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A thermodynamic analysis of hydrogen in metals in the presence of an applied stress field
A
THERMODYNAMIC
ANALYSIS
PRESENCE
OF
J. O’M.
OF
AN
HYDROGEN
APPLIED
BOCKRISt
and
P.
K.
IN
STRESS
METALS
IN
THE
FIELD*
SUBRAMANYANQ
The thermodynamics of a two component system of metal-hydrogen, oriented towards obtaining a suitable relationship for the evaluation of the pmv of the mobile component (H), in presence of an externally applied uniform hydrostatic stress field has been considered in detail. In the present analysis the central difficulty of maintaining constancy of the amount of the mobile component (H) in the system for obtaining the pmv has been overcome, for those metals which have very small solubility of hydrogen. ANALYSE
THERMODYNAMIQUE PRESENCE
DE
DUN
L’HYDROGENE
CHAMP
DANS
DE CONTRAINTES
LES
METAUX
EN
EXTERNE
Pour un systema iL deux constituants, metal-hydrogene, on a fait l’etude thermodynamique du volume partiel moleculeire du constituent mobile en presence de constreintes hydrostatiques exterieurement appliquees. De cette analyse, la principale difficult6 d’avoir dans le syst6me une quantite constente en constituent mobile (H) a et6 resolve pour les metaux dans lesquels l’hydrogene est tres peu soluble. EINE
THERMODYNAMISCHE IN
ANALYSE
GENENWART
VON
EINES
WASSERSTOFF
SPANNUNGS
IN
Die thermodynamische Behandlung eines systemes von zwei Komponenten mit dem Ziel gemacht worden, das pertielmolar Volum des Wessertoffes, aus Der Vorteil Koncentratior% vom Buaseren hydrostatischen Drucke, abeuleiten. in diesem: die Schwierigkeit, die Gesamtzahl der Wrtsserstoffatome musste derungen konstant gehalten werden, ist im Betracht genommen worden. Die mit einem kleinen Losslichkeit von Wasserstoff beschriinkt.
INTRODUCTION
The response steel
to
of dissolved
applied
compressive)
elastic
is found
diffusing situation,
constant
hydrogen
stresses to
in iron
(both
tensile
be reversib1e.o~3)
temperature.
and
solubility of H (~1 ppm). Li et al. have considered
In
a
to apply
interstitial
The effect of stress acts on the solubility,
potential
dynamic
analysis
hydrogen
and metal under the action
of
a two-component
stress is more involved component
than that
to evaluate
the
system
of
of an applied
of a usual two-
gaseous or liquid system.
is oriented
and hence
The thermo-
partial
for iron
species
contribution
of hydrogen.
However,
easy
the elastic stress does not produce
potential,
(Metall, Wasserstoff) ist der Abhiingigkeit seines dieser An&se existiert wihrend der DruckiinAbleitung wird Metallen
and
any change in the diffusion coefficient of hydrogen.(2*3) on the chemical
METALLEN
FELDES
due to the small
the thermodynamics
in stressed
solids
in a valuable
species.(4)
may not be complete
However,
as independent
component
(i.e.
the
systems, metal)
their
in a thermodynamic
sense, because they treat the interstitial C, H)
of
and obtained expressions for the chemical
of the dissolved
analysis
this is not at all
or steel,
species (e.g.
leaving
out
of
the
other
consideration.
If the analysis
Because of this, certain conditions
molar
[e.g. those stated in the present paper by equations
volume
are not made clear
(pmv) of H in the metal by the application of an external pressure, one is struck with the very stringent
(14) and (26) below and the importance of the conditions originating from these] in obtaining a relation
condition
for the experimental
must able
that the total amount
be preserved to get
chemical
constant
a measurement with
of the [cf.
in its
equation
(13)]. of the of the
as a function
stress
change
be
An alternative approach to the determination pmv of H is to observe the change in volume system
potential
of H in the system
and yet one must
of the amount
of H in it at
* Received July 29, 1970. This paper represents work done by P. K. Subramanyan in partial fulfillment of the requirements for the Ph.D degree of the University of Pennsylvania, Philadelphia (1970). t The Electrochemistry Laboratory, the University of Pennsylvania, Philadelphia. $ Now at: Materiels Technology Division, Gould Laboratories, 640 East 105 Street, Cleveland, Ohio 44108. ACTA
METALLURGICA,
VOL.
19, NOVEMBER
1971
cally
meaningful
hydrogen, meet
the
hydrogen
evaluation pmv.
of a thermodynami-
Further,
with
they have not established condition content
stress to obtain
of
maintaining
in the system
constancy
during
an experimental
respect
to
how one could change
evaluation
of of
of the
pmv when the system is open with respect to hydrogen [cf. equations (13), (16) and (23)], present problem. In
the
following
we
present
the crux of the
a thermodynamic
analysis of metal-hydrogen systems under the action of an external non-shear stress, in which we have tried to meet these difficulties. 1205
1206
ACTA
METALLURGICA,
ANALYSIS
A given amount of metal (M) is equilibrated with hydrogen gas at constant pressure PHB, at a constant temperature T (Fig. 1). Suppose there arem*, g , atoms of M and nn g . atoms of dissolved hydrogen (H) in the metal-hydrogen system. The total volume V of the metal with dissolved hydrogen is:
VOL.
1971
Differentiating equation (6) with respect to all the variables except temperature, we get : nM
@M
+
PM
anEn, +
nH @H
+
fh3
dnH 2
=
nM
d_@M$-
i[
+
V = (mMv,%1+ nnvn)
19,
~hd~~v~-
TdflM-_KdvM
(
?
TflAr, - $
-@,+ohF,-
vM
i
(1)
?
dn,w
where p is the partial molar volume. Let the metalhydrogen alloy be subjected to elastic stresses. Under such stresses, a solid undergoes elastic deformation which produces changes in the energy content. Let o, represent the hydrostatic equivalent of the stresses acting on the metal-hydrogen alloy. In general, Oh =
-$(0,,
+
(TYY
+
cZZ)
+
(2)
where 0xX, uyy and ozz are the components of stress in the X, Y and Z directions acting on planes normal to these directions. The elastic energy stored in the material as a result of application of a stress system defined by equation (2) can be obtained a@)
~2nHlHdK+nHPHd~~-~nHVHdo,
?I (7)
Subtracting and adding n&&u dnM and nnpn dnn to the rhs of equation (7) and transferring the terms ,& dnM and pn dnn to the rhs, we get nM
w=$V
(
$M
+
nH
@H
(3)
where K is the bulk modulus of the alloy. The modified Euler equation for a solid system under stress can be written as:* E+o,B-TS-~p,ni-~V=6
(4)
Here E and S are the internal energy and entropy of the solid, pi is the chemical potential of the ith component, and ni the number of g . atoms of the ith component. For the two-component system of metal-hydrogen alloy, n.Wpu,W +
nHpH
=
E + a,V -
TR - g
V
(5)
Expanding the rhs of equation (5) in terms of partial properties,
-
$ (no VM +
nH
pEI)
(‘-4
* The effect of the action of the three shear stress components is to be separately shown in the Euler equation (4) to make it general. However, in the present treatment we restrict to the effects of non-shear stresses and, hence, the corresponding terms do not appear in the analysis.
For an isothermal reversible process, the sum of the terms in the first and fourth sets of simple brackets in equation (8) reduces to zero (cf. first law of thermodynamics). Again, the sum of the terms in the second
BOCKRIS
: HYDROGEN
AND SUBRAMANYAN
1
temperature and pressure
and fifth sets of simple brackets reduces to zero [cf. equation (4)]. Therefore, +M
+
nH
fJh2
-
@H
uh
- - ni+f~M da, + K
IzMpM
Oh
K
n
H vHaoh +
%pE
anH
(9’
Let us impose the following restrictions on the system. The amount of metal nM is conserved, i.e. dnM = 0. The amount of hydrogen in the system is allowed to be so small that it does not produce any change in the bulk modulus K. The applied stresses are below the elastic limit so that K is independent of stress. Hence,
= +
[(
nMVM
nHvH
dah
da,
+
-
Oh nMVM
K
nHpE
da,
1
an
(10) 11
or,
=
1E,.?Z2
= vi ,....
12,*...
(13)
T
Using equations (13) and (la), we eliminate terms containing pM from equation (12).
(15)
or,
(16)
When the interchange of H between metal and the surroundings occurs, as it will do on application of s, hydroststic stress to the system open with respect to hydrogen, the change in the chemical potential dpu produced by da, is compensated by adding or removing a certain amount of H equal to (a+&, where the subscript eq denotes the equilibrium value, i.e. 0. Therefore, when the system (apH/aoh),,,K,T = attains a new equilibrium, after the application of a stress, equation (15) becomes : = 0
[(l-g)nMVM+
(l-2)n,FH
-t%PH 5
ap
anM
gn ~',dK+n,~-,du, 2K2 a
+
($ >
Since the amount of H in the system is very much smaller than the amount of metal in the system, the effect of addition or removal of a very small amount, dnu, of H will leave ,uM unperturbed. Hence,
dK + nMTM da,,
n,vM
2K2
-(-
1207
FIELD
where V is the total volume of the system before the application of stress oh and (1 - oh/K) V, the volume of the system under the stress oh. By definition, the partial molar volume (pmv) of the ith component of a system is given byt6)
FIG. 1. Schematic representation of the metal-hydrogen system.
nM
A STRESS
H
H
H2
IN
M with
dissolved
Hz
METALS
where the subscripts on the lhs represent the conditions imposed on the system. Suppose that, in addition, we restrict the transfer of H out of, and into, the metal, we have dnn = 0. Hence,
_l-l, at constant
-Metal
IN
(dnH)I db,
(17)
or
(11)
(18)
ACTA
1208
METALLURGICA,
The chemical potential of H when c,, = 0 may be expressed as ,&.o = Eln” + RT ln
(19)
cH,O
(as the concentration C,,, is very small, the concentration is set equal to activity). However, when a stress is applied, ,%,o,, f
lUu” + RT ln
(20)
cH,O
After the system has been subjected to applied stress o,, the change in ,un at first caused by the stress [cf. equation (4)] is exactly compensated by the transfer of H from the surface where the chemical potential of H, ,uH.s, is invariant with respect to the application of stress,* until ~n,~ = ,un,o=o. As a result of this transfer of hydrogen, Cn,, changes to Cn,+. Therefore, the work SghdpH represents work done on the system by the external stress. The change of concentration of it which occurs within the metal to compensate this work then by the system is, hence, the negative of RT In G,.,h/Cn,o. Thus, Oh
d,u, = -RT
c ins
JO
(21)
cjH.~
However, this latter work must be equal to the change of free energy during transfer of it from or to the bulk, to or from the surface. This can be represented by: ,uH.s(dna),g. Differentiating equation (21) with respect to ch, we get:
VOL.
19,
1971
It will be noted that this equation does not involve the maintenance of constant nn, the total number of H atoms in the system. In the experimental measurements of the changes in concentration of H in the system, upon application of an external stress, there could be a contribution from the change of the chemical potential of the metal [which would be measured along with the change originating from (a&&s,)]. This arises because of the following equilibrium : (25)
H,+MsJf,H,
Since the solubility of H in Fe or steel is of the order of ppm, the effect arising from a small change in pM (with stress) on the solubility of H can be neglected. However, this situation may not exist in the case of metals which have a high solubility of H. Equation (25) holds true only if there is a source of H at constant chemical potential at the surface. By generating hydrogen electrochemically at a constant overpotential, the constancy of the fugacity of H adsorbed on the electrode or its chemical potential is easily obtained. Beck et aZ.t2)and Bockris et aLc3) have used equation (25) to evaluate the pmv of hydrogen in iron and steel.* The results obtained are consistent with the theory. The same magnitude of pmv is obtained whether it is calculated from the effect of compressive stresses or tensile stresses. ACKNOWLEDGEMENTS
(22) pn.s, the chemical potential of hydrogen atoms adsorbed on the surface of the metal, is equal to the chemical potential of H in the bulk when ch = 0, i.e. ,uH+,, = ,uH (before application of stress). Therefore, from equations (16), (17) and (22), we obtain
= RT
(23) nM.K,T
For a-iron, the maximum value of o, is 4 x IO* dyn/ cm2, while K = 1.67 x lOi dyn/cm2. Hence,
Thanks are due to the Naval-Air Engineering Centre for financial support under Contract No. NO0 156-67-C-1941; to Dr W. Beck of that Organization for discussion and stimulation; to Professor John G. Miller, University of Pennsylvania for critical discussion of some thermodynamic points; and to Dr. Richard Oriani, United States Steels for discussion. REFERENCES 1. F. DE KAZINCZY, Jernkont. An& 189,885(1955). 2. W. BECK, J. O’M. BOCKRIS, J. MCBREEN and L. NANIS, Proc. R. Sot. A!ZGQ, 220 (1966). 3. J. O’M. BOCKRIS, W. BECK, M. A. GENSHAW, P. K. SUBRAMANYAN 8ndF. S. WILLIAMS, ActaMetl9,1209 (1971). 4. J. M. C. LI, R. A. ORIANI and L. S. DARKEN, 2. phya. Chem. 49, 271 (1966). 5. N. H. POLAKOWSKI and E. J. RIPPLING, rStrength and Structure of Engineering Materials. Prentice-Hall (1966). 6. S. GLASTONE, Textbook Nostrand (1940).
RT
z=z
VH
(24)
?Z,.k’.T
* This implies that when hydrogen is generated electrolytically, thi rate constants -for -the hfdrogen evolution reaction are not affected bv stress.
* The measurements
of Physical Ckmistq.
by Beck
were the
D.
van
first measurements
in which V, was determined from the effect of stress upon Ca. It is not practical to obtain the partial molar volume of H in materials such as Fe by the X-ray method because the
volume expenslon 1s too smell.
THERMODYNAMIC
ANALYSIS
PRESENCE
OF
J. O’M.
OF
AN
HYDROGEN
APPLIED
BOCKRISt
and
P.
K.
IN
STRESS
METALS
IN
THE
FIELD*
SUBRAMANYANQ
The thermodynamics of a two component system of metal-hydrogen, oriented towards obtaining a suitable relationship for the evaluation of the pmv of the mobile component (H), in presence of an externally applied uniform hydrostatic stress field has been considered in detail. In the present analysis the central difficulty of maintaining constancy of the amount of the mobile component (H) in the system for obtaining the pmv has been overcome, for those metals which have very small solubility of hydrogen. ANALYSE
THERMODYNAMIQUE PRESENCE
DE
DUN
L’HYDROGENE
CHAMP
DANS
DE CONTRAINTES
LES
METAUX
EN
EXTERNE
Pour un systema iL deux constituants, metal-hydrogene, on a fait l’etude thermodynamique du volume partiel moleculeire du constituent mobile en presence de constreintes hydrostatiques exterieurement appliquees. De cette analyse, la principale difficult6 d’avoir dans le syst6me une quantite constente en constituent mobile (H) a et6 resolve pour les metaux dans lesquels l’hydrogene est tres peu soluble. EINE
THERMODYNAMISCHE IN
ANALYSE
GENENWART
VON
EINES
WASSERSTOFF
SPANNUNGS
IN
Die thermodynamische Behandlung eines systemes von zwei Komponenten mit dem Ziel gemacht worden, das pertielmolar Volum des Wessertoffes, aus Der Vorteil Koncentratior% vom Buaseren hydrostatischen Drucke, abeuleiten. in diesem: die Schwierigkeit, die Gesamtzahl der Wrtsserstoffatome musste derungen konstant gehalten werden, ist im Betracht genommen worden. Die mit einem kleinen Losslichkeit von Wasserstoff beschriinkt.
INTRODUCTION
The response steel
to
of dissolved
applied
compressive)
elastic
is found
diffusing situation,
constant
hydrogen
stresses to
in iron
(both
tensile
be reversib1e.o~3)
temperature.
and
solubility of H (~1 ppm). Li et al. have considered
In
a
to apply
interstitial
The effect of stress acts on the solubility,
potential
dynamic
analysis
hydrogen
and metal under the action
of
a two-component
stress is more involved component
than that
to evaluate
the
system
of
of an applied
of a usual two-
gaseous or liquid system.
is oriented
and hence
The thermo-
partial
for iron
species
contribution
of hydrogen.
However,
easy
the elastic stress does not produce
potential,
(Metall, Wasserstoff) ist der Abhiingigkeit seines dieser An&se existiert wihrend der DruckiinAbleitung wird Metallen
and
any change in the diffusion coefficient of hydrogen.(2*3) on the chemical
METALLEN
FELDES
due to the small
the thermodynamics
in stressed
solids
in a valuable
species.(4)
may not be complete
However,
as independent
component
(i.e.
the
systems, metal)
their
in a thermodynamic
sense, because they treat the interstitial C, H)
of
and obtained expressions for the chemical
of the dissolved
analysis
this is not at all
or steel,
species (e.g.
leaving
out
of
the
other
consideration.
If the analysis
Because of this, certain conditions
molar
[e.g. those stated in the present paper by equations
volume
are not made clear
(pmv) of H in the metal by the application of an external pressure, one is struck with the very stringent
(14) and (26) below and the importance of the conditions originating from these] in obtaining a relation
condition
for the experimental
must able
that the total amount
be preserved to get
chemical
constant
a measurement with
of the [cf.
in its
equation
(13)]. of the of the
as a function
stress
change
be
An alternative approach to the determination pmv of H is to observe the change in volume system
potential
of H in the system
and yet one must
of the amount
of H in it at
* Received July 29, 1970. This paper represents work done by P. K. Subramanyan in partial fulfillment of the requirements for the Ph.D degree of the University of Pennsylvania, Philadelphia (1970). t The Electrochemistry Laboratory, the University of Pennsylvania, Philadelphia. $ Now at: Materiels Technology Division, Gould Laboratories, 640 East 105 Street, Cleveland, Ohio 44108. ACTA
METALLURGICA,
VOL.
19, NOVEMBER
1971
cally
meaningful
hydrogen, meet
the
hydrogen
evaluation pmv.
of a thermodynami-
Further,
with
they have not established condition content
stress to obtain
of
maintaining
in the system
constancy
during
an experimental
respect
to
how one could change
evaluation
of of
of the
pmv when the system is open with respect to hydrogen [cf. equations (13), (16) and (23)], present problem. In
the
following
we
present
the crux of the
a thermodynamic
analysis of metal-hydrogen systems under the action of an external non-shear stress, in which we have tried to meet these difficulties. 1205
1206
ACTA
METALLURGICA,
ANALYSIS
A given amount of metal (M) is equilibrated with hydrogen gas at constant pressure PHB, at a constant temperature T (Fig. 1). Suppose there arem*, g , atoms of M and nn g . atoms of dissolved hydrogen (H) in the metal-hydrogen system. The total volume V of the metal with dissolved hydrogen is:
VOL.
1971
Differentiating equation (6) with respect to all the variables except temperature, we get : nM
@M
+
PM
anEn, +
nH @H
+
fh3
dnH 2
=
nM
d_@M$-
i[
+
V = (mMv,%1+ nnvn)
19,
~hd~~v~-
TdflM-_KdvM
(
?
TflAr, - $
-@,+ohF,-
vM
i
(1)
?
dn,w
where p is the partial molar volume. Let the metalhydrogen alloy be subjected to elastic stresses. Under such stresses, a solid undergoes elastic deformation which produces changes in the energy content. Let o, represent the hydrostatic equivalent of the stresses acting on the metal-hydrogen alloy. In general, Oh =
-$(0,,
+
(TYY
+
cZZ)
+
(2)
where 0xX, uyy and ozz are the components of stress in the X, Y and Z directions acting on planes normal to these directions. The elastic energy stored in the material as a result of application of a stress system defined by equation (2) can be obtained a@)
~2nHlHdK+nHPHd~~-~nHVHdo,
?I (7)
Subtracting and adding n&&u dnM and nnpn dnn to the rhs of equation (7) and transferring the terms ,& dnM and pn dnn to the rhs, we get nM
w=$V
(
$M
+
nH
@H
(3)
where K is the bulk modulus of the alloy. The modified Euler equation for a solid system under stress can be written as:* E+o,B-TS-~p,ni-~V=6
(4)
Here E and S are the internal energy and entropy of the solid, pi is the chemical potential of the ith component, and ni the number of g . atoms of the ith component. For the two-component system of metal-hydrogen alloy, n.Wpu,W +
nHpH
=
E + a,V -
TR - g
V
(5)
Expanding the rhs of equation (5) in terms of partial properties,
-
$ (no VM +
nH
pEI)
(‘-4
* The effect of the action of the three shear stress components is to be separately shown in the Euler equation (4) to make it general. However, in the present treatment we restrict to the effects of non-shear stresses and, hence, the corresponding terms do not appear in the analysis.
For an isothermal reversible process, the sum of the terms in the first and fourth sets of simple brackets in equation (8) reduces to zero (cf. first law of thermodynamics). Again, the sum of the terms in the second
BOCKRIS
: HYDROGEN
AND SUBRAMANYAN
1
temperature and pressure
and fifth sets of simple brackets reduces to zero [cf. equation (4)]. Therefore, +M
+
nH
fJh2
-
@H
uh
- - ni+f~M da, + K
IzMpM
Oh
K
n
H vHaoh +
%pE
anH
(9’
Let us impose the following restrictions on the system. The amount of metal nM is conserved, i.e. dnM = 0. The amount of hydrogen in the system is allowed to be so small that it does not produce any change in the bulk modulus K. The applied stresses are below the elastic limit so that K is independent of stress. Hence,
= +
[(
nMVM
nHvH
dah
da,
+
-
Oh nMVM
K
nHpE
da,
1
an
(10) 11
or,
=
1E,.?Z2
= vi ,....
12,*...
(13)
T
Using equations (13) and (la), we eliminate terms containing pM from equation (12).
(15)
or,
(16)
When the interchange of H between metal and the surroundings occurs, as it will do on application of s, hydroststic stress to the system open with respect to hydrogen, the change in the chemical potential dpu produced by da, is compensated by adding or removing a certain amount of H equal to (a+&, where the subscript eq denotes the equilibrium value, i.e. 0. Therefore, when the system (apH/aoh),,,K,T = attains a new equilibrium, after the application of a stress, equation (15) becomes : = 0
[(l-g)nMVM+
(l-2)n,FH
-t%PH 5
ap
anM
gn ~',dK+n,~-,du, 2K2 a
+
($ >
Since the amount of H in the system is very much smaller than the amount of metal in the system, the effect of addition or removal of a very small amount, dnu, of H will leave ,uM unperturbed. Hence,
dK + nMTM da,,
n,vM
2K2
-(-
1207
FIELD
where V is the total volume of the system before the application of stress oh and (1 - oh/K) V, the volume of the system under the stress oh. By definition, the partial molar volume (pmv) of the ith component of a system is given byt6)
FIG. 1. Schematic representation of the metal-hydrogen system.
nM
A STRESS
H
H
H2
IN
M with
dissolved
Hz
METALS
where the subscripts on the lhs represent the conditions imposed on the system. Suppose that, in addition, we restrict the transfer of H out of, and into, the metal, we have dnn = 0. Hence,
_l-l, at constant
-Metal
IN
(dnH)I db,
(17)
or
(11)
(18)
ACTA
1208
METALLURGICA,
The chemical potential of H when c,, = 0 may be expressed as ,&.o = Eln” + RT ln
(19)
cH,O
(as the concentration C,,, is very small, the concentration is set equal to activity). However, when a stress is applied, ,%,o,, f
lUu” + RT ln
(20)
cH,O
After the system has been subjected to applied stress o,, the change in ,un at first caused by the stress [cf. equation (4)] is exactly compensated by the transfer of H from the surface where the chemical potential of H, ,uH.s, is invariant with respect to the application of stress,* until ~n,~ = ,un,o=o. As a result of this transfer of hydrogen, Cn,, changes to Cn,+. Therefore, the work SghdpH represents work done on the system by the external stress. The change of concentration of it which occurs within the metal to compensate this work then by the system is, hence, the negative of RT In G,.,h/Cn,o. Thus, Oh
d,u, = -RT
c ins
JO
(21)
cjH.~
However, this latter work must be equal to the change of free energy during transfer of it from or to the bulk, to or from the surface. This can be represented by: ,uH.s(dna),g. Differentiating equation (21) with respect to ch, we get:
VOL.
19,
1971
It will be noted that this equation does not involve the maintenance of constant nn, the total number of H atoms in the system. In the experimental measurements of the changes in concentration of H in the system, upon application of an external stress, there could be a contribution from the change of the chemical potential of the metal [which would be measured along with the change originating from (a&&s,)]. This arises because of the following equilibrium : (25)
H,+MsJf,H,
Since the solubility of H in Fe or steel is of the order of ppm, the effect arising from a small change in pM (with stress) on the solubility of H can be neglected. However, this situation may not exist in the case of metals which have a high solubility of H. Equation (25) holds true only if there is a source of H at constant chemical potential at the surface. By generating hydrogen electrochemically at a constant overpotential, the constancy of the fugacity of H adsorbed on the electrode or its chemical potential is easily obtained. Beck et aZ.t2)and Bockris et aLc3) have used equation (25) to evaluate the pmv of hydrogen in iron and steel.* The results obtained are consistent with the theory. The same magnitude of pmv is obtained whether it is calculated from the effect of compressive stresses or tensile stresses. ACKNOWLEDGEMENTS
(22) pn.s, the chemical potential of hydrogen atoms adsorbed on the surface of the metal, is equal to the chemical potential of H in the bulk when ch = 0, i.e. ,uH+,, = ,uH (before application of stress). Therefore, from equations (16), (17) and (22), we obtain
= RT
(23) nM.K,T
For a-iron, the maximum value of o, is 4 x IO* dyn/ cm2, while K = 1.67 x lOi dyn/cm2. Hence,
Thanks are due to the Naval-Air Engineering Centre for financial support under Contract No. NO0 156-67-C-1941; to Dr W. Beck of that Organization for discussion and stimulation; to Professor John G. Miller, University of Pennsylvania for critical discussion of some thermodynamic points; and to Dr. Richard Oriani, United States Steels for discussion. REFERENCES 1. F. DE KAZINCZY, Jernkont. An& 189,885(1955). 2. W. BECK, J. O’M. BOCKRIS, J. MCBREEN and L. NANIS, Proc. R. Sot. A!ZGQ, 220 (1966). 3. J. O’M. BOCKRIS, W. BECK, M. A. GENSHAW, P. K. SUBRAMANYAN 8ndF. S. WILLIAMS, ActaMetl9,1209 (1971). 4. J. M. C. LI, R. A. ORIANI and L. S. DARKEN, 2. phya. Chem. 49, 271 (1966). 5. N. H. POLAKOWSKI and E. J. RIPPLING, rStrength and Structure of Engineering Materials. Prentice-Hall (1966). 6. S. GLASTONE, Textbook Nostrand (1940).
RT
z=z
VH
(24)
?Z,.k’.T
* This implies that when hydrogen is generated electrolytically, thi rate constants -for -the hfdrogen evolution reaction are not affected bv stress.
* The measurements
of Physical Ckmistq.
by Beck
were the
D.
van
first measurements
in which V, was determined from the effect of stress upon Ca. It is not practical to obtain the partial molar volume of H in materials such as Fe by the X-ray method because the
volume expenslon 1s too smell.