e l s e n *-3 and W hiteway, MacLennan and Cof
fin*-5 . The tim e o f r e l a x a t i o n e f f e c t i s n o t c o n s id e r e d
o p e r a t i v e h e r e f o r th e r e a s o n t h a t i n a homogeneous
e l e c t r o l y t e s o l u t i o n th e i o n i c a tm o s p h e re s o f each io n
s h o u ld be s y m m e tr ic a l. In th e a b s e n c e o f asym metry
t h e r e s h o u ld be no e l e c t r i c a l d r a g o f t h e a tm o sp h ere
and hence no tim e o f r e l a x a t i o n e f f e c t .
The m o b i l i t y te r m , how ever, i s n o t a c o n s t a n t b u t
m ust depend in some m anner upon t h e v i s c o s i t y o f th e
s o l u t i o n and p o s s i b l y on o t h e r v a r i a b l e s . These e f f e c t s
a r e n e g l e c t e d i n th e l i m i t i n g s lo p e e q u a t i o n .
11
The ”C onductance" L im i ti n g Law fo x S e l f D i f f u s i o n .
H u iz e n g a , G r ie g e r and W all h o ld t h a t th e a b se n c e of
b u lk c o n c e n t r a t i o n g r a d i e n t s c a u s e s th e a c t i v i t y c o e f f
i c i e n t term t o d i s a p p e a r . I f t h i s term i s d ro p p ed from
e q u a t io n ( 6 ) , t h e i r e q u a t i o n i s o b t a i n e d . They c o n s i d e r
th e d i f f u s i o n m o b i l i t y to be th e same a s th e c o n d u c ta n c e
m o b i l i t y and c a l c u l a t e t h e i r d i f f u s i o n c o e f f i c i e n t s from
t h e e q u a t io n
D = 0 .0 2 3 * 1 (10)
\z ±\
w here i s th e o b se rv e d e q u i v a l e n t c o n d u c ta n c e o f th e
io n and i t s v a l e n c e . In t h i s e q u a t io n th e d i f f u s i o n
c o e f f i c i e n t v a r i e s o n ly a s t h e c o n d u c ta n c e m o b i l i t y .
T h is c a n n o t be t h e c a s e s i n c e th e p r o c e s s o f e l e c t r i c a l
con ductance, d i f f e r s m ark e d ly from s e l f d i f f u s i o n . Under
i
t h e i n f l u e n c e o f th e e l e c t r i c c u r r e n t th e a tm o sp h e re s a re j
i
made u n sy m m e tric a l in f lo w in g p a s t each o t h e r and a tim e !
i
o f r e l a x a t i o n e f f e c t a p p e a r s . The e l e c t r i c a l f o r c e w i l l '
te n d to move th e a tm o sp h e re and t h e l i q u i d c o n t a i n i n g i t
t
in th e d i r e c t i o n o f th e f o r c e . T h is w i l l b r i n g an elect*-!
r i c a l f o r c e to b e a r on th e c e n t r a l io n i n th e o p p o s i t e
d i r e c t i o n to i t s m o tio n . N e i t h e r o f th e s e f o r c e s p l a y s
a p a r t in s e l f d i f f u s i o n . B ecause o f t h e s e f o r c e s th e
c o n d u c ta n c e m o b i l i t y i s l e s s th a n t h e d i f f u s i o n m o b i l i t y
13 I
and th e d i f f u s i o n c o e f f i c i e n t s c a l c u l a t e d by e q u a tio n ( 1 0 )
w i l l be to o s m a ll .
The ”R e l a x a t i o n ” L im i ti n g Law f o r S e l f D i f f u s i o n .
O nsager 1 9 h a s t r e a t e d t h e c a s e o f d i f f u s i o n i n a s o l u t i o n
c o n t a i n i n g more t h a n two k in d s o f i o n s . In a d d i t i o n to i
t
♦
a g e n e r a l t r e a t m e n t he d e r i v e d a s im p le l i m i t i n g law f o r I
i
th e d i f f u s i o n o f an io n o f v e ry low c o n c e n t r a t i o n i n a !
s o l u t i o n o f o t h e r w is e c o n s t a n t c o m p o s itio n w ith o n ly
t h r e e k in d s o f io n s p r e s e n t . G o s tin g and E arn e d 1 7 have
a p p l i e d t h i s e q u a t io n to th e s p e c i a l c a s e o f s e l f d i f f
u s i o n by t r e a t i n g th e t r a c e r io n as a n o th e r s p e c i e s o f
io n p r e s e n t in t r a c e q u a n t i t y , b u t w ith t h e same p r o p e r
t i e s a s i t s i s o t o p e . j
I
I f d i f f u s i o n t a k e s p l a c e in a s o l u t i o n c o n t a i n i n g ;
more th a n -tw o i o n s t h e s e io n s need n o t have th e same v e l
o c i t y . The d i f f u s i n g io n p r e s e n t in o n ly a t r a c e amount
has a v e l o c i t y r e l a t i v e to th e o t h e r io n s and hence has
an a s y m m e tric a l a tm o s p h e re . T h is g i v e s r i s e to a tim e o f
r e l a x a t i o n f o r c e on th e io n w hich c h an g es i t s v e l o c i t y .
(19) L. O n sa g e r, Ann. H.Y, Acad. S c i . , 4 6 , 241 (1 9 4 5 ).
13
F o r t h i s c a s e O nsager h as found
° j = * 10-7 - ^ lz4 y -£ * io~9T / ^ A i ~
3 I2 # 3D v 1000DRT
( i d
w here Aj i s th e l i m i t i n g i o n i c c o n d u c ta n c e o f th e
io n w hich i s t h e io n p r e s e n t i n t r a c e am ount, R i s th e
g a s c o n s t a n t , T _ t h e a b s o l u t e t e m p e r a t u r e , z,j i s th e v a l
ence o f th e i o n , F i s t h e F a r a d a y , v th e v e l o c i t y o f
l i g h t , € t h e e l e c t r o n i c c h a r g e , D t h e d i e l e c t r i c c o n s ta n t,
c,^ t h e c o n c e n t r a t i o n o f th e i tJl io n in m oles p e r l i t e r
a n d z. i s t h e v a l e n c e . The i ^ 11 io n s a r e a l l th o s e io n s
p r e s e n t e x c e p ti n g th e t r a c e ( j t h ) i o n . The q u a n t i t y ,
d C ^ j) , i s r e p r e s e n t e d by th e e q u a t io n
d(OJ .) — — 1 g.. ^ ,Pi }„ zjl ( 1 2 )
3 £ c i 2i 1 Al Z Ai U '
i iz-jl ■ * " »Z£|
F o r t h e c a s e o f t h r e e io n s p r e s e n t w i t h j a 1 and i « 3
f o r t h e io n f o r w hich i s t h e t r a c e r and i = 3 f o r th e
io n o f o p p o s i t e c h a r g e , e q u a t io n 0£ ) re d u c e s to
d(£V ~ (\z2\ + |2l)A3^ ) (13)
G o s tin g and H arned a p p l i e d t h i s e q u a t io n to s e l f d i f f
u s i o n by c o n s i d e r i n g 1 t h e r a d i o a c t i v e t r a c e r i o n ,
14
2 th e n o rm al io n c o r r e s p o n d i n g to 1 , and _3 th e io n o f
o p p o s i t e c h a r g e . F or s e l f d i f f u s i o n i n 1 -1 e l e c t r o l y t e s
e q u a t io n ( 1 1 ) r e d u c e s to
a * )
G o s tin g and H arned u s e t h i s e q u a t io n f o r d (h ^ ) to s o lv e
e q u a t io n ( 1 1 ) f o r t h e l i m i t i n g s l o p e o f t h e i r t h e o r y .
The weak p o i n t i n G o s tin g and H a rn e d 1s t r e a t m e n t
a p p e a r s t o be t h a t i t r e q u i r e s t h a t a t r a c e r io n have
an a s y m m e tric a l a tm o sp h e re in a homogeneous e l e c t r o l y t e
s o l u t i o n c o n t a i n i n g i t s i s o t o p e . Yet su c h s o l u t i o n s
have s y m m e tric a l io n a tm o s p h e r e s . S in c e t h e two io n s
a r e c h e m ic a lly i d e n t i c a l t h i s does n o t seem p o s s i b l e .
Each o f th e t h e o r i e s c o n s i d e r e d above i s to some \
i
i
e x t e n t f u n d a m e n ta lly u n s a t i s f a c t o r y . T h is w ork, h o w e v e r,!
d i d n o t c o n c e rn i t s e l f w ith th e b a s i c s o l u t i o n th e o r y
I
and so no d e v elo p m e n t o f a more fu n d a m e n ta l t h e o r y was ;
a tt e m p t e d .
THEORY OF THE DIAPHRAGM METHOD i
l
The e q u a t io n f o r d i f f u s i o n c o e f f i c i e n t s from d i a - !
phragm c e l l m easurem en ts i s o b t a i n e d by t h e d i r e c t a p p l i - j
c a t i o n o f F i c k ’ s f i r s t la w . T h is i s th e d e f i n i n g equa
t i o n f o r t h e d i f f u s i o n c o e f f i c i e n t and t h u s i s in d e p e n d
15 |
e n t o f t h e o r i e s o f th e mechanism o f th e d i f f u s i o n p ro
c e s s . The d iap hragm i s assum ed t o c o n s i s t o f a number o f
p o r e s o f l e n g t h , and t o t a l c r o s s s e c t i o n a l a r e a , A. j
Two s o l u t i o n s o f c o n c e n t r a t i o n s , c^ a n ^ £ 2 * a r e b r o u g h t |
\
i n t o c o n t a c t th r o u g h t h i s d iap h ra g m . I f ci^and Cg change |
s lo w ly th e c o n c e n t r a t i o n g r a d i e n t , (c]_ - c 2 ) , th r o u g h th e ;
d iap hragm w i l l be e s s e n t i a l l y l i n e a r . From F i c k 's law
t h e am ount, d £ , o f a s o l u t e d i f f u s i n g i n tim e , d t , i s
In o r d e r t o e x te n d t h e m easurem ents o v e r a p r a c t i c a l
tim e i n t e r v a l t h e i n t e g r a l form o f t h i s e q u a t io n m ust be j
»
u s e d . I f a mass b a la n c e i s w r i t t e n f o r th e d i f f u s i o n j
c e l l i
w here and V2 a r e th e volum es o f th e co m p a rtm e n ts.
T h is e q u a tio n becomes
I f D i s assum ed in d e p e n d e n t o f c o n c e n t r a t i o n t h i s equa
t i o n can be i n t e g r a t e d from t h e i n i t i a l c o n c e n t r a t i o n
d i f f e r e n c e , (c-j_ - c o n c e n t r a t i o n d i f f
(15)
V ld o l * JT^0! - ° 3 ^ d t = v 3 d c 3 * f f i ° l ~ °3 )d t 06)
(IV)
e r e n c e , (c-^ - c 2 ) f
(18) j
_ J
16
S in c e A g e n e r a l l y c a n n o t be d e te r m in e d , th e c o n s
t a n t , ^ (L + i. \ , i s d e s i g n a t e d K and d e te r m in e d by
2 .3 0 3 ^ \Y]_ ““
c a l i b r a t i o n w ith a s a l t whose d i f f u s i o n c o e f f i c i e n t i s
known. T h is g i v e s
The c o n c e n t r a t i o n u n i t s c a n c e l i n t h i s e q u a t i o n and so
t h e c o u n ts p e r m in u te p e r u n i t volume can be u s e d d i r e c t
l y i n p l a c e o f t h e t r a c e r c o n c e n t r a t i o n i n more conven
t i o n a l u n i t s . S in c e t h e t r a c e r was ad d ed t o o n ly one
com partm ent t h e i n i t i a l v a lu e o f c_ 2 i s z e r o . By making
a t r a c e r mass b a l a n c e t h e i n i t i a l v a lu e o f c j can be ob
t a i n e d in te rm s o f th e f i n a l v a lu e s o f _ci and eg and th e
v o lu m e s, V ]_ and Vg.
In th e d e r i v a t i o n t h e a s s u m p tio n i s made t h a t th e d i f f
u s i o n c o e f f i c i e n t i s in d e p e n d e n t o f c o n c e n t r a t i o n . I f
t h e d i f f u s i o n c o e f f i c i e n t o f th e s p e c i e s i n q u e s t i o n
v a r i e s w ith i t s c o n c e n t r a t i o n , th e u s e o f e q u a t io n ( 2 0 )
y i e l d s a ty p e o f a v e r a g e o r i n t e g r a l d i f f u s i o n c o e f f i c
i e n t . T h is i s an i m p o r t a n t p o i n t i n mean d i f f u s i o n
s t u d i e s b u t i s n o t in v o lv e d i n s e l f d i f f u s i o n e x p e r i
m en ts s i n c e th e o v e r a l l c o n c e n t r a t i o n o f t h e l a b e l l e d
(19)
( 2 0 )
'n
17 !
s p e c i e s does n o t c h a n g e . |
Stokes® h a s r e p o r t e d d i f f u s i o n d a t a f o r sodium j
c h l o r i d e a t 25°C. d i f f u s i n g i n t o w a t e r . Under t h e s e j
c o n d i t i o n s th e Na+ and Cl" io n s d i f f u s e t o g e t h e r t o p r e - \
s e r v e e l e c t r i d a l n e u t r a l i t y . These d i f f u s i o n c o e f f i c - j
i e n t s were o b t a i n e d by th e diap h rag m c e l l method and !
-. : L - I
would seem t o r e p r e s e n t some .mean v a lu e o f th e Na*~ and
• r i
C l” io n s e l f d i f f u s i o n c o e f f i c i e n t s r e p o r t e d i n t h i s I
w ork. The N e r n s t e x p r e s s i o n s a t i n f i n i t e d i l u t i o n a r e
% a = ^ a kT DC1“ X C lkT DNa01 T % 1kT
S o lv in g f o r Djja Qi i n term s o f E%a and g i v e s
n ° - DNa Dei
% a C l Dtfa + 'Drfx (2S)
T h is e q u a t io n i s e x a c t a t i n f i n i t e d i l u t i o n and s h o u ld
b e a c l o s e a p p ro x im a tio n a t f i n i t e c o n c e n t r a t i o n s . How
e v e r , b e f o r e t e s t i n g t h i s e q u a tio n i t i s a t l e a s t n e c
e s s a r y to b r i n g th e two s e t s o f m easurem en ts t o t h e same
b a s i s i n one im p o r ta n t r e s p e c t . In th e c a s e o f d i f f u s
io n i n t o w a t e r , t h e r e i s a n e t flo w o f s a l t and o f
w a te r ; t h e volume flo w o f th e two m ust be e q u a l in mag
n i t u d e and o p p o s i t e in s ig n s i n c e t h e system a s a w hole
h as a c o n s t a n t volum e. The m easu red d i f f u s i o n c o e f f i c
i e n t i s t h e r e f o r e an a v e ra g e o f t h e i n t r i n s i c c o e f f i c
i e n t s f o r t h e two com ponents, g iv e n by e q u a t io n ( 2 3 ) .
18
The d i f f u s i o n c o e f f i c i e n t o f w a te r i s 2 .3 8 5ra2 p e r day 1 0
and r e l a t i v e l y in d e p e n d e n t o f s a l t c o n c e n t r a t i o n . S in c e
t h i s d i f f u s i o n c o e f f i c i e n t i s g r e a t e r t h a n t h a t f o r th e
io n s h e re c o n s i d e r e d , D0^,B w i l l be l a r g e r t h a n Ds a ^ ^ .
^ o b s ~ ^salt^ H g O “ ^HgO^ions (23)
w here and Z io n s a r e " fciie volume f r a c t i o n s o f w a te r
and i o n s . T h is c o r r e c t i o n to D ^ g s h o u ld i n c r e a s e w ith
i n c r e a s i n g c o n c e n t r a t i o n , b u t i t s v a lu e i s u n c e r t a i n b e
c a u s e t h e number o f w a te r s o f h y d r a t i o n t o be u s e d i n
co m p u tin g F£ons i s n o t known. Only a f t e r c o r r e c t i o n f o r
t h e above volume e f f e c t i n mean d i f f u s i o n e x p e rim e n ts
c an a c o m p a riso n be made w ith s e l f d i f f u s i o n c o e f f i c i e n t s
s i n c e t h e l a t t e r a r e , o f c o u r s e , u n a f f e c t e d by any such
volume f o r c e s .
The diaphragm c e l l was c h o sen f o r t h i s work b e
c a u s e o f i t s s i m p l i c i t y and due to th e f a c t t h a t o t h e r
m ethods w ere deemed to o d i f f i c u l t o r in a d e q u a te f o r th e
s p e c i a l c a s e o f s e l f d i f f u s i o n . . In th e d iaphragm c e l l
m ethod, d i f f u s i o n t a k e s p l a c e o n ly w i t h i n t h e c o n f i n e s
o f th e diaph ragm p o r e s w here m e c h a n ic a l and th e r m a l v a r
i a t i o n s a r e u n a b le t o a f f e c t t h e d i f f u s i o n , r u n s may be
c o m p le te d i n 3 o r 4 d ay s and th e a p p a r a t u s i s s im p le and
c o n v e n ie n t t o u s e . O p t i c a l m ethods c o u ld n o t b e u s e d
19
s i n c e t h e r e a r e no c o n c e n t r a t i o n g r a d i e n t s . F red d i f f
u s i o n was a tte m p te d w ith an a p p a r a t u s i n which a bound
a r y c o u ld be form ed betw een th e norm al and th e ta g g e d
s o l u t i o n s to g iv e a column 10 cm, i n l e n g t h and 1 cm. i n
d i a m e t e r . T h is m ethod c o u ld n o t be s u c c e s s f u l l y u s e d a t
ou r l a b o r a t o r y f o r no m o unting c o u ld be c o n s t r u c t e d t h a t
was u n a f f e c t e d by t h e b u i l d i n g ' s v i b r a t i o n . F re e d i f f
u s i o n , how ever, h as s i n c e b e en d e v e lo p e d 1 ^ to o f f e r a
c o n v e n ie n t and a c c u r a t e m ethod.
c
FIG. I. D IA P H R A G M DIFFUSIO N C E L L
EXPERIMENTAL METHODS
Use o f th e Diaphragm C e l l . F ig u r e 1 shows th e
d i f f u s i o n c e l l u se d i n t h i s w ork. I t c o n s i s t e d o f a
c y l i n d r i c a l g l a s s v e s s e l d i v i d e d i n t o two com partm ents by
a s i n t e r e d g l a s s d iap h ra g m . Each com partm ent had a v o l
ume o f a b o u t 25 m l. and had two s to p c o c k s t o a llo w th e
com partm ent t o be e m p tied o r f i l l e d w ith o u t d i s t u r b i n g
t h e l i q u i d i n t h e d iap h rag m . One com partm ent was f i l l e d
w i t h a s o l u t i o n c o n t a i n i n g t r a c e r io n and th e o t h e r w ith
a s o l u t i o n o f t h e same c o n c e n t r a t i o n b u t u n ta g g e d . The
c e l l was t h e n m ounted in a w a t e r t h e r m o s t a t a t 25 - 0 .0 £ c
The ty p e o f m ou n tin g was v a r i e d d e p e n d in g on th e ty p e o f
s t i r r i n g u s e d . A f t e r two t o f o u r h o u rs o f p r e l i m i n a r y
d i f f u s i o n to e s t a b l i s h th e d i f f u s i o n g r a d i e n t i n t h e d i a
phragm , t h e s o l u t i o n i n t o w hich t h e t r a c e r i s d i f f u s i n g
was r e p l a c e d w ith n o n - ta g g e d s o l u t i o n a f t e r c a r e f u l r i n s
i n g . T h is s o l u t i o n was t h e r m o s t a t t e d t o 25°c . b e f o r e
u s e . The c e l l was a g a i n p l a c e d i n p o s i t i o n and d i f f u s i o n
a llo w e d t o p r o c e e d u n t i l a p p ro x im a te ly 25$ o f t h e t r a c e r
had p a s s e d th r o u g h t h e d iap h rag m . T h is to o k from two to
f i v e d a y s . The s o l u t i o n s w ere s t o r e d in s t o p p e r e d t e s t
t u b e s f o r c o u n ti n g .
32
P r e p a r a t i o n o f S o l u t i o n s , A ll s o l u t i o n s w ere p r e
p a re d u s i n g d o u bly d i s t i l l e d w a te r w ith a s p e c i f i c con
d u c ta n c e o f l e s s t h a n 3 x 10” ® mho cm. A n a l y t i c a l q u a l
i t y sodium and p o ta s s iu m c h l o r i d e s and sodium io d id e were
i
u s e d w ith o u t f u r t h e r p u r i f i c a t i o n . The sodium s u l f a t e of
C. P . q u a l i t y was r e c r y s t a l l i z e d b e f o r e u s e . The s o l u
t i o n s were p r e p a r e d from s a l t s w hich had b een d r i e d f o r
24 h o u rs a t 1 1 0 °0 . and c o o le d i n a d e s i c c a t o r . A ll s o l u
t i o n s o f c o n c e n t r a t i o n l e s s th a n 0 . 0 1 N. were p r e p a r e d by
d i l u t i o n o f m easu red volum es from more c o n c e n t r a t e d s o l u
t i o n s . D is s o lv e d g a s e s w ere removed from th e s o l u t i o n s
p r i o r t o u s e by b o i l i n g f o r one m in u te i n t h e vacuum o f
a w a te r pump.
P r e p a r a t i o n o f t h e T r a c e r s . Two r a d i o a c t i v e i s o -
j to p e s o f sodium w ere u se d d u r i n g t h e c o u rs e o f t h i s r e
s e a r c h , Na2 2 (3 y e a r h a l f l i f e ) and Ha2 4 ( 1 4 .8 h o u r h a l f
' 1
l i f e ) . The Na2 2 was o b t a i n e d from th e M .I .T . R a d ia ti o n
L a b o r a to r y and th e C a rn e g ie I n s t i t u t e o f T e r r e s t r i a l
I
j M agnetism . I t was p r e p a r e d by d e u te r o n bombardment of
m agnesium . The t r a c e r was r e c e i v e d a s sodium c h l o r i d e
s o l i d o r s o l u t i o n o f v e ry h ig h s p e c i f i c a c t i v i t y . These
j w ere d i l u t e d w ith a s m a ll volume o f w a te r t o make th e
i
s t o c k s o l u t i o n . The Na2 4 from n e u tr o n bom barded sodium i
i
j c a r b o n a t e was o b t a i n e d from th e Oak Ridge N a t io n a l L abor
33 t
a t o r y . The t r a c e r a r r i v e d a s sodium c a r b o n a te o f v e ry
i
h ig h s p e c i f i c a c t i v i t y . The t r a c e r was c o n v e r te d t o u s e !
a s sodium c h l o r i d e by a d d in g e x c e s s h y d r o c h l o r i c a c i d ,
e v a p o r a t i n g t o d r y n e s s to e l i m i n a t e HG1 and r e d i s s o l v i n g
i
i n a s m a ll amount o f w a te r t o form a s t o c k s o l u t i o n . Sod-1
ium s u l f a t e was p r e p a r e d by t i t r a t i o n w ith s u l f u r i c a c i d i
1
u s i n g m e th y l r e d a s i n d i c a t o r . T h is s o l u t i o n was e v a p o r - ,
a t e d to a s m a ll volume and u se d a s t h e s t o c k s o l u t i o n . I
Upon u s e t h e Na2 4 was fo u n d t o c o n t a i n r a d i o a c t i v e im pur
i t i e s w hich gave i t an a p p a r e n t h a l f l i f e o f 1 5 .6 h o u r s ,
how ever, on co m p a rin g d u p l i c a t e r u n s t h e r e was no a p p a r e n t
j
d i f f e r e n c e b etw een t h e r e s u l t s o b t a i n e d u s i n g t h i s t r a c e r
and t h o s e made u s i n g Ha2 2 .
The S3 5 and C l3® t r a c e r s w ere made from a n e u tr o n
bom barded sam ple o f p o ta s s iu m c h l o r i d e o b t a i n e d from th e
Atom ic E nergy Com mission. S in c e t h i s sam ple had aged
s e v e r a l m onths a l l a c t i v i t i e s o f s h o r t h a l f l i v e s had de
c a y e d l e a v i n g 8 7 .1 day S3® and 10® y e a r C l3 6 . An a b s o r p
t i o n c u rv e on a t h i n sam ple o f t h i s m a t e r i a l a f f i r m e d t h a t
o n ly t h e s e two a c t i v i t i e s w ere p r e s e n t , w ith th e c h l o r i n e
a c t i v i t y l e s s th a n one p e r c e n t o f t h e t o t a l . The C l3® was
p r e s e n t in th e sam ple e n t i r e l y a s c h l o r i d e b u t th e was
d i s t r i b u t e d among s e v e r a l v a le n c e s t a t e s i n c l u d i n g th e !
s u l f i d e and th e s u l f a t e . j
A s o l u t i o n o f t h e n e u tr o n bom barded p o ta s s iu m
c h l o r i d e was p a s s e d th r o u g h an io n exchange column con
t a i n i n g th e H* form o f Dowex 50 , y i e l d i n g an HC1 s o l u
t i o n . T h is HC1 s o l u t i o n c o n t a i n i n g t h e r a d i o s u l f u r and
t h e r a d i o c h l o r i n e was t h e n t r e a t e d w ith b rom ine w a te r to
o x i d i z e t h e s u l f u r t o s u l f a t e . A f t e r e x p e l l i n g th e b r o
m ine by b o i l i n g , th e c h l o r i d e was p r e c i p i t a t e d by th e add-;
i t i o n o f t h e s t o i c h i o m e t r i c amount o f s i l v e r n i t r a t e .
A lth o u g h th e s i l v e r c h l o r i d e p r e c i p i t a t e c o n ta in e d some
C13 S , r e c o v e r y was n o t f e a s i b l e f o r th e s p e c i f i c a c t i v
i t y was n o t s u f f i c i e n t l y g r e a t .
The f i l t r a t e from t h e s i l v e r c h l o r i d e p r e c i p i t a t e j
c o n ta in e d r a d i o s u l f a t e i n n i t r i c a c i d . T h is was a g a in j
' p a s s e d th r o u g h t h e io n e x c h a n g e r to e n s u r e th e a b se n c e
j o f a l l . p o s i t i v e io n s o t h e r th a n K+ , and was th e n b o i l e d
I I
! down t o ' n e a r d r y n e s s t o e x p e l most o f th e a c i d and t h e n .
one d ro p o f 1 0 “ ® H- sodium h y d ro x id e was added and th e
e v a p o r a t i o n c o n tin u e d t o d r y n e s s . The t r a c e o f s o l i d s
i
j r e m a in in g was ta k e n up i n w a te r t o g iv e th e s t o c k s o l u
t i o n o f ta g g e d sodium s u l f a t e .
A seco n d sh ip m en t o f s u l f u r t r a c e r was o b t a i n e d
from th e Atomic E nergy Commission a s cadmium s u l f i d e
w ith b a riu m io n in s o l u t i o n t o p r e c i p i t a t e any s u l f a t e j
form ed. T h is s o l u t i o n was p a s s e d th r o u g h t h e H* io n
25 |
I
e x c h a n g e r , o x i d i z e d by brom ine w a te r a s b e f o r e and evap
o r a t e d a f t e r a d d in g a d rop o f 10” ® N sodium h y d r o x id e .
A s to c k s o l u t i o n wa8 made from t h e sodium s u l f a t e rem ain-j
i n g . ,
I n o r d e r t o t e s t t h e c o m p le te n e s s o f th e c o n v e r-
s i o n o f r a d i o a c t i v e s u l f u r t o s u l f a t e a sodium s u l f a t e ^
s o l u t i o n was ta g g e d w i t h t h i s t r a c e r . T h is ta g g e d s o l u - -
t i o n was a llo w e d to d i f f u s e i n t o w a te r i n a diaphragm
c e l l u n t i l 25$ o f th e s a l t had p a s s e d th r o u g h . Now
w eig h ed sam p les o f s a l t from b o th com partm ents were
c o u n te d . The s p e c i f i c a c t i v i t i e s w ere t h e same. S in c e
I
t h e d i f f u s i o n m o b i l i t i e s o f th e v a r i o u s s u l f u r a n io n s j
a r e d i f f e r e n t , t h e d i f f u s a t e would c o n t a i n a d i f f e r e n t i
i
r a t i o o f t h e s e io n s and hen ce have a d i f f e r e n t s p e c i f i c
a c t i v i t y i n th e e v e n t r a d i o a c t i v e s u l f u r was p r e s e n t in
o t h e r v a le n c e fo rm s. T h is t e s t showed t h a t no im p u ri
t i e s w ere p r e s e n t i n am ounts s u f f i c i e n t t o c a u s e th e
t r a c e r t o be s i g n i f i c a n t l y d i f f e r e n t from s u l f a t e io n i n
i t s b e h a v i o r .
C h lo r in e t r a c e r s u i t a b l e f o r work i n v e ry d i l u t e
s o l u t i o n s was o b t a i n e d from th e Oak R idge N a t io n a l Lab
o r a t o r y a s sodium c h l o r i d e s o l u t i o n o f m o d era te a c t i v
i t y . The s o l u t i o n a s r e c e i v e d was u s e d a s s to c k t r a c e r
s o l u t i o n . i
In a l l c a s e s th e s t o c k t r a c e r s o l u t i o n s w ere o f
su c h s p e c i f i c a c t i v i t y t h a t o n ly a few lambda, n eed be i
added t o t a g t h e s o l u t i o n . T h is added n e i t h e r enough j
s a l t n o r w a te r to m e a su ra b ly change th e c o n c e n t r a t i o n o f I
any s o l u t i o n so ta g g e d .
C o u n tin g t h e R a d i o a c t i v e T r a c e r s . At f i r s t one !
m l. a l i q u o t s o f N a ^ and N a ^ .and C l 3 3 ta g g e d s o l u t i o n s (
i
were e v a p o r a te d to d r y n e s s in sm a ll w atch g l a s s e s u n d e r j
I
an i n f r a r e d h e a t lamp and w ere c o u n te d u n d e r a m ica
window c o u n t e r . L a t e r s m a ll g l a s s j a c k e t e d G e ig er c o u n t
e r tu b e s were u se d i n w hich s o l u t i o n s d i r e c t from th e com
p a r t m e n t s w ere p i p e t t e d i n t o th e c o u n tin g tu b e and t h r e e 1
!
f i v e - m i n u t e c o u n ts w ere made. Thus c o u n ti n g c o u ld be
i
c o m p le te d i n t h i r t y m in u te s w hereas i f e v a p o r a t i o n were
u s e d , more th a n two h o u rs was n eeded t o e v a p o r a te t h r e e
sam p les from e a c h com partm ent and c o u n t t h r e e f i v e - m in
u t e c o u n ts o f e a c h . The o b s e rv e d c o u n ts were c o r r e c t e d
f o r b a ck g ro u n d and c o in c id e n c e and th e W a ^ sa m p le s were
c o r r e c t e d f o r d e c a y .
The 0 .1 6 8 mev. b e t a r a y s from S3 3 a r e n o t o f h ig h
enough e n erg y t o p e r m it c o u n tin g t h e n i n a s o l u t i o n
c o u n t e r . E v a p o ra te d o r p r e c i p i t a t e d sam p les w ere c o u n t
ed w i t h a m ica window c o u n ti n g tu b e o r w ith an a tm o s-
t
p h e r i c p r e s s u r e flo w g a s c o u n te r tu b e i n t o w hich t h e j
27 I
i
sam ple c o u ld be p l a c e d . Due t o t h e weak r a d i a t i o n th e
sam ple m ust b e o f u n ifo rm c r o s s s e c t i o n and a s e l f a b -
w o r p tio n c o r r e c t i o n made. I t i s im p o r ta n t t h a t th e s o l i d
m a t e r i a l be o f u n if o r m , v e ry s m a ll p a r t i c l e s i z e and n o t
d e l i q u e s c e n t . Barium s u l f a t e was t h e form cho sen f o r !
i
i
t h i s p r o c e d u r e .
P o r t i o n s o f th e sodium s u l f a t e s o l u t i o n s from t h e
o om p artm en ts o f t h e d i f f u s i o n c e l l were made a p p ro x .
0 . 1 N i n n i t r i c a c i d and p r e c i p i t a t e d by a d d in g a s l i g h t
i
e x c e s s o f b a riu m n i t r a t e s o l u t i o n . The b ariu m s u l f a t e
p r e c i p i t a t e was c e n t r i f u g e d , th e s u p e r n a t a n t l i q u i d d e -
l
! c a n t e d and t h e n th e p r e c i p i t a t e was washed tw ic e w ith
v e r y d i l u t e n i t r i c a c i d fo llo w e d by two w a sh in g s w i t h ab
s o l u t e e t h y l a l c o h o l . The r e s u l t i n g b a riu m s u l f a t e was
i s l u r r i e d i n e t h y l a l c o h o l and p l a c e d i n g l a s s c y l i n d e r s
i
■ clam ped upon w eighed a lu m in u n d i s k s . The p r e c i p i t a t e
j was a llo w e d to s e t t l e o n to th e d i s k and th e e t h y l a l c o h o l
i
e v a p o r a t e d i n an oven a t 75 C. The d i s k w ith t h e sam ple
j was th e n w e ig h ed , and f i n a l l y c o u n te d . I t was found
I n e c e s s a r y t o d ry t h e sam p les u n d e r a h e a t lamp j u s t p r i o r
I t o c o u n ti n g t o g e t r e p r o d u c i b l e r e s u l t s .
i The s e l f a b s o r p t i o n c o r r e c t i o n i s made by a f a c t o r
i w h ich c o r r e c t s eac h c o u n te d v a l u e t o t h e e x t r a p o l a t e d
v a lu e f o r z e r o sam ple t h i c k n e s s . In o r d e r to d e te r m in e ■
28
t h e s e f a c t o r s t r a c e r sodium s u l f a t e was added t o a so d
ium s o l u t i o n t o g iv e a s o l u t i o n h a v in g a g iv e n s p e c i f i c
a c t i v i t y . P o r t i o n s o f t h i s s o l u t i o n w ere p r e c i p i t a t e d
a s b a riu m s u l f a t e , s l u r r i e d and a s e r i e s o f d i s k s p r e
p a r e d e a c h w ith a d i f f e r e n t t h i c k n e s s b u t. o f th e same
a r e a . These d i s k s w ere c o u n te d and th e s e l f a b s o r p t i o n
f a c t o r s o b t a i n e d a c c o r d i n g to t h e g e n e r a l m ethod o f
Yankwich, N o r r i s and Huston^® a r e t a b u l a t e d i n T ab le I
and p l o t t e d i n F ig . 3.
(SO) P . E. Y ankw ich, T. H. N o r r i s and J . H u sto n , A n al.
C h em ., 1 9 , 439 ( 1 9 4 7 ).
TABLE I
ABSORPTION CURVE OF SULFUR3 5 IN BARIUM SULFATE
W eight o f
Sample
C ounts p e r
M inute
S e l f A b s o rp tio n
F a c to r
1 .7 5 171 0 .9 9 4
2 .8 5 282 0 .9 8 7
3 .4 342, 0 .9 8 3
8 .7 823 0 .9 3 7
9 .2 870 0 .9 3 2
1 2 .5 1127 0 .8 9 8
1 4 .8 1310 0 .8 7 3
2 0 .4 1625 0 .8 1 0
2 1 .3 1720 0 .8 0 4
2 6 .4 1955 0 .7 5 0
3 2 .6 2214 0 . 6 8 6
3 9 .9 2393 0 .6 1 1
4 2 .5 2450 0 .5 8 4
5 5 .4 2639 0 .4 7 9
6 2 . 3a 2651 0 .4 3 7
a A ll a r e a s 5 .7 2 9 cm** with, t h i s e x c e p ti o n w hich i s
5 .5 2 9 cm2.
SELF ABSORPTION FACTOR
.0
0.9
0.8
0 .7
0.6
0 .5
0 .4
0 .3
0.2
0 , 1
4 0 5 0 6 0 0 20 3 0 1 0
MG/CM
FIG. 2 . S E L F A B S O R P T I O N F A C T O R S OF IN B a S O *
1
THE EFFECT OF STIRRING
l
The diaphragm c e l l h as b een c r i t i c a l l y re v ie w e d by j
Pi !
Gordon^ . T h is re v ie w p o i n t s o u t th e n e e d f o r f u r t h e r I
i
i n v e s t i g a t i o n o f t h i s m ethod a s to th e e f f e c t o f s t i r r i n g , ]
m ethod o f c a l i b r a t i o n o f th e c e l l s and t h e s p e c i f i c e f f - j
e c t o f t h e l a r g e s u r f a c e a r e a o f t h e diaphragm on d i f f u s - ,
io n . These p ro b lem s a r e o f g r e a t im p o rta n c e to t h e a p p l i
c a t i o n o f t h i s m ethod to s e l f d i f f u s i o n w ith r a d i o a c t i v e j
t r a c e r s f o r t h e r e a r e no d e n s i t y d i f f e r e n c e s betw een t h e j
t
two s o l u t i o n s and more d i l u t e s o l u t i o n s may be u s e d w ith
t r a c e r a n a l y s i s t e c h n i q u e . These c o n d i t i o n s a re su c h a s
t o e x a g g e r a t e any s t i r r i n g o r s u r f a c e e f f e c t s and hence
th e d iaphragm m ethod had t o be c r i t i c a l l y exam ined f o r
e v id e n c e o f su c h e f f e c t s . Much o f th e m a t e r i a l t h a t f o l -
j low s c o n s i s t s o f th e p r o c e d u r e s ta k e n to h a n d le th e s e
p ro b lem s t h a t d e v e lo p e d due to u s e o f th e d iaphragm c e l l
m ethod.
1 I m p l i c i t i n th e e q u a t io n f o r c a l c u l a t i o n o f th e
i
d i f f u s i o n c o e f f i c i e n t from d iap h rag m c e l l r e s u l t s i s th e
J a s s u m p tio n t h a t b o th s o l u t i o n s a r e o f u n ifo rm c o n c e n t r a -
j t i o n r i g h t up t o t h e d ia p h ra g m . Any s t a g n a n t l a y e r s w hich
| m ig h t e x i s t a d j a c e n t t o t h e d iap hragm c o u ld n o t be e x p e c t- ,
ed to rem a in t h e same f o r a l l s u b s ta n c e s i n a l l c o n c e n tr a - ;
. 32
t i o n s and so e r r o n e o u s v a lu e s would r e s u l t . U niform con
c e n t r a t i o n s i n th e com p artm en ts a r e o b t a i n e d in p r a c t i c e
th ro u g h s t i r r i n g by v a r i o u s m eans. I t w ould be e x p e c te d
t h a t th e amount o f s t i r r i n g n eed ed s h o u ld be t o some de
g re e d e p e n d e n t on th e p a r t i c u l a r c o n d i t i o n s o f th e s o l u
t i o n s , su c h a s v i s c o s i t y . T h e r e f o r e , s t i r r i n g w hich i s
s u f f i c i e n t f o r th e d i l u t e s o l u t i o n s u s e d i n c a l i b r a t i o n
may n o t be a d e q u a te f o r o t h e r , more c o n c e n t r a t e d s o l u
t i o n s .
N o n -m ech an ical S t i r r i n g . I t h a s b e e n c o n s i d e r e d
t h a t i f t h e c e l l i s m ounted v e r t i c a l l y w i t h t h e more d en se
s o l u t i o n i n th e u p p e r com partm ent s t i r r i n g i s a cco m p lish ed
s u f f i c i e n t t o p e r m i t no s t a g n a n t l a y e r s a s t h e more d en se
l a y e r s i n eac h com partm ent w ould be u p p e rm o st and hen ce i n
SX I
j an u n s t a b l e c o n d i t i o n . Gordon com pared h i s d e n s i t y !
! s t i r r e d r e s u l t s on p o ta s s iu m c h l o r i d e w i t h t h e r e s u l t s ob-
* 22
t a i n e d w i t h m e c h a n ic a l s t i r r i n g by H a r t l e y and R u n n ic le s
from w hich he c o n c lu d e d t h a t i f t h e c e l l c a l i b r a t i o n and
I
i s u b s e q u e n t d i f f u s i o n m easurem ents w ere made a t n e a r l y th e !
same d e n s i t y and v i s c o s i t y d i f f e r e n c e s e i t h e r d e n s i t y o r
m e c h a n ic a l s t i r r i n g c o u ld be u se d w ith c o m p a ra b le r e s u l t s . '
I f , how ever, t h e r e s h o u ld be d i f f e r e n c e s in th e d e n s i t i e s
" 1 ..... ' ’' 1 "'' i
(23) G. S. H a r t l e y and D. F. R u n n i c l e s , P r o c . Roy.
S o c . , A 1 6 8 . 401 (1 9 3 8 ).
33 j
an d v i s c o s i t i e s , m e c h a n ic a l s t i r r i n g s h o u ld he u s e d . In
s e l f d i f f u s i o n t h e r e i s no d e n s i t y d i f f e r e n c e o r v i s c o s - j
i t y d i f f e r e n c e b etw een t h e s o l u t i o n s i n th e two c o m p a rt- \
m ents and so d e n s i t y s t i r r i n g c a n n o t keep th e s o l u t i o n s
hom ogeneous.
Adamson2 3 h as made a t e s t o f th e e f f e c t o f s t i r - ■
r i n g in diap h rag m c e l l s i n th e d i f f u s i o n o f sodium t r a c e r :
i n 0 .0 4 7 8 H. sodium i o d i d e a t 2 5 ° C ., in w hich he fo und
t h a t th e d i f f u s i o n c o e f f i c i e n t o b t a i n e d w ith s t i r r i n g was \
1 . 9 # h i g h e r t h a n t h a t o b t a i n e d w i t h o u t s t i r r i n g . P r e l i m - j
S
i n a r y i n v e s t i g a t i o n s f o r t h i s r e s e a r c h w ere made by d e te r - :
I
m in in g t h e d i f f u s i o n c o e f f i c i e n t s o f sodium t r a c e r i n eod-j
ium s u l f a t e s o l u t i o n s up t o l.Q If. a t 25°C. These d e t e r - j
I
j
m in a tio n s w ere made w i t h o u t m e c h a n ic a l s t i r r i n g . The
!
c e l l s w ere su sp en d e d i n th e t h e r m o s t a t on r u b b e r t u b i n g .
Each c e l l gave v a l u e s o f th e d i f f u s i o n c o e f f i c i e n t w hich
upon p l o t t i n g ( F i g . 3) seemed to be c o n s i s t e n t w i t h o t h e r
v a lu e s o b t a i n e d f o r t h e same c e l l , b u t t h e c u rv e o b t a i n e d :
f o r each c e l l was d i f f e r e n t . A c o r r e l a t i o n w ith t h e c e l l I
c o n s t a n t was fo u n d . The d i f f u s i o n c o e f f i c i e n t s w ere in
eac h c a s e lo w e r t h a n . t h o s e o b t a i n e d u s i n g s t i r r i n g . At
h i g h e r c o n c e n t r a t i o n s t h e d e v i a t i o n s became more m arked.
(23) A. W. Adamson, P r i v a t e co m m u n icatio n .
i
STIRRED
K = 0 ,0 7 7 6
0.6
K = 0 .0 9 2 7
I
0 .4
0.2
0 0 .4 0 .8 1.2 M
-fc
FIG. 3. DEPENDENCE OF SELF DIFFUSION COEFFICIENTS OF N
IN N a ^ O * ON CELL CONSTANT IN UNSTIRRED R U NS
w
35
I
The lo w e s t v a l u e s f o r t h e d i f f u s i o n c o e f f i c i e n t s were J
I
o b t a i n e d w ith t h e c e l l w ith t h e l a r g e s t c e l l c o n s t a n t , !
w h ile th e h i g h e s t v a l u e s were o b ta in e d w ith t h e c e l l j
w i t h th e s m a l l e s t c e l l c o n s t a n t . T h is means t h a t th e I
more p o ro u s d iap h ra g m s g iv e t h e lo w e s t d i f f u s i o n c o e f f
i c i e n t s w hich i s e x a c t l y a s would be e x p e c te d i f t h e s o l
u t i o n s in t h e co m p artm en ts w ere n o t homogeneous up t o
t h e diaphragm w ith r e s u l t a n t d i f f u s i o n g r a d i e n t s i n t h e j
m ain body o f t h e s o l u t i o n s . T hese g r a d i e n t s would be j
more p ro n o u n ce d in th e more p o ro u s d iap hragm c e l l s w here i
i
t h e io n s a r e t r a n s p o r t e d i n g r e a t e r num bers th r o u g h t h e !
d iap h rag m p e r u n i t t im e . T h is e f f e c t i v e l y i n c r e a s e s t h e !
d i s t a n c e th ro u g h w hich t h e d i f f u s i o n o c c u r s and r e s u l t s
in low v a lu e s f o r t h e d i f f u s i o n c o e f f i c i e n t s . ■
M e c h a n ic a l S t i r r i n g . D u rin g th e e a r l y work on
s t i r r i n g i n t h i s r e s e a r c h i t was f e l t d e s i r a b l e to de
v i s e a s t i r r i n g m ethod w hich d i d n o t r e q u i r e any s t i r r i n g
d e v ic e t o come i n t o c o n t a c t w ith t h e diap h rag m so a s . t o
a v o id any p o s s i b l e pum ping e f f e c t s . F i r s t th e c e l l s
w ere m ounted w i t h t h e lo n g a x e s h o r i z o n t a l and r o t a t e d *
a b o u t t h i s a x i s . Sm all 1 t o 2 mm. d i a m e te r g l a s s ro d s
were p l a c e d i n e a c h com partm ent to move a lo n g th e w a l ls
a s t h e c e l l r o t a t e d . The m easu red d i f f u s i o n c o e f f i c i e n t
was found t o i n c r e a s e a s t h e r a t e o f r o t a t i o n was i n - '
36
c r e a s e d and no l i m i t was r e a c h e d up t o a b o u t 50 rpm. I t
was now a r r a n g e d t o r e v e r s e th e d i r e c t i o n o f r o t a t i o n
e v e r y f o u r r e v o l u t i o n s . W ith t h i s a rra n g e m e n t i n c r e a s i n g
t h e r a t e o f r o t a t i o n i n c r e a s e d t h e d i f f u s i o n c o e f f i c i e n t
o f 0 .0 2 K sodium s u l f a t e u n t i l a p p ro x . 25 rpm. was r e a c h e d
an d t h e n no f u r t h e r i n c r e a s e was n o te d up t o 60 rpm. w hich
was t h e h i g h e s t r a t e t e s t e d . T h is i n d i c a t e d t h a t c o m p le te
s t i r r i n g was o b t a i n e d above 25 rpm. S in c e t h i s f i g u r e
m ig h t be e x p e c te d t o i n c r e a s e i f th e s o l u t i o n s w ere more
c o n c e n t r a t e d t h e p r o c e s s was r e p e a t e d on 1 .5 .N sodium
s u l f a t e w ith c o m p le te s t i r r i n g r e a c h e d a t a b o u t 40 rpm.
At t h i s tim e a p r i v a t e com m unication was r e c e iv e d •
from R. H. S to k e s c o n t a i n i n g i n f o r m a ti o n l a t e r p u b l i -
q
sh e d on a m a g n e tic s t i r r i n g m ethod. S t i r r i n g was accom
p l i s h e d by s t i r r e r s o p e r a t e d m a g n e t i c a l l y from o u t s i d e
th e c e l l . These s t i r r e r s c o n s i s t e d o f g l a s s t u b i n g 3 ram.
i n d i a m e te r and s l i g h t l y s h o r t e r th a n t h e diap hragm d i a - j
I
i
m e t e r . I n t o e a c h s t i r r e r was s e a l e d a l e n g t h o f ' i r o n \
I
w ir e and t h e s t i r r e r s w ere so a d j u s t e d t h a t th e s t i r r e r ;
i
i n t h e u p p e r com partm ent sank and t h a t i n t h e lo w e r com- j
I
p a r tm e n t f l o a t e d so t h a t t h e s t i r r e r s were a lw ay s a d j a c - j
e n t t o t h e d ia p h ra g m . The s t i r r e r s were r o t a t e d by a
U -shaped magnet r o t a t e d a b o u t t h e c e l l . S to k e s t e s t e d
t h e m ethod t o f i n d t h a t pum ping and b u lk flo w e f f e c t s
. - . n
37 I
w ere n e g l i g i b l e . D i f f u s i o n c o e f f i c i e n t s o f p o ta s s iu m
c h l o r i d e o b t a i n e d by t h i s m ethod w ere com pared w ith th o s e
o f H arned and B u t t a l l 7 w hich a r e a v a i l a b l e up t o 0 . 5 N.
and were fo u n d t o a g r e e v e ry w e ll a t c o n c e n t r a t i o n s above-
a p p ro x . 0 .0 5 N. Below t h i s c o n c e n t r a t i o n s u r f a c e e f f e c t s
o f t h e d iaphragm g iv e c o e f f i c i e n t s h i g h e r t h a n t h e f r e e !
i
d i f f u s i o n e x p e r im e n ts . In 0 .0 0 5 S . p o ta s s iu m b ro m ide th e
d i f f u s i o n c o e f f i c i e n t was a l r e a d y 3 .5 $ abo v e t h e O n sa g e r- j
i
Fuoss t h e o r e t i c a l v a lu e f o r t h a t c o n c e n t r a t i o n and i n
0 .0 0 3 5 N. p o ta s s iu m b ro m id e , w hich was t h e lo w e s t concen
t r a t i o n m e a su re d , t h e d i f f u s i o n c o e f f i c i e n t was 6 $ above
t h e t h e o r e t i c a l .
A m o d i f i c a t i o n o f t h e S t o k e s ' m ethod was made.
M ag n etic s t i r r e r s w ere p l a c e d i n t h e c o m p artm en ts and
s i n c e i t was more c o n v e n ie n t t o r o t a t e t h e c e l l s w ith th e
e x i s t i n g e q u ip m e n t, t h e m ag n ets were p e rm a n e n tly mounted
i n t h e t h e r m o s t a t and t h e c e l l s w ere r o t a t e d b etw een them .
T a b le I I shows t h e r e s u l t s o f a c o m p a riso n betw een m agnet
i c s t i r r i n g and a l t e r n a t e r o t a t i o n s t i r r i n g a t SO rpm. f o r
a s e r i e s o f f i v e sodium s u l f a t e s o l u t i o n s . The two m eth
ods a g r e e w i t h i n 1 $ e x c e p t a t th e m ost c o n c e n t r a t e d s o l
u t i o n .
The a p p e a ra n c e o f t h e p a p e r o f Wang an d Kennedy ^ 4
a t t h i s tim e p r o v i d e d t r a c e r s e l f d i f f u s i o n r e s u l t s i
38
o b t a i n e d by an a b s o l u t e m ethod w hich made p o s s i b l e a d i r
e c t check o f t h e s u i t a b i l i t y o f t h e s e s t i r r i n g m ethods
f o r s e l f d i f f u s i o n . T a b le I I shows t h e r e s u l t s o f r a t e
o f s t i r r i n g t e s t s on t h e s o l u t i o n s . Sodium t r a c e r d i f f -
i
u s i o n was m easured f o r 0 . 0 5 , 0 . 6 , 2 .0 and 3 .5 H sodium ]
i o d i d e u s i n g t h e m a g n e tic s t i r r i n g m ethod a t 60 rpm. In
t h e two lo w e r c o n c e n t r a t i o n s t h e d i f f u s i o n c o e f f i c i e n t s j
i
a g r e e d w i t h i n 1$ o f th o s e by t h e a b s o l u t e m ethod. At
2 .0 IT. th e m a g n e t i c a l l y s t i r r e d v a l u e was 5$ low and a t
3 .5 N. t h e v a lu e was 27$ lo w . Runs were made on 2 .0 and
3 .5 N. s o l u t i o n s a t 80 and 100 rpm. V alu es o b t a i n e d from
t t h e s e r u n s a g r e e d w i t h i n one p e r c e n t o f th o s e o f Wang and
K ennedy. I n t h e more c o n c e n t r a t e d s o l u t i o n s o f sodium
I
s u l f a t e and sodium c h l o r i d e i t was a l s o found n e c e s s a r y
j t o s t i r a t a more r a p i d r a t e t o r e a c h a v a lu e f o r t h e
d i f f u s i o n c o e f f i c i e n t in d e p e n d e n t o f t h e r a t e . The a g r e e
m ent o f t h e m a g n e t i c a l l y s t i r r e d r u n s on sodium io d id e
s o l u t i o n s w ith th o s e o f t h e a b s o l u t e m ethod o f Wang and
j Kennedy i n d i c a t e d t h a t th e d iap h rag m c e l l m ethod c a n be
j u s e d t o g iv e a c c u r a t e io n d i f f u s i o n c o e f f i c i e n t s o v e r a
J wide ra n g e o f c o n c e n t r a t i o n s p r o v i d e d s t i r r i n g i s done
I
i a t a r a t e a d e q u a te f o r t h e s o l u t i o n t e s t e d .
The d a t a i n T a b le I I a r e n o t s u f f i c i e n t t o a c t u a l - ,
i
l y a n a l y z e t h e r e a s o n f o r t h e n e ed f o r i n c r e a s e d s t i r r i n g ]
39 '
l
1
i n c o n c e n t r a t e d s o l u t i o n s . C e r t a i n l y v i s c o s i t y s h o u ld j
he a f a c t o r , t h u s t h e v i s c o s i t y i n c r e a s e s by 50$ o v e r t h e j
ra n g e 0 .0 3 t o 3 .0 N. sodium s u l f a t e . However, v a r i a t i o n s j
i n t h e bo uyancy o f t h e s t i r r e r r o d s w ith t h e d e n s i t y may 1
a l s o have an e f f e c t . I
A ll d i f f u s i o n c o e f f i c i e n t s m easu red f o r t h i s r e - i
s e a r c h u s i n g m a g n e tic s t i r r i n g on s o l u t i o n s above 1 .5 H.
w ere done a t 100 rpm. A t e s t o f t h e a l t e r n a t e r o t a t i o n
s t i r r i n g method a t 60 rpm. on a 3 .5 H. sodium io d id e s o l
u t i o n gave a r e s u l t a l s o low by 16 $. T h is m ethod was n o t
u s e d f o r r u n s i n c o n c e n t r a t i o n ra n g e s g r e a t e r t h a n 1 . 0 N. j
S to k e s r e p o r t e d no r a t e o f s t i r r i n g m easurem ents I
on c o n c e n t r a t e d s o l u t i o n s . I t would seem from h i s
g
p a p e r s t h a t 50 rpm. was u se d e x c l u s i v e l y f o r h i s meas
u r e m e n ts . As s e e n i n t h i s r e p o r t t h e r a t e o f s t i r r i n g
n e e d e d t o p ro d u c e a d e q u a te s t i r r i n g d epended v e ry g r e a t
l y on c o n c e n t r a t i o n . A c r i t i c a l e x a m in a tio n o f t h i s e f f
e c t s h o u ld be made on t h e s o l u t i o n s u s e d by S to k e s t o de
te r m in e i f an i n c r e a s e d s t i r r i n g r a t e m ig h t n o t be n e c e s s
a r y f o r h i g h e r c o n c e n t r a t i o n s .
C o n s ta n t c o n t a c t o f t h e s t i r r e r s has been r e p o r t e d
by S to k e s 9 to c a u s e w ear o f th e diaph ragm r e s u l t i n g in
a s lo w ly i n c r e a s i n g c e l l c o n s t a n t . D u rin g th e p e r i o d o f
some 4 5 d ay s o f u s e of t h i s m ethod th e c e l l c o n s t a n t s i
re m a in e d c o n s t a n t w i t h i n one p e r c e n t and no c o r r e c t i o n s
w ere made.
41
TABLE II
Com parison o f t h e e f f e c t o f r a t e o f s t i r r i n g on ,
t h e a l t e r n a t e r o t a t i o n and m a g n e tic s t i r r i n g m eth o d s.
I
Na4" i n sodium i o d i d e ,
D ( c m " /d a y )
c Wang & A l t . r o t M a g n etic s t i r r i n g
Worm. Kennedy 60 rpm 60 rpm 80 rpm 100 rpm
0 .0 5 1 .1 0 7 1 .1010.3*
0 . 6 1 .0 9 8 1 .1 0 ± 0 .5
2 .0 1 .0 7 1 1 .0 2 1 .0 6 1 .0 8
3 .5 1 .0 7 0 0 .9 1 + 1 .5 0 . 7 8 H . 1 1 .0 8 1 .0 8
Na4- in sodium s u l f a t e
C A l t . r o t . M agnetic s t i r r i n g
Norm. 60 rpm 25 rpm 35 rpm 48 rpm 60 rpm 100 rpm
0 .0 2 1 .0 7 1 0 .2 1 .0 3 1 .1 3 1 . 1 2 1 .1 2 * 0
1 . 2
0 .8 4 1 0 .6 0 .8 5 1 0 .8
1 . 5
0 .8 0 1 0 .3 0 . 8 1 1 0 . 3 0 .8 0 + 0 .5
2 .0 0 .7 3 1 0 .4 0 .7 3 0.7710.4*
3 .0 0 .6 3 1 0 .8 0 .5 7 1 1 .0 0 .7 3 1 1 .2
Na+ i n sodium c h l o r i d e
C
Norm.
1.0
1 . 5
2.0
3 .5
A l t . r o t .
60 rpm
0 .9 7 1 0 .4
0 .6 9 1 0 .7
60 rpm
1 .0 4
1 .0 2 1 0 .5
0 .9 7 ± 0 .6
0 .7 6
D (cm ^ /d ay )
M ag n etic s t i r r i n g
80 rpm 1 0 0 rpm
1.01
0 .9 9
1 .0 3 1 0 .6
1 .0 1 1 0 .4
1 .0110.3
* D e v i a t i o n s a r e g iv e n i n p e r c e n t .
CALIBRATION OF THE DIAPHRAGM CELLS
The c e l l c o n s t a n t , K, i n th e d iap h rag m d i f f u s i o n
e q u a t io n i s r e l a t e d t o t h e d im e n s io n s o f th e f r e e s p a c e
i
o f t h e d iap hrag m by th e e q u a t io n
K “ 2 .3 0 3 J jA (vu +‘ t L) |
w here and Vl a r e 'blie v° l umes ° f th e u p p e r and lo w e r
1
c o m p artm en ts and k i s t h e r a t i o o f t h e mean p o r e l e n g t h
t o t h e t o t a l p o re a r e a . T h is r a t i o has n o t b een e x p e r i
m e n t a l l y d e te r m i n a b le and so t h e d iap h rag m c e l l method
h a s r e q u i r e d c a l i b r a t i o n w i t h a s o l u t i o n of known d i f f
u s i o n c o e f f i c i e n t t o d e te rm in e t h e c e l l c o n s t a n t .
The s o l u t i o n g e n e r a l l y u s e d f o r c a l i b r a t i o n i s j
o . l N. p o ta s s iu m c h l o r i d e w hich i s a llo w e d t o d i f f u s e
! i n t o w a te r u n t i l 25$ h as p a s s e d t h r o u g h t h e diap h rag m ,
j S to k e s 9 h as c r i t i c a l l y c o n s i d e r e d t h e d i f f u s i o n c o e f f i c -
j i e n t o f t h i s s a l t by t h e a b s o l u t e m ethods o f Harned a,nd
i N u t t a l l 7 , Lamm2 4 , and Cohen and B r u i n s 2^ t o a r r i v e a t
t h e v a lu e o f 1 .6 1 3 cm2 p e r day a s t h e i n t e g r a l d i f f u s i o n
i c o e f f i c i e n t f o r th e above c o n d i t i o n s . T h is v a lu e was
l
I _ _ _
J
(24) 0 . Lamm, Nova A cta Regiaw Soc. S c i . U p s a l i e n s i s ,
IV 1 0 . 6 ( 1 9 3 ? ) .
(2 5 ) E. Cohen and H. R. B r u i n s , Z. p h y s i k . C hem ., j
1 0 3 , 337 (1 9 2 3 ).
43
u s e d i n th e c a l i b r a t i o n o f t h e c e l l s em ployed i n t h i s r e
s e a r c h . The a g re em e n t o f my diaphragm c e l l r e s u l t s w ith
t h e a b s o l u t e d i f f u s i o n c o e f f i c i e n t s o f Na+ i n sodium i o
d id e a s d e te r m in e d by Wang and Kennedy * 4 c o n s t i t u t e s a
f u r t h e r ch eck on t h e v a l i d i t y o f th e c h o s e n v a l u e , 1 .6 1 3
cm ^/day, a s w e ll a s th e E e l f c o n s i s t e n c y o f t h e s t i r r i n g
i n t h e c a l i b r a t i o n and su b s e q u e n t s e l f d i f f u s i o n r u n s . j
A f u r t h e r c a l i b r a t i o n was made p o s s i b l e by the.
m ethod o f Mehl and Schmidt^® i n which $ /A i s d e te rm in e d
by f i l l i n g th e d iaph rag m w ith a s o l u t i o n o f known s p e c -
I i f i c c o n d u c ta n c e and d e te r m i n i n g t h e a p p a r e n t r e s i s t a n c e ,
j F i g u r e s 4 and 5 show t h e a p p a r a t u s u s e d i n t h i s d e te r m i n - ;
| ' j
a t i o n . I t was c o n s t r u c t e d from s t a n d a r d t a p e r g l a s s !
j o i n t s t o have a rem o vable c e n t e r c y l i n d e r c o n t a i n i n g a j
j s i n t e r e d g l a s s d iap h ra g m . By u s i n g a p p r o p r i a t e end s e c t - j
| i o n s , c o n d u c ta n c e , d i f f u s i o n and t r a n s f e r e n c e m easurem ents
c o u ld be made i n t h e same d iap h rag m .
The p la tin u m e l e c t r o d e s were mounted on s h o r t
|
j c o i l s o f p la t i n u m w ire w hich were s o l d e r e d to t u n g s t e n
w ir e and s e a l e d th r o u g h a g l a s s tu b e c o n t a i n i n g m ercury
t o make t h e c o n t a c t . The e l e c t r o d e s w ere c u t th e same
I
I d ia m e te r a s th e d ia p h ra g m , p l a t i n i z e d and s l i g h t l y d e n te d
1 1 '■ ■ 1 1 1 i
(2 6 ) J . W. Mehl and C. L. A. S c h m id t, U n iv . C a l i f . *
P u b l . P h y s i o l . , 8 _, 165 (1 9 3 7 ).
C O N D U C T A N C E END S E C T I O N S
D IA P H R A G M S E C T IO N
i
\
I
I
i
\
D IFFU SIO N E N D S E C T I O N S
FIG.4 . SECTIONAL CONDUCTANCE AND DIFFUSION C E L L S
BALL
JOINT
W j mh
[.liH m u i.iM iif.i.n .ti DIAPHRAGM
PIG. 5. TRANSFERENCE APPARATUS
46
i n t h r e e p l a c e s a lo n g t h e c ir c u m f e r e n c e so t h a t when t h e j
c e l l was a sse m b le d t h e p la t i n u m b l a c k d i d n o t come i n t o |
I
d i r e c t c o n t a c t w i t h t h e d iap h rag m b u t was some few te n th s ]
o f a m i l l i m e t e r rem oved. In o r d e r t o e v a l u a t e th e p o s s - j
i b l e e r r o r in v o lv e d in t h i s s e p a r a t i o n r e a d i n g s w ere madej
I
on a s o l u t i o n a s t h e e l e c t r o d e s w ere moved p r o g r e s s i v e l y !
\
b a c k from t h e d ia p h ra g m . The p e r c e n t change i n A was ;
n u m e r i c a l l y a p p r o x im a te ly f i v e tim e s t h e sum o f t h e d i s - '
i
t a n c e s betw een th e e l e c t r o d e s and t h e d ia p h ra g m , i n m il
l i m e t e r s . The few t e n t h s o f a mm. s e p a r a t i o n due to th e
d e n ts would t h e r e f o r e c o rr e s p o n d t o c a . one p e r c e n t
i
c n a n g e , and a l l t h e m easu rem ents w ere c o r r e c t e d by t h i s j
a m o u n t. 1
i
R e s i s t a n c e m easurem ents were made w ith a Leeds
and N o rth ru p J o n e s -D ik e b r i d g e , a u d io f r e q u e n c y o s c i l l a - :
t o r , n u l l d e t e c t o r and an o s c i l l o s c o p e . Due t o th e
low r e s i s t a n c e o f t h e c e l l a l l m easurem ents w ere made
w ith a s t a n d a r d r e s i s t a n c e i n s e r i e s w i t h t h e c e l l . A ll
s o l u t i o n s u s e d i n t h e c e l l were p r e p a r e d u s i n g w a te r
— R
w i t h a s p e c i f i c c o n d u c ta n c e o f 2 x 1 0 ° mho cm. and were
d e g a s s e d b e f o r e u s e by b o i l i n g i n th e vacuum o f a w a te r
pump. The c e l l was f i l l e d by s u c t i o n an d b r o u g h t to
te m p e r a tu r e e q u i l i b r i u m i n an o i l t h e r m o s t a t m a in ta in e d
i
a t 35 - 0 .0 0 2 ° 0 . R e ad in g s w ere ta k e n a t te m p e r a t u r e j
47 I
I
e q u i l i b r i u m and th e n more s o l u t i o n ( a t t e m p e r a t u r e ) was
drawn th r o u g h th e d iap hragm and th e r e a d i n g s r e p e a t e d
u n t i l a c o n s t a n t v a lu e was o b t a i n e d . C o nductance c e l l i
I
i I
| c o n s t a n t s o f t h e two d iap h ra g m s w ere d e te r m in e d u s i n g
:
s e v e r a l sodium c h l o r i d e s o l u t i o n s from 0 .0 0 4 t o 0 .1 ET. i
t 4
T hese two d ia p h ra g m s were th e n f i t t e d w ith end
i
s e c t i o n s t o make d i f f u s i o n c e l l s an d c e l l c o n s t a n t s w ere ;
»
d e te r m in e d by p o ta s s iu m c h l o r i d e d i f f u s i o n , f o r e a c h , ;
i
f i r s t , w i t h d e n s i t y s t i r r i n g and t h e n w ith m a g n e tic s t i r -
i
r i n g a t 60 rpm. T hese c e l l c o n s t a n t s a r e com pared in
T a b le I I I w ith th e c o n s t a n t s c a l c u l a t e d from th e c o n d u c t-
!
a n c e c e l l c o n s t a n t s i
TABLE I I I
! ~ *
D i f f u s i o n C e l l C f l i b r a t i o n by
; C o n d u ctan ce M easurem ents
C e l l J /k K ( l f f ? | S a )
M -l 0 .3 0 8 + 0 .8 * 0 .0 4 8 0 1 0 .8 0 .0 4 4 8 + 0 .0 0 .0 4 8 2 * 0 .0
Mg 0 .4 6 4 1 0 .5 0 .0327 tO.5 0 .0 3 0 9 1 0 .3 0 . 0 3 2 6 i 0 . 5
* D e v i a t io n s g iv e n i n p e r c e n t .
T h is p r o v i d e s a n o th e r in d e p e n d e n t v e r i f i c a t i o n of
t h e c e l l c a l i b r a t i o n p r o c e d u r e u s i n g 0 . 1 N. p o ta s s iu m
c h l o r i d e and t h e v a l u e , 1 .6 1 3 c m 2 /d a y , w ith m a g n e tic
\
j
s t i r r i n g . . ___ ______ ______ _ J
The c e l l c o n s t a n t f o r u se o f t h e c e l l w ith a l t e r
n a t e r o t a t i o n was o b t a i n e d by c a l i b r a t i o n w ith a l t e r n a t e
r o t a t i o n a t 60 rpm. o f t h e c e l l mounted v e r t i c a l l y w ith
0 . 1 N. p o ta s s iu m c h l o r i d e i n t h e lo w e r com partm ent and
w a te r i n t h e u p p e r . F or any g iv e n c e l l t h e s e two c a l i b
r a t i o n v a l u e s a g r e e d iw ithin Vfo.
For f u t u r e d i f f u s i o n work c e l l s can be c a l i b r a t e d
u s i n g one o f t h e io n s s t u d i e d in t h i s w ork. T h is method
p e r m i t s c a l i b r a t i o n i n a s o l u t i o n w ith no b u lk c o n c e n tr a
t i o n g r a d i e n t s and so i s a d v a n ta g e o u s i n t h a t t h e c o m p li
c a t i o n s o f c o m p a riso n w ith mean d i f f u s i o n v a l u e s can be
a v o id e d .
SURFACE EFFECTS
P r e l i m i n a r y work on d i l u t e s o l u t i o n s y i e l d e d
d i f f u s i o n c o e f f i c i e n t s w hich d id n o t a g r e e w ith th o s e
e x p e c te d from t h e U e r n s t e q u a t io n f o r d i f f u s i o n c o e f f i c - 1
i
i e n t s a t i n f i n i t e d i l u t i o n or w ith t h e O n sa g e r-F u o ss |
e q u a t i o n f o r t h e l i m i t i n g s l o p e i n d i l u t e s o l u t i o n s . The :
t
e x p e r i m e n t a l e x t r a p o l a t e d l i m i t i n g v a l u e s w ere h i g h e r j
th a n th e K e r n s t v a lu e and th e s lo p e was g r e a t e r th e n t h a t
c a l c u l a t e d by th e O n s a g e r-F u o ss t h e o r y .
I f t h i s d e v i a t i o n from t h e o r y i s due t o th e u s e I
i
i
o f a g l a s s d ia p h ra g m , t h e n some t r a n s p o r t p r o c e s s i s
p r e s e n t w hich i n c r e a s e s t h e a p p a r e n t m o b i l i t y o f t h e io n
i n t h e d ia p h ra g m . T h is c o u ld be d i f f u s i o n o f th e a d - !
s o r b e d io n on th e s u r f a c e o f t h e p o r e s i n t h e d iap h rag m .
Such a m echanism would a l s o i n c r e a s e th e c o n d u c ta n c e o f
t h e s o l u t i o n i n t h e d iaph rag m and t h i s i n c r e a s e h a s b e en
o b s e r v e d i n t h e c a s e o f p o ta s s iu m c h l o r i d e by M ysels and
McBa.in^7 . They m easured the c o n d u c ta n c e o f a p p ro x im a t
e l y 0 . 1 and 0 .0 0 0 5 N. p o ta s s iu m c h l o r i d e s to c k s o l u t i o n s
in a c o n d u c ta n c e c e l l w hich h ad b e e n m o d ifie d by t h e i n
s e r t i o n o f a P y re x s i n t e r e d g l a s s d iap h rag m b etw een th e
(2 7) K. J . M ysels and J . W. McBain, J . C o l l . S c i . ,
3 , 45 (1 9 4 8 ) . !
50
two e l e c t r o d e s . C om parison o f t h e s e v a l u e s showed t h a t
th e d i l u t e s o l u t i o n a p p e a re d t o c o n d u c t 5 - 10$. b e t t e r
i n t h e c e l l w ith th e d ia p h ra g m . T h is i n c r e a s e th e y s t a t e
m ust be due t o t h e e l e c t r i c a l s u r f a c e c o n d u c ta n c e o f th e
d o u b le l a y e r a d s o r b e d on t h e l a r g e g l a s s - p o t a s s i u m c h l o r
id e s o l u t i o n i n t e r f a c e in t h e d iap h rag m .
i
In o r d e r to d e te r m in e a t what d i l u t i o n t h e a d s o r b - {
ed l a y e r would l e a d t o a m e a s u ra b le e r r o r on t h e d ia p h
ragm , r e s i s t a n c e m easu rem en ts w ere made on a s e r i e s o f
s o l u t i o n s o f sodium c h l o r i d e and sodium s u l f a t e i n th e
s e c t i o n a l c o n d u c ta n c e c e l l ( F i g . 4 ) . C o nd u ctan ce c e l l
c o n s t a n t s were c a l c u l a t e d from t h e s e d a t a u s i n g c o n d u c t-
j p q
an c e v a lu e s from M aclnnes . I f c o n d u c ta n c e and t r a n s -
s
[ f e r e n c e m easurem ents on v e r y d i l u t e s o l u t i o n s were ob
t a i n e d i t w ould be p o s s i b l e t o e v a l u a t e t h e e f f e c t o f t h e 1
S
s u r f a c e on t h e c o n d u c ta n c e m o b i l i t i e s a t e a c h c o n c e n t r a - i
i
i
t i o n . T hese m o b i l i t i e s a r e n o t e q u i v a l e n t t o d i f f u s i o n :
m o b i l i t i e s , how ever, and h en ce t h e r e s u l t s c a n n o t be u se d
a c c u r a t e l y ' t o c o r r e c t s e l f d i f f u s i o n c o e f f i c i e n t s f o r :
t h i s s u r f a c e e f f e c t . F i g u r e s - 6 and 7 show how t h e s e
c e l l c o n s t a n t s v a ry w ith s a l t c o n c e n t r a t i o n below
i
n n m * T » (
f
(28) D. A. M a cln n e s, "The P r i n c i p l e s o f E le c tro c h e m
i s t r y " , R e in h o ld P ub . C o r p ., p . 339 ( 1 9 3 9 ). J
0.46
0 .4 2
0 .3 8
0 .3 4
0 .3 0
0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0.10 0.12
. VC
FIG. 6. CONDUCTANCE CELL CONSTANTS OF NaCI
SOLUTIONS IN DIAPHRAGM CELL
52
0.34
0.32
0.30
k
0.28 -
0.26
0.2 0.1 0.3
/ C
FIG. 7. CONDUCTANCE CELL CONSTANTS OF NajSO ^
SOLUTIONS IN DIAPHRAGM CELL
0 .0 0 3 H. f o r sodium c h l o r i d e and 0 .0 1 N f o r sodium s u l f - '
»
a t e . Below t h e s e v a l u e s t h e s i n t e r e d g l a s s d iaphragm
g i v e s e r r o n e o u s r e s u l t s .
S i n t e r e d s t a i n l e s s s t e e l d iap h ra g m s w ere m ounted !
i n c e l l s and u s e d i n d i f f u s i o n s t u d i e s i n an a tt e m p t t o j
e l i m i n a t e t h i s a d s o r p t i o n e r r o r w ith o u t a b a n d o n in g t h e !
i
d iap h ra g m m ethod. In T a b le IV a r e g iv e n th e d i f f u s i o n '
d a t a in v e r y d i l u t e s o l u t i o n s u s i n g b o th t h e g l a s s and
s t a i n l e s s s t e e l d ia p h ra g m s . I n e a c h c a s e t h e g l a s s d i a
phragm r e s u l t s a r e h i g h e r t h a n t h o s e e x p e c te d from t h e
o r y . The s t a i n l e s s s t e e l diap h rag m r e s u l t s a r e i n e v e r y
c a s e lo w e r th a n th o s e w i t h t h e g l a s s d ia p h ra g m . I n th e
1
c a s e o f Na+ io n i n sodium c h l o r i d e t h e r e seems t o be no !
a d s o r p t i o n e f f e c t s i n c e t h e r e s u l t s a r e v e ry n e a r l y ;
t h o s e e x p e c te d from t h e o r y , h ow ever, th o s e f o r Cl" and
SO^ io n s do show an a p p r e c i a b l e e f f e c t . In gen era/1 ,
t h e r e f o r e , th e s t a i n l e s s s t e e l d iap hragm c a n n o t be con
s i d e r e d any more s a t i s f a c t o r y th a n th e P y re x ones a n d ,
m o re o v e r, a r e more d i f f i c u l t t o h a n d le e x p e r i m e n t a l l y .
The m easurem ent o f t r a n s f e r e n c e num bers o f Na+
and SOJ io n s was a t t e m p t e d , t o p r o v i d e d a t a t o e s t i m a t e
t h e r e l a t i v e c o n t r i b u t i o n s o f t h e s e io n s t o t h e a d s o r p
t i o n e r r o r . For t h i s p u r p o s e t h e ’’a n a l y t i c a l b o u n d a ry ” j
m ethod f o r t r a n s f e r e n c e number m easurem ent d e v e lo p e d by
54
TABLE IV
Com parison o f th e e f f e c t o f a d s o r p t i o n
on g l a s s a n d s t e e l d iap h ra g m s on d i f f u s i o n c o e f f i c i e n t s
Ion C
Form.
N e rn s t
t h e o r y
D ( c m ^ /d a y )
G l a s s
d i a p h .
S t e e l
d i a p h .
F a* in
F aC l
C l ' i n
FaC l
Na+ i n
FagSC^
c a 1 0 °
0 .0 0 0 3 4 4
0 .003
0.02
1 .1 5 4
c a IQ" 5
c a 1 0 **^
0 .0 0 0 3 3 4
0 . 0 0 3.:
0.01
SOJ in c a 1 0 -5
F a |S 0 4
0 .0 0 3 5
1 .7 5 4
1 .1 5 4
0 .9 1 9
1 .4 7 1 0 .8 *
1 .2 5 1 0 .8
1 .1 3 1 0 .3
2 .7 0 1 1 .0 '
1 .4 7 1 0 .8
1 .3 8 1 0 .4
0 .9 8 1 0 .4
1 .1 5 6t 0 . 4
1 .1 3 1 0 .3
1 .1 0 t o . 4
2 .4 1 ± 0 .9
1 .1 5 s t 0 . 4
1 .2 7
1 .1 7 ± 0 .6
0 .9 4 ± 0 .6
0.88
* D e v i a t io n s g iv e n i n p e r c e n t .
55
p q
Brady and S a l l e y was u s e d w ith t h e a p p a r a t u s m o d ifie d
by i n s e r t i n g g ro u n d g l a s s j o i n t s t o a llo w th e i n t r o d u c t
io n o f t h e same diaphragm s e c t i d n u s e d f o r d i f f u s i o n and
c o n d u c ta n c e . The c u r r e n t p a s s e d was m easured p e r i o d i c a l
l y w i t h a L eeds and l o r t h r u p P o r t a b l e P r e c i s i o n P o t e n t i o
m e te r i n te rm s o f t h e v o l t a g e a c r o s s a 1 0 ohm r e s i s t o r
p l a c e d in s e r i e s w ith t h e t r a n s f e r e n c e c e l l . D i f f i c u l t y
was e x p e r ie n c e d i n s e a l i n g t h e v a r i o u s j o i n t s o f t h e app
a r a t u s t o be n o n - c o n d u c t in g . The t h e r m o s t a t l i q u i d
c o u ld n o t be w a te r f o r a t 0 .0 0 3 N. sodium s u l f a t e a b o u t
50$ o f t h e c u r r e n t fo u n d i t s way th r o u g h th e sy s te m w ith
o u t p a s s i n g th r o u g h t h e d ia p h ra g m . Upon u s i n g o i l a s t h e
t h e r m o s t a t l i q u i d t h i s s o u r c e o f e r r o r seemed t o b e e lim -
i
i n a t e d s i n c e r e a d i n g s ta k e n w i t h t h e a p p a r a t u s i n t h e
t h e r m o s t a t w ere unchanged i f t h e c e l l w ere l i f t e d from
l
t h e b a t h d u r i n g t h e r e a d i n g . I n t h e o i l t h e r m o s t a t t h e 1
t r a n s f e r e n c e num bers o f Na4' and SO4 i n 0 .0 0 3 N. sodium !
s u l f a t e w ere fo u n d t o be 0 .3 8 9 and 0 .5 6 1 . These two v a l -
i i
1 u e s sum t o 0 .9 5 0 r a t h e r th a n u n i t y w hich i n d i c a t e s t h a t |
! t
one o r b o th a r e i n e r r o r . The v a l u e s o b t a i n e d w ith o u t
a diaphragm by L ongsw orth*^ a r e 0 .3 8 6 and 0 .6 1 4 . T hese ;
...........i n ■ » " I ■ * " ■ ■ ■ ■ " ■
(29) A. P . Brady and D. J . S a l l e y , J . Am. Ohem. S o c .,
70 . 914 (1 9 4 8 ).
(3 0 ) L. Cr. L o n g sw o rth , J . Am. Chera. S o c ., 57_, 1185
( 1 9 3 5 ) .
56
s t u d i e s w ere d i s c o n t i n u e d f o r l i t t l e o r no i n f o r m a ti o n
seem ed f o r th c o m in g a t t h e e x p en se o f g r e a t e f f o r t .
Assuming t h a t t h e t r a n s f e r e n c e num bers f o r th e
io n s i n th e a d s o rb e d l a y e r a r e th e same a s i n b u lk s o l u
t i o n , t h e s u r f a c e e f f e c t may b e a p p ro x im a te d by a p p ly
i n g t o eac h d i f f u s i o n c o e f f i c i e n t d e t e r m i n a t i o n t h e cond
u c t a n c e c e l l c o n s t a n t f o r t h a t c o n c e n t r a t i o n . F o r Na+
i n 0 .0 0 0 3 4 4 N. sodium c h l o r i d e th e c o r r e c t e d D i s 1 .0 5
cm ^/day and f o r Na+ i n 0 .0 0 0 3 2 4 N. sodium s u l f a t e th e
c o r r e c t e d D i s 1 .2 0 cm ^/day. At t h i s v e ry low c o n c e n tr a
t i o n t h e v a lu e o f t h e d i f f u s i o n c o e f f i c i e n t s h o u ld be
w i t h i n a few t e n t h s o f a p e r c e n t o f t h e i n f i n i t e d i l u t i o n
( N e r n s t) v a lu e o f 1 .1 5 4 . Thus t h i s s im p le c o r r e c t i o n i s
i n s u f f i c i e n t .
RESULTS AHD SUMMARY
The d i f f u s i o n c o e f f i c i e n t s f o r Na+ and Cl” i n so d
ium c h l o r i d e and Na+ and SO4 i n sodium s u l f a t e a t 25°C.
a r e t a b u l a t e d i n T a b le s V and V I . I n g e n e r a l eac h v a lu e
i s a mean o f two o r more in d e p e n d e n t d e t e r m i n a t i o n s .
The d e v i a t i o n s i n p e r c e n t a r e g iv e n f o r t h o s e v a l u e s
w h ic h a r e an a v e r a g e o f more t h a n one v a l u e . The d a t a
i n T a b le s V and VI a r e p l o t t e d i n F i g u r e s 8 and 9 . I n
t h e s e f i g u r e s t h e c u rv e s a r e o r i g i n a t e d a t th e N e rn s t
c a l c u l a t e d v a l u e . Due t o t h e e f f e c t o f s u r f a c e a d s o r p
t i o n in d i l u t e s o l u t i o n s , a c c u r a t e i n v e s t i g a t i o n s c o u ld
t
n o t be made i n t h i s r e g i o n b u t i n t h e a b s e n c e o f v a l i d !
e v id e n c e to th e c o n t r a r y th e c a l c u l a t e d v a lu e s w ere I
a c c e p t e d .
As a r e s u l t o f t h i s work t h e l i m i t a t i o n s o f t h e
i
d iaphragm m ethod f o r m e a su rin g d i f f u s i o n c o e f f i c i e n t s
have been made more c l e a r . I t had been p o i n t e d o u t by
G o r d o n ^ t h a t work was n e c e s s a r y t o d e te r m in e t h e e f f e c t
o f s t i r r i n g an d t h e l a r g e s u r f a c e a r e a o f th e d iap hragm
on t h e m easurem ent o f d i f f u s i o n c o e f f i c i e n t s . He r e p o r t s
a n o m a lie s w hich i n d i c a t e t h a t t h e s e e f f e c t s m ight be con
s i d e r a b l e b u t d e c r i e s th e l a c k o f d e f i n i t i v e d a t a . Ex-
i
p e r i m e n t a l d a t a a r e p r e s e n t e d i n t h i s r e s e a r c h on t h e ;
58
TABLE V
S e l f d i f f u s i o n c o e f f i c i e n t s o f Na± and 01“
i n a q ueou s sodium c h l o r i d e s o l u t i o n s a t 25°C
C
Norm.
JL
C2
0 .0 0 5 0 .0 7 0 7
0 .0 2 0 .1 4 1
0 . 0 4 0 .2 0 0
0 . 1 0 0 .3 1 6
0 . 1 6 0 .4 0 0
0 . 2 0 0 .4 4 7
0 .3 0 0 .5 4 8
0 .5 0 0 .7 0 7
0 . 6 0 0 .7 7 5
0 . 7 0 0 .8 3 7
0 . 8 0 0 .8 9 4
1.00 1.000
1 .4 0 1 .1 8 3
2 .0 0 1 .4 1 4
3 .5 0 1 .8 7 1
% a
cm2/ d a y
DC1
cm2/ d a y
1 .1 3 5*2T0.9* 1 .7 0 5± 0 .4
1 .1 3 1 .6 6 ± 0 .1
1 .1 2 t o . 2
1 .6 3 ± 0 .1
1 .0 9 ± 0 .0
1 .6 3 ± 0 .3
1 . 0 8 ? ±0.3
1 .5 8 ± 0 .7
1 .0 5 ±1.0
1 .5 0 ± 0 .4
1 .0 5 ± 0 .3
1 .4 6 ± 0 .2
1 . 0 3 5 ± 0 .6 1 .4 3 ±0.4
1 . 0 1 5 ± 0.4 1 .4 1
1 .0 1 ± 0 .3 1 .4 1 ± 0 .5
* D e v i a t io n s a r e g iv e n i n p e r c e n t .
c
Norm.
0.01
0.02
0 . 0 3
0 . 0 5
0.10
0 . 2 0
0 .4 0
0 .5 5
0 .7 0
1.20
1 . 5 0
2.00
3 .0 0
59
TABLE VI
S e l f d i f f u s i o n c o e f f i c i e n t s o f Ha"*" and SOJ
i n aq u eo u s sodium s u l f a t e s o l u t i o n s a t 25°C.
' A Ds04
0 2 cm2/ d a y cm2/d a y
0 .1 0 0 1 .1 2 5 ±0.8* 0 .8 3 ± 1.0
0 .1 4 1 1 .1 2 ± 0 .0
0 .1 7 3 0 .7 5 ± 0 .4
0 .2 2 4 1 .0 9 s ± 0 .6 0 .7 2 ± 0 .7
0 .3 1 6 “ 0 .6 5
0 .4 4 7 1 .0 5 ± 0 .0 0 .6 1 ± 0 .3
0 .6 3 3 1 .0 0 ± 0 .5 0 .5 7 ± 0 .4
0 .7 4 2 0 .5 7
0 .8 3 7 0 . 9 3 5 ± 0 .3 0 . 5 5 5 ± 0.6
1 .0 9 5 0 . 8 4 5± 0 .8 0 .5 3 ± 0 .0
1 .2 2 5 0 .8 0 ± 0 .5 0 .5 0
1 .4 1 4 0 .7 7 ± 0 .4 0 .5 2 5 ±0.4
1 .7 3 2 0 .7 3 ± 1 .2 0 .5 4 ± 0 .6
* D e v i a t io n s a r e g iv e n i n p e r c e n t .
EQ.II
EQ, 10
EQ.9
o -
1.4
N b l
0.8
» 0 .4
/ C
FIG. 8 . SE L F D IFFU SIO N C O E F F IC IE N T S OF N a+ AND Cl" AT 2 5 °C .
0.8
i.4
EQ.I I
EQ.9
0.8
0 . 6 - EQ*9
EQ.I I
jO-
s o ;
0 .4
0.2
0 0 .4 0.8 1.2
1.6
■/c"
FIG. 9 . S E L F DIFFUSION COEFFICIENTS O F N * + AND S 0 7 AT 2 5 ° C .
** o>
63 ;
e f f e c t o f s t i r r i n g , c o r r e c t m ethod o f c a l i b r a t i o n and |
s u r f a c e e f f e c t f o r t h e s p e c i a l c a s e o f s e l f d i f f u s i o n . j
i
The r e s u l t s a r e a l s o a p p l i c a b l e i n g e n e r a l t o any d i f f u s - j
io n m easu rem en ts i n d iap h rag m c e l l s . !
i
M e c h a n ic a l s t i r r i n g was fo u n d t o be n e c e s s a r y f o r '
I
a l l c o n c e n t r a t i o n s . The amount n e c e s s a r y t o e n s u r e homo- 1
g e n e i t y i n t h e co m p artm en ts seems t o i n c r e a s e w ith t h e
i
c o n c e n t r a t i o n o f th e s o l u t i o n . T h is e f f e c t i s n o t c l e a r - j
I
l y u n d e r s t o o d b u t seems t o be in p a r t o r w h o lly due t o j
t h e i n c r e a s e i n v i s c o s i t y .
S u p p le m e n ta l p r o o f was o b t a i n e d v e r i f y i n g t h e e x - j
a c t n e s s o f t h e m ethod o f c a l i b r a t i o n s u g g e s te d by S to k e s^ .j
I
Two a d d i t i o n a l m ethods o f c a l i b r a t i o n w ere d e v e lo p e d .
I t was shown t h a t t h e r e e x i s t s a s u r f a c e e f f e c t in
s e l f d i f f u s i o n o f i o n s a t low c o n c e n t r a t i o n s . T h is e f f
e c t v a r i e s w ith th e io n and w ith t h e d iap h ra g m . I t was
n o t found p o s s i b l e t o c o r r e c t f o r t h i s e f f e c t b u t a m eth
od was d e v e lo p e d to d e te r m in e a t w hat lo w e r c o n c e n t r a t i o n
t h i s e f f e c t becom es a p p r e c i a b l e a n d hen ce s e t s a lo w e r
c o n c e n t r a t i o n l i m i t f o r th e u s e f u l n e s s o f d iap hragm d i f f
u s i o n c e l l s f o r t h a t p a r t i c u l a r i o n .
The p r e s e n t f i n d i n g s on s t i r r i n g needd and s u r f a c e
e f f e c t s s h o u ld make i t p o s s i b l e t o c r i t i c a l l y a p p r a i s e
diaphragm d e te r m in e d d i f f u s i o n c o e f f i c i e n t s r e p o r t e d i n j
63
t h e l i t e r a t u r e . In p a r t i c u l a r a l l v a lu e s f o r d i l u t e s o l
u t i o n s and a l l th o s e o b t a i n e d w hich had no m e c h a n ic a l
s t i r r i n g a r e s u s p e c t .
DISCUSSION
The l i m i t i n g s l o p e s from e q u a t i o n s (9 ) and (11)
| and t h e c u r v e s from e q u a t io n ( 1 0 ) i n t h e d i l u t e c o n c e n -
i
t r a t i o n r e g i o n a r e shown w ith t h e d i f f u s i o n c o e f f i c i e n t s j
|
i n F i g u r e s 8 and 9 t o p r o v i d e a t e s t o f th e v a r i o u s t h e - i
o r i e s . The l i n e s l a b e l l e d "Eq. 9 ” r e p r e s e n t th e s l o p e s !
i
c a l c u l a t e d a c c o r d i n g t o t h e Adamson, C obble and N i e ls e n
and W hitew ay, MacLennan an d C o f f i n t h e o r y . F o r sodium
c h l o r i d e
Djfa (cm2 / d a y ) * 1 .1 5 4 ( 1 - 0.586V 5) (9 a)
Dc l (cm2 / d a y ) = 1 .7 5 8 ( 1 - 0.586V c) (9b) j
i
! F o r sodium s u l f a t e '
D jjfa (cm 2/day) - 1 .1 5 4 ( 1 - Q.715-/c) (9c) |
• D gg^fcm ^/day) = 0 .9 1 9 ( 1 - 2 . 8 6 Vc) (9d)
! Only th e d i f f u s i o n c o e f f i c i e n t c u rv e f o r Cl~ in
sodium c h l o r i d e seems t o a p p ro a c h th e l i m i t i n g s lo p e eq
u a t i o n a t low c o n c e n t r a t i o n s . T h is l a c k o f g e n e r a l
1 a g re e m e n t may n o t d i s p r o v e th e t h e o r y f o r t h e v a r i a t i o n
o f m o b i l i t y w ith c o n c e n t r a t i o n was n e g l e c t e d in t h i s
i
i t h e o r e t i c a l t r e a t m e n t . The m o b i l i t y may be s t r o n g l y c o n -
i
c e n t r a t i o n d e p e n d e n t and hen ce c o u ld b r i n g t h e c u rv e s
i n t o a g re e m e n t. A t h e o r e t i c a l a p p ro a c h t o t h e f o r c e s
a c t i n g upon io n s m oving i n a medium w ith no c o n c e n t r a t i o n i
65
g r a d i e n t s i s n e e d e d . As t h e t h e o r y now s t a n d s th e a g r e e
m ent i s n o t s a t i s f a c t o r y .
The c u r v e s m arked "Eq. 10" p r o v i d e a c o m p a ris o n o f
t h e H u iz e n g a , G r i e g e r and W all t h e o r y w i t h e x p e r im e n t.
In e v e r y c a s e t h e t h e o r e t i c a l c u rv e l i e s below th e e x p e r i
m e n t a l . For t h e s u l f a t e i o n , how ever’, t h e a g re e m e n t may
b e c o n s i d e r e d f a i r . In view o f th e b a s i c d i f f e r e n c e b e
tw een c o n d u c ta n c e a n d s e l f d i f f u s i o n i t d o e s n o t seem p r o
f i t a b l e t o a tte m p t t o a p p ly c o r r e c t i o n s t o t h i s e q u a t i o n .
More s h o u ld b e g a in e d by r e c o n s i d e r i n g t h e b a s i c f a c t o r s
i n v o lv e d i n s e l f d i f f u s i o n .
| G o s tin g and H a r n e d 's t h e o r y i s r e p r e s e n t e d on F i g
u r e s 8 and 9 by t h e l i m i t i n g s l o p e s m arked "Eq. 1 1 " . The
n u m e r ic a l e q u a t i o n s f o r t h e i n d i v i d u a l io n s w ere c a l c u l a -
; t e d from e q u a t i o n ( 1 1 ) . For sodium c h l o r i d e
Djja (cm ^ /d ay ) = 1*154 - 0.23181^"
j DQiCcm^/day) « 1 .7 5 6 - 0.4543V5*
' F or sodium s u l f a t e
i Djja (cm ^ /d ay ) « 1 .1 5 4 - 0 .3 5 3 6 t€T
| DS04 ( cm3 / d ay) “ O * 0 ! 9 “ 0 .9 8 0 t 6 ”
! The e x p e r i m e n t a l c u rv e s f o r Na+ i n sodium c h l o r i d e ,
I Na* i n sodium s u l f a t e and SO^ i n sodium s u l f a t e a p p ro a c h
I
j t h e l i m i t i n g s l o p e s v e r y c l o s e l y and t h u s seem t o i n d i c -
! a t e t h e c o r r e c t n e s s of th e t h e o r y . The c u rv e f o r C l” i n
( 1 1 a)
( l i b )
( 1 1 c)
( l i d )
66
sodium c h l o r i d e d oes n o t f i t s o c l o s e l y . G o s tin g and
Harned^-^ a p p l i e d t h i s t h e o r y to Ua+ and 1 “ i n sodium i o
d id e a s m easured by Wang and Kennedy1^ and Ag+ i n s i l v e r
n i t r a t e a s m easu red by W hitew ay, MacLennan and C o f f i n 1 5 .
The c u rv e f o r E a+ i n sodium i o d i d e a p p ro a c h e d t h e l i m i t
in g s l o p e v e ry s a t i s f a c t o r i l y . The d a t a f o r Ag+ i n s i l
v e r n i t r a t e was so s c a t t e r e d a s t o make i t d i f f i c u l t to
d e f i n i t e l y p l a c e th e c u r v e , b u t G o s tin g and E arn ed con
s i d e r e d t h i s t o be a p o s s i b l e v e r i f i c a t i o n . The d a t a f o r
I " i n sodium i o d i d e , w hich w ere p r e c i s e , d ip p e d somewhat
belo w t h e l i m i t i n g s l o p e c u rv e b e f o r e r i s i n g t o t h e i n
f i n i t e d i l u t i o n v a l u e . From F ig u r e 8 i t can be se e n t h a t !
I
t h e c u rv e f o r C l” i n sodium c h l o r i d e l i k e w i s e d i p s belo w j
t h e l i m i t i n g s l o p e c u r v e . I n g e n e r a l , h ow ever, a p p a r e n t
a g re e m e n t h as b e e n fo u n d w ith t h e l i m i t i n g s l o p e p r e d i c t - :
ed by t h e O n sag er e q u a t i o n a s a p p l i e d by G o s tin g and H ar-
n e d . A greem ent h as b e e n o b s e r v e d f o r Ea+ i n sodium i o
d i d e , sodium c h l o r i d e and sodium s u l f a t e an d f o r s u l f a t e
i o n in sodium s u l f a t e . A p p a ren t l a c k o f a g re em e n t h as
b e e n o b se rv e d f o r I ” and Cl” i n sodium i o d i d e and sodium
c h l o r i d e .
I
B I B L I 0 G R A P H Y
BIBLIOGRAPHY
BOOKS
D. A. M a cln n e s, “P r i n c i p l e s o f E l e c t r o c h e m i s t r y . ” R e in
h o ld P u b l i s h i n g C o r p ., New Y ork, p . 339 ( 1 9 3 9 ).
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69 <
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UNPUBLISHED WORK

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Nielsen, Julian M (author)

Core Title A determination of ionic diffusion coefficients

Degree Doctor of Philosophy

Degree Program Chemistry

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag chemistry, inorganic,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Adamson, A.W. (committee chair), [illegible] (committee member), Copeland, C.S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-619861

Unique identifier UC11354654

Identifier DP21752.pdf (filename),usctheses-c17-619861 (legacy record id)

Legacy Identifier DP21752.pdf

Dmrecord 619861

Document Type Dissertation

Rights Nielsen, Julian M.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA