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Reply to comments on “A thermodynamic analysis of hydrogen in metals in the presence of an applied stress field”
Scripta
METALLURGICA
Vol. 6, pp. 9 4 7 - 9 5 4 , 1972 P r i n t e d in the U n i t e d States
Pergamon
Press,
R E P L Y TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H Y D R O G E N IN M E T A L S IN T H E P R E S E N C E OF AN A P P L I E D S T R E S S F I E L D "
J.O'M. 8ockris School of Physical Sciences, The Flinders University, Adelaide, Australia. P.K. Subramanyan Materials Technology Department, Gould Laboratories, Cleveland, Ohio, U.S.A.
(Received I.
August
16,
1972)
INTRODUCTION The partial molar volume of hydrogen in metals relates
the solubility at stress, a, to the solubility at zero stress through the equation Sa = S o e~°/RT.
A numerical value of it is hence the
foundation of knowledge concerning the permeation proportional to S o of H into metals at local stress points. The partial molar volume of H was firstly related to the variation of the solubility with stress by Beck, Bockris, Nannis and McBreen in 1966*(1);and rededuced by Bockris and Subramanyan in 1970 (2) . Parts of this latter deduction have been criticized by Chopra and Li (3). The assumed independence of the rate constant of the hydrogen evolution reaction on stress has been questioned (4).
* The numerical evaluation of pmv from thisf~lationship was later corrected for inversion of the factor 2.303 "-~.
947
Inc
948
REPLY
If.
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
Vol.
EARLIER DEDUCTIONS OF VH In 1966, Beck et al (I) published a derivation of the
expression:
---RT
d%
b u t t h e d e d u c t i o n was s i m p l i s t i c ,
j
and d i d n o t d e f i n e w e l l c o n d i t i o n s
under which t h e e x p r e s s i o n i s a p p l i c a b l e . Li, Oriani and Barken discussed expressions for the chemical potential of dissolved species in solids, as a function of stress (6)
They stopped before deriving a specific ~xplession
relating the partial molar volume to the solubility as a function of stress.
III.
~THODOLOGY OF THE 1970 DEDUCTION OF BOCKRIS AND SUBRAMANYAN (2) Bockris and Subramanyan write an expression for the energy
of a multi-component system as a function of stress.
Equations are
developed to obtain the change in free energy of a metal in which is dissolved hydrogen, upon reversible application of stress.
A condi-
tion is deduced whereby the change of chemical potential of the metal with stress can be eliminated.
Thereafter, the work is calculated to
take the necessary quantity of hydrogen at constant overpotential from or to the surface of the metal (to or from the bulk, respectively) to compensate the change of the chemical potential of hydrogen in the bulk, due to the effects of stress. Equilibrium at the surface with the bulk is assumed, and the effect of conditions of constant overpotential is taken into account.
6, NO.
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Vol.
6, No.
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REPLY
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
949
The concentration of hydrogen in the metal changes with change of stress as if the external "pressure" of hydrogen remains constant. The result obtained is:
d°h
JN, K,T :(l
H
I t i s s i m p l e t o show t h a t Oh/K << 1. The d e r i v a t i o n
shows t h a t c o n d i t i o n s t o make t h e
equation v a l i d are constancy of o v e r p o t e n t i a l ;
zero change o f r a t e
c o n s t a n t o f t h e h y d r o g e n e v o l u t i o n r e a c t i o n due t o s t r e s s ; sufficiently
IV.
low s o l u b i l i t y
and a
o f hydrogen in the metal.
CRITICISM OF CHOPRA AND LI We reply on a point to point basis: o2 h (I) The term ~ V is not an unnecessary quantity.
Since
E, S and V in our equations represent the internal energy, entropy and volume in the unstressed state, the term representing the strain energy of deformation is a legitimate one.
Such a term does not
appear in Gibbs's equations because his reference state is the stressed state (7) . Q
However, a similar term does appear in the work
of Li, Oriani and Darken (6). oh The factor (i - K ~ This can be proved as below:
takes care of the compressibility, B.
950
REPLY TO COMMENTS ON "A THERMODYNAMIC ANALYSIS OF H IN METALS" 1
dV
Vol.
1
Hence: (1
.
°h ~ K') . .
[i .
{_ i .dV= . "O "V V t'~h'h)Tr hJ = -V°'h"
dr" [V + (~hh)TOh] -- Voh.
Where Voh i s the v a l u e o f t h e pmv a t a s t r e s s (2)
That a chemical p o t e n t i a l
l e v e l o f o h.
o f an immobile component
can be d e f i n e d at the s u r f a c e o n l y under a h y d r o s t a t i c is incorrect.
It arises
from an o v e r l i b e r a l
e q u a t i o n s 393 to 395 o f Gibbs ( 7 ) . conditions of precipitation
e x t e n s i o n o f the
These e q u a t i o n s s p e c i f y the
o f s o l i d from a s u p e r s a t u r a t e d s o l u t i o n
on t h e s u r f a c e o f a n o n - h y d r o s t a t i c a l l y n o t i n v o l v e the chemical p o t e n t i a l
stressed
solid.
of the triaxially
n o r o f a component d i s s o l v e d in i t and i t s
They do
stressed
solid,
surface activity.
o t h e r hand, t h e y only c o n t a i n the chemical p o t e n t i a l precipitated
stress,
On t h e
o f t h e newly
s o l i d which i s s u b j e c t e d to the h y d r o s t a t i c
stress
of
the s o l u t i o n . (3) in the first
The o b j e c t i o n o f Chopra and Li t h a t t h e sum o f terms
and f o u r t h s e t s o f s i m p l e b r a c k e t s o f (2) does n o t reduce
to z e r o i s wrong.
The f i r s t
and second laws o f thermodynamics
a p p l i e d to an i s o t h e r m a l r e v e r s i b l e r e d u c e s to z e r o .
p r o c e s s does i n d e e d show t h e sum
The term (o2/2K)V c o r r e s p o n d s t o e n t r o p y (as s t r a i n -
ing the s o l i d i s c a r r i e d out
isothermally
and r e v e r s i b l y ) .
Hence,
6, No.
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6
No.
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REPLY
'
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
) (~/2K)dV corresponds to a change in entropy and is additive with TdS (of the unstressed condition). a point here°
However, we do have to clarify
We added and subtracted terms like nM~ M dn M and
ntt~tt dn H in e q u a t i o n (8) o f o u r p a p e r .
These dn M and dn H s h o u l d
have been r e p r e s e n t e d by dn~ and dn~ b e c a u s e they r e p r e s e n t change i n number o f mole and are d i f f e r e n t from d i f f e r e n t i a t i o n and dn H = n t t dn~l. (4) infer
from dn M and dntt r e s u l t i n g
o f n M and n H by a f a c t o r o f n o r dn M - n M dn~ After equation (8),
only dn' a p p e a r s .
I t i s Chopra and Li who are i n c o r r e c t when t h e y
t h a t one cannot w r i t e a c h e m i c a l p o t e n t i a l
e n e r g y terms i n thermodynamic e q u a t i o n s .
along with o t h e r
It is a standard procedure
i n thermodynamics to i n c l u d e such t e r m s a s , e . g o , g r a v i t a t i o n a l energy, surface energy, product of electric density,
along with chemical p o t e n t i a I o
potential
and c h a r g e
The term ~Hs(dn~)eq i s
a measure o f t h e f r e e e n e r g y change o c c u r r i n g when one mole o f d i s s o l v e d hydrogen s u f f e r s
an a p p l i c a t i o n
moles o f H p e r mole t r a n s p o r t e d t h e bulk to t h e s u r f a c e ) (5)
of stress.
(dn~)eq i s t h e number o f
( i n t o t h e b u l k from t h e s u r f a c e o r from
to a t t a i n
e q u i l i b r i u m a t t h e new s t r e s s .
A previous satisfactory derivation of V H in terms
of stress effects on permeation has not been given.
Equation (2) of
Chopra and Li does not bring out the conditions to be imposed for the equation to be valid.
For example, it does not point out the condition
of maintaining constant the chemical potential of H 2 outside the solid metal during the application of stress. A correct criticism of Chopra and Li is that there is a change in sign in our equation (24), which occurred in our analysis at equation
951
952
REPLY TO COMMENTS ON "A THERMODYNAMIC ANALYSIS OF H IN METALS"
Vol.
6, No.
(21), where we omitted to take account of the fact o h is negative when is positive. Further on differentiation
of their equation (3) with
respect to ~h:
d -
RT
in iC--~?-~I
ida. 1
dh
Now:
%t
J
where 8 is the compressibility. Hen ce:
l
Therefore,
d
RT
1
ln/c~ I dC~h
which is our equation (24).
Since ah/K is << i, we put it equal to
~H (in the unstressed condition).
However, we consider Chopra and
Li err here, because they may not consistently differentiate ~H with respect to (~h because in (6) the integration f(~h ~H d(~h was carried o out by assuming that ~H was incompressible.
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6, No.
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REPLY
(6)
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
Our fuller analysis cannot be adequately replaced
by the intuitive equations (2) and (3) which Chopra and Li write. The happenings occurring involve conditions for V and nH, which are not discussed in the very simple approach which Chopra and Li advocate.
Vo
TO WHAT EXTENT DOES THE RATE CONSTANT OF THE HYDROGEN EVOLUTION
REACTION DEPEND UPON STRESS? The proof given by Bockris and Subramanyan implicitly assumes that there is zero variation of the rate constant of hydrogen evolution with stress. Despic et al (8) observed that at potentials close to the corrosion potential of ~ - Fe in H2SO 4 solution (ncath = - 0.002 volt), the rate of hydrogen evolution increased on application of an elastic tensile stress.
They considered this as a catalytic effect of stress
on the hydrogen evolution reaction (h.e.r.).
Based on this, Townsend (4)
suggested that the increase in permeation of hydrogen on application of a tensile stress may simply be due to the catalytic effect of stress on the hydrogen evolution reaction.
The possible contribution of this
effect to the pmv of hydrogen can be examined as follows:
The rate of
the h.e.r, on ~-Fe in H2SO 4 at ~cath = - 0.002 volt is 56Vamp/cm -2. On application of a stress of 3000 Kg/cm 2, it increases to 58~amp/cm -2. (Figs. 2 and 3 of Despic et al). - 0.OO18 volt of potential.
This is equivalent to an additional
The stress permeation study of Bockris et
al (8) was carried out in a NaOH ambient.
Were the work of Despic et
9S3
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REPLY
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
a l i n H2SO4 and h i g h e r s t r e s s
l e v e l s to apply t o i t ,
Vol.
then in t h e
dncath c u r r e n t d e n s i t y range o f t h e p e r m e a t i o n s t u d y , d In P = - (O.6 to 0 . 7 ) v o l t where P •i s t h e p e r m e a b i l i t y .
For ~ = 3000 Kg/cm2 , dDcath =
P ~---- 7.10 - 6 . o C o r r e s p o n d i n g l y , ~H i s t h e n g i v e n by:
4.10 -6 v o l t .
Therefore, din
Here ~oh = - 1OOO Kg/cm2. in ~-Fe i s 2°66cc.
The v a I u e o f the pmv o f hydrogen a t 27°C
Hence, t h e p o s s i b l e e r r o r a r i s i n g
of stress
on t h e r a t e c o n s t a n t i s n e g l i g i b l e .
of stress
on r a t e a t c o n s t a n t o v e r p o t e n t i a l
the permeation studies
from an e f f e c t
No d e t e c t a b l e
effect
was, however, o b s e r v e d in
carried out.
ACKNOWLEDGMENT The authors acknowledge helpful discussion of the work of Chopra and Li with Professor JoGo Miller, University of Pennsylvania. REFERENCES:
(1)
W. Beck, JoO'M. B o c k r i s ,
J . McBreen and L. N a n i s , Proc. Roy°Soc°, A290, 220 (1966).
(2)
J.O'M. Bockris and P.Ko Subramanyan, Acta M e t . , 1_99, 1205 (1971).
(3)
R. Chopra and JoCoM° L i , Acta M e t . , 20, J u l y (1972).
(4)
H.Eo Townsend Jr., Corrosion, 26, 361 (1970).
(5)
R°A. Oriani, Trans° AIME, 236, 1368 (1966).
(6)
J.C.Mo Li, R.Ao Oriani and LoS. Darken, Z.Physik. Chem. N°F., 49, 271 (1966).
(7)
JoW° Gibbs, "The Collected Works", Vol° 1 (Yale University Press, New Haven, Conn., 1948) .
(8)
A°R. Despic, R.Go Raicheff and J°O'M° Bockris, J.Chem. Phys.,
4._9.9,926 (1968).
6, No.
i0
METALLURGICA
Vol. 6, pp. 9 4 7 - 9 5 4 , 1972 P r i n t e d in the U n i t e d States
Pergamon
Press,
R E P L Y TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H Y D R O G E N IN M E T A L S IN T H E P R E S E N C E OF AN A P P L I E D S T R E S S F I E L D "
J.O'M. 8ockris School of Physical Sciences, The Flinders University, Adelaide, Australia. P.K. Subramanyan Materials Technology Department, Gould Laboratories, Cleveland, Ohio, U.S.A.
(Received I.
August
16,
1972)
INTRODUCTION The partial molar volume of hydrogen in metals relates
the solubility at stress, a, to the solubility at zero stress through the equation Sa = S o e~°/RT.
A numerical value of it is hence the
foundation of knowledge concerning the permeation proportional to S o of H into metals at local stress points. The partial molar volume of H was firstly related to the variation of the solubility with stress by Beck, Bockris, Nannis and McBreen in 1966*(1);and rededuced by Bockris and Subramanyan in 1970 (2) . Parts of this latter deduction have been criticized by Chopra and Li (3). The assumed independence of the rate constant of the hydrogen evolution reaction on stress has been questioned (4).
* The numerical evaluation of pmv from thisf~lationship was later corrected for inversion of the factor 2.303 "-~.
947
Inc
948
REPLY
If.
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
Vol.
EARLIER DEDUCTIONS OF VH In 1966, Beck et al (I) published a derivation of the
expression:
---RT
d%
b u t t h e d e d u c t i o n was s i m p l i s t i c ,
j
and d i d n o t d e f i n e w e l l c o n d i t i o n s
under which t h e e x p r e s s i o n i s a p p l i c a b l e . Li, Oriani and Barken discussed expressions for the chemical potential of dissolved species in solids, as a function of stress (6)
They stopped before deriving a specific ~xplession
relating the partial molar volume to the solubility as a function of stress.
III.
~THODOLOGY OF THE 1970 DEDUCTION OF BOCKRIS AND SUBRAMANYAN (2) Bockris and Subramanyan write an expression for the energy
of a multi-component system as a function of stress.
Equations are
developed to obtain the change in free energy of a metal in which is dissolved hydrogen, upon reversible application of stress.
A condi-
tion is deduced whereby the change of chemical potential of the metal with stress can be eliminated.
Thereafter, the work is calculated to
take the necessary quantity of hydrogen at constant overpotential from or to the surface of the metal (to or from the bulk, respectively) to compensate the change of the chemical potential of hydrogen in the bulk, due to the effects of stress. Equilibrium at the surface with the bulk is assumed, and the effect of conditions of constant overpotential is taken into account.
6, NO.
i0
Vol.
6, No.
I0
REPLY
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
949
The concentration of hydrogen in the metal changes with change of stress as if the external "pressure" of hydrogen remains constant. The result obtained is:
d°h
JN, K,T :(l
H
I t i s s i m p l e t o show t h a t Oh/K << 1. The d e r i v a t i o n
shows t h a t c o n d i t i o n s t o make t h e
equation v a l i d are constancy of o v e r p o t e n t i a l ;
zero change o f r a t e
c o n s t a n t o f t h e h y d r o g e n e v o l u t i o n r e a c t i o n due t o s t r e s s ; sufficiently
IV.
low s o l u b i l i t y
and a
o f hydrogen in the metal.
CRITICISM OF CHOPRA AND LI We reply on a point to point basis: o2 h (I) The term ~ V is not an unnecessary quantity.
Since
E, S and V in our equations represent the internal energy, entropy and volume in the unstressed state, the term representing the strain energy of deformation is a legitimate one.
Such a term does not
appear in Gibbs's equations because his reference state is the stressed state (7) . Q
However, a similar term does appear in the work
of Li, Oriani and Darken (6). oh The factor (i - K ~ This can be proved as below:
takes care of the compressibility, B.
950
REPLY TO COMMENTS ON "A THERMODYNAMIC ANALYSIS OF H IN METALS" 1
dV
Vol.
1
Hence: (1
.
°h ~ K') . .
[i .
{_ i .dV= . "O "V V t'~h'h)Tr hJ = -V°'h"
dr" [V + (~hh)TOh] -- Voh.
Where Voh i s the v a l u e o f t h e pmv a t a s t r e s s (2)
That a chemical p o t e n t i a l
l e v e l o f o h.
o f an immobile component
can be d e f i n e d at the s u r f a c e o n l y under a h y d r o s t a t i c is incorrect.
It arises
from an o v e r l i b e r a l
e q u a t i o n s 393 to 395 o f Gibbs ( 7 ) . conditions of precipitation
e x t e n s i o n o f the
These e q u a t i o n s s p e c i f y the
o f s o l i d from a s u p e r s a t u r a t e d s o l u t i o n
on t h e s u r f a c e o f a n o n - h y d r o s t a t i c a l l y n o t i n v o l v e the chemical p o t e n t i a l
stressed
solid.
of the triaxially
n o r o f a component d i s s o l v e d in i t and i t s
They do
stressed
solid,
surface activity.
o t h e r hand, t h e y only c o n t a i n the chemical p o t e n t i a l precipitated
stress,
On t h e
o f t h e newly
s o l i d which i s s u b j e c t e d to the h y d r o s t a t i c
stress
of
the s o l u t i o n . (3) in the first
The o b j e c t i o n o f Chopra and Li t h a t t h e sum o f terms
and f o u r t h s e t s o f s i m p l e b r a c k e t s o f (2) does n o t reduce
to z e r o i s wrong.
The f i r s t
and second laws o f thermodynamics
a p p l i e d to an i s o t h e r m a l r e v e r s i b l e r e d u c e s to z e r o .
p r o c e s s does i n d e e d show t h e sum
The term (o2/2K)V c o r r e s p o n d s t o e n t r o p y (as s t r a i n -
ing the s o l i d i s c a r r i e d out
isothermally
and r e v e r s i b l y ) .
Hence,
6, No.
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Vol.
6
No.
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REPLY
'
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
) (~/2K)dV corresponds to a change in entropy and is additive with TdS (of the unstressed condition). a point here°
However, we do have to clarify
We added and subtracted terms like nM~ M dn M and
ntt~tt dn H in e q u a t i o n (8) o f o u r p a p e r .
These dn M and dn H s h o u l d
have been r e p r e s e n t e d by dn~ and dn~ b e c a u s e they r e p r e s e n t change i n number o f mole and are d i f f e r e n t from d i f f e r e n t i a t i o n and dn H = n t t dn~l. (4) infer
from dn M and dntt r e s u l t i n g
o f n M and n H by a f a c t o r o f n o r dn M - n M dn~ After equation (8),
only dn' a p p e a r s .
I t i s Chopra and Li who are i n c o r r e c t when t h e y
t h a t one cannot w r i t e a c h e m i c a l p o t e n t i a l
e n e r g y terms i n thermodynamic e q u a t i o n s .
along with o t h e r
It is a standard procedure
i n thermodynamics to i n c l u d e such t e r m s a s , e . g o , g r a v i t a t i o n a l energy, surface energy, product of electric density,
along with chemical p o t e n t i a I o
potential
and c h a r g e
The term ~Hs(dn~)eq i s
a measure o f t h e f r e e e n e r g y change o c c u r r i n g when one mole o f d i s s o l v e d hydrogen s u f f e r s
an a p p l i c a t i o n
moles o f H p e r mole t r a n s p o r t e d t h e bulk to t h e s u r f a c e ) (5)
of stress.
(dn~)eq i s t h e number o f
( i n t o t h e b u l k from t h e s u r f a c e o r from
to a t t a i n
e q u i l i b r i u m a t t h e new s t r e s s .
A previous satisfactory derivation of V H in terms
of stress effects on permeation has not been given.
Equation (2) of
Chopra and Li does not bring out the conditions to be imposed for the equation to be valid.
For example, it does not point out the condition
of maintaining constant the chemical potential of H 2 outside the solid metal during the application of stress. A correct criticism of Chopra and Li is that there is a change in sign in our equation (24), which occurred in our analysis at equation
951
952
REPLY TO COMMENTS ON "A THERMODYNAMIC ANALYSIS OF H IN METALS"
Vol.
6, No.
(21), where we omitted to take account of the fact o h is negative when is positive. Further on differentiation
of their equation (3) with
respect to ~h:
d -
RT
in iC--~?-~I
ida. 1
dh
Now:
%t
J
where 8 is the compressibility. Hen ce:
l
Therefore,
d
RT
1
ln/c~ I dC~h
which is our equation (24).
Since ah/K is << i, we put it equal to
~H (in the unstressed condition).
However, we consider Chopra and
Li err here, because they may not consistently differentiate ~H with respect to (~h because in (6) the integration f(~h ~H d(~h was carried o out by assuming that ~H was incompressible.
i0
Vol.
6, No.
I0
REPLY
(6)
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
Our fuller analysis cannot be adequately replaced
by the intuitive equations (2) and (3) which Chopra and Li write. The happenings occurring involve conditions for V and nH, which are not discussed in the very simple approach which Chopra and Li advocate.
Vo
TO WHAT EXTENT DOES THE RATE CONSTANT OF THE HYDROGEN EVOLUTION
REACTION DEPEND UPON STRESS? The proof given by Bockris and Subramanyan implicitly assumes that there is zero variation of the rate constant of hydrogen evolution with stress. Despic et al (8) observed that at potentials close to the corrosion potential of ~ - Fe in H2SO 4 solution (ncath = - 0.002 volt), the rate of hydrogen evolution increased on application of an elastic tensile stress.
They considered this as a catalytic effect of stress
on the hydrogen evolution reaction (h.e.r.).
Based on this, Townsend (4)
suggested that the increase in permeation of hydrogen on application of a tensile stress may simply be due to the catalytic effect of stress on the hydrogen evolution reaction.
The possible contribution of this
effect to the pmv of hydrogen can be examined as follows:
The rate of
the h.e.r, on ~-Fe in H2SO 4 at ~cath = - 0.002 volt is 56Vamp/cm -2. On application of a stress of 3000 Kg/cm 2, it increases to 58~amp/cm -2. (Figs. 2 and 3 of Despic et al). - 0.OO18 volt of potential.
This is equivalent to an additional
The stress permeation study of Bockris et
al (8) was carried out in a NaOH ambient.
Were the work of Despic et
9S3
954
REPLY
TO C O M M E N T S ON "A T H E R M O D Y N A M I C A N A L Y S I S OF H IN M E T A L S "
a l i n H2SO4 and h i g h e r s t r e s s
l e v e l s to apply t o i t ,
Vol.
then in t h e
dncath c u r r e n t d e n s i t y range o f t h e p e r m e a t i o n s t u d y , d In P = - (O.6 to 0 . 7 ) v o l t where P •i s t h e p e r m e a b i l i t y .
For ~ = 3000 Kg/cm2 , dDcath =
P ~---- 7.10 - 6 . o C o r r e s p o n d i n g l y , ~H i s t h e n g i v e n by:
4.10 -6 v o l t .
Therefore, din
Here ~oh = - 1OOO Kg/cm2. in ~-Fe i s 2°66cc.
The v a I u e o f the pmv o f hydrogen a t 27°C
Hence, t h e p o s s i b l e e r r o r a r i s i n g
of stress
on t h e r a t e c o n s t a n t i s n e g l i g i b l e .
of stress
on r a t e a t c o n s t a n t o v e r p o t e n t i a l
the permeation studies
from an e f f e c t
No d e t e c t a b l e
effect
was, however, o b s e r v e d in
carried out.
ACKNOWLEDGMENT The authors acknowledge helpful discussion of the work of Chopra and Li with Professor JoGo Miller, University of Pennsylvania. REFERENCES:
(1)
W. Beck, JoO'M. B o c k r i s ,
J . McBreen and L. N a n i s , Proc. Roy°Soc°, A290, 220 (1966).
(2)
J.O'M. Bockris and P.Ko Subramanyan, Acta M e t . , 1_99, 1205 (1971).
(3)
R. Chopra and JoCoM° L i , Acta M e t . , 20, J u l y (1972).
(4)
H.Eo Townsend Jr., Corrosion, 26, 361 (1970).
(5)
R°A. Oriani, Trans° AIME, 236, 1368 (1966).
(6)
J.C.Mo Li, R.Ao Oriani and LoS. Darken, Z.Physik. Chem. N°F., 49, 271 (1966).
(7)
JoW° Gibbs, "The Collected Works", Vol° 1 (Yale University Press, New Haven, Conn., 1948) .
(8)
A°R. Despic, R.Go Raicheff and J°O'M° Bockris, J.Chem. Phys.,
4._9.9,926 (1968).
6, No.
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