Hysteresis in Metal Hydrides An Illustration of Entropy Production Ted B. Flanagan Department of Chemistry, University of Vermont. Burlington. VT 05405 C. N. Park Department of Metallurgical Engineering, Chonnam Natlonal University. Kwangju, Korea D. H. Everett Department of Physical Chemistry, University of Bristol, Bristol, U.K. Hysteresis is a rather neglected phenomenon which, according to Neumann ( I ) , "has been avoided by chemists and physicists as a scientific curiosity". I t is, however, important in bioloeical Processes (2). nermanent maenetism (3). adsorptioi(4), A d solid s&ieactions (5).1n;rder to develop the thermodvnamics of hvsteresis we will emolov . . a metal hydride as the thermodynamic system because hysteresis is invariably observed during the formation and decomposition of the hydride phase. The existence of hysteresis requires that irreversibility be present, but, in contrast to other irreversible processes, the extent of the irreversibility is repeatable through many cycles of hysteresis. This paper will. we hone. serve the dual nurnoses of drawine attention to a reproduiible process where thk irreversibility-can be readiIv evaluated and of orovidine a thermodvnamic descrintion df the important, b i t neglectkd, phenomenon of hysteresis. Before discussina the thermodvnamics of hvsteresis. i t must be carefully defined ( 6 ) .when a process is-carried out dona a aiven ~ a t from h A to B hv- varvina . -the inde~endent variable, the dependent variable assumes a certa& set of values, hut, when the process is reversed by allowing the independent variable to assume the same set of values in the reverse sequence, the values of the dependent variable that are observed will differ from those found along the forward path from A to B, that is, it is impossible to go from B to A followine the exact reverse of the ~ a t from h A to B. All noints along b&h paths must corresp&d to stable, reproducible values. The system must be restored toits original state after completion of a hysteresis cycle. Irreversibility is an inherent feature of a system that exhibits hysteresis and, in contrast to other irreversible processes, reversible behavior cannot be obtained even if the chanees are carried out infinitesimallv slowly.
Hysteresis is shown for a real metal-hydrogen system, palladium-hydrogen, in Figure 1 (7). When hydrogen gas initially dissolves in the palladium lattice, it enters into solid solution (Fig. 1) until the hydride phase first appears a t pf. When more hydrogen enters the solid, the dilute phase converts into the hydride phase a t a constant hydrogen pressure of pfuntil the hydride phase fully forms. When the pressure exceeds pr, further hydrogen dissolves in the hydride phase and, for example, a t point B (Fig. l ) , a single-phase, nonstoichiometric hydride phase of composition H/Pd = 0.53 is formed that is in equilibrium with hydrogen a t 1atm (373 K). If the pressure is decreased starting from point B, the system does not follow the reverse of the forward pathway, but, instead, the hydride phase does not start to decompose until the pressure falls to pd (Fig. 1). The pressure then remains constant while the hydride and dilute phases coexist during the decomposition of the hydride at pd. The experimental results shown in Figure 1conform to the definition of hysteresis given above where the independent variable is taken as HE'd and the dependent variable as the hydrogen pressure. The cycle shown in Figure 1 is experimentally reproducible and independent of time. The thermodynamic treatment of hysteresis, which we will present below, does not depend upon the origin of hysteresis; however, its origin is of interest, and one of the authors and his co-workers have suggested (8)that i t is caused by plastic deformation, which is irreversible; the sample deforms plastically because of the need to accommodate the abrupt volume change that occurs on hydride formation or decomposition. For palladium-hydrogen (Fig. 1) the molar volume change for the dilute hydride phase transition is 10% (9). I t is easy tocalculate the entropy production for a hysteresis cycle using elementary thermodynamic relationships. Figure 2 shows a simplified, schematic hysteresis cycle for
-
1
Figure 1. Data of Steverts and Danr (71for paliadium-hydrogen at 373 K.
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a
HIM-
b
Fiaure 2. A schematic isothermal hvstereris cvcis fwa metal hvdride. For this scnemalc cycle lne phase ooundary composlt ons a and o for the dolute and nyarlde phases, resDectove y. are the same for hydr da lormat~onand decom position.
the isothermal formation and decomposition of a metal hydride; pe, is the equilibrium hydrogen pressure a t which the hydride phase would form and decompose reversibly, that is, if hysteresis were to he absent. I t is not directly attainable experimentally, but, fortunately, its precise location, somewhere between pd and p , is not needed for the thermodynamic treatment of the hysteresis cycle. The differential entropy change for the system, dS, is given by
where d,S, the entropy change sensed from heat exchange with the surroundings, is given hy dQ1Tand diS is the entropy internally produced within the system and which measures the irreversihility of the process (10). The differential entropy change of the universe, the system and surroundings, is equal to diS; however, the advantage of describing this irreversihility in terms of diS rather than hy the entropy change of the universe, is that with this treatment the origin of the irreversihility is correctly identified with the system. The system illustrated in Figure 2 is composed of 1mol of metal and 1mol of H where the latter is distributed hetween the gas phase as Hz and the solid phase as dissolved H. The system is connected to the surroundings by a frictionless piston and enclosed in a constant-temperature reservoir. The volume of the hydrogen in the gas phase is (1- r)RT/2p where r = Hlmetal atom ratio and p is the hydrogen pressure. The entropy production for the cycle $diS will now he 1, which evaluated using Figure 2. For steps 2 3 and 4 correspond to the reversible isothermal expansion and compression of the gas phase, respectively, AiS = 0. This can be shown by calculating ASS for each of these two steps and showing that these values are equal to A S and therefore AiS must be zero. This demonstration is unnecessary, however, because AiS must vanish for any step in the cycle which involves only reversihle changes of the gas phase such as 1; i t is assumed that the solid phase is steps 2 3 and 4 incompressible. This is a good assumption for the pressure range that we will consider, that is, the range where the hydrogen gas behaves ideally. The entropy changes for steps 1 2 and 3 4, which are the irreversihle steps of hydride formation and decomposition, can he evaluated in the usual manner by following alternative, reversihle pathways. For example, with reference to Figure 2,
AH(1-2) and AH(3-4) can also he evaluated by following reversihle pathways, that is
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For any step in the cycle (Fig. 2) the entropy production can he written from eq 1as From eq 13 together with eqs 5 and 11(or eqs 6 and 12) the entropy production for the two irreversible steps of hydride formation (or decomposition) can be written as
- -
- -
- -
-
the entropy and enthalpy changes for which are generallv negative. The hydride decomposition reaction is given by
and proceeding similarly for decomposition as we did to obtain eq 3 for the formation reaction, we obtain
A.S can be evaluatedfor these two steps from QIT, and eqs 7 and 8 are obtained from Q = AH, whichisvalid for a constant pressure process provided that any non-pV work, for example, plastic work, is fully converted into heat; this will he the case for hysteresis.
-
Since steps 1 2 and 3 4 are the only irreversible steps in the cycle, the entropy production for the cycle is given by
Since $dS for the cycle must be zero, i t follows from eq 1 that §d,S = -f djS= - X ( b
- 4 R 1" bfIpdl
(17)
There is a net entropy production in the hysteresis cycle given by eq 16 and since p f > pd the entropy produced in the cycle is always positive in the presence of hysteresis. Hysteresis requires that the net work done on the system he degraded into heat, which is transferred to the surroundings. Since the only changes which occur in steps 2 3 and 4 1are the reversihle, isothermal expansion and compression of the ideal gas, W = -Q for each of these steps. The work done on the system can he readily calculated for these isothermal, reversihle steps, that is,
-
-
-
where 1 A&, 1 = 1 AHe, I IT; the absolute value sign has heen used for clarity. Step 1 2 corresponds to hydride formation,
-
The enthalpy changes for the reversible expansion and compression of the gas, such as steps 2 3 and 4 1,are zero in the ideal pressure range. Using eqs 9 and 10, &S(1-2) and A,S(3-4) in eqs 7 and 8 can be expressed as
W(4-1)
= - 141pdv=X ( 1
- o)RT
C
dlnp
The net work done on the system and the heat transferred to the surroundings during the hysteresis cycle is given by eq 20 which follows from eqs 18 and 19 since the sum of the two 4, involves neither the irreversible steps, 1 2 and 3 transfer of heat nor work to the system.
-
-
fdW = X ( b - a)RT In bf/pdl = -fdQ
(20)
I t is surprising that the net work done on the system and the transfer of the equivalent amount of heat to the surroundings occurs during the reversihle steps, eqs 18 and 19, rather than during the two irreversihle steps. Some insight into this apparent incongruity can he obtained from the following arguments, which are based on Figure 2. Volume 64
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November 1967
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The expression to the right of the first equality sign can be rewritten using eq 13 as
where d i e is the counterpart to diS, that is, i t would be zero if the process were to be reversible. The total free energy change for the cycle is zero leading to the result that
+ AG(4-1)
AG(2-3)
which can be rearranged to
where the first expression in brackets is the entropy change of the system, which is zero for the cycle; the second expression in brackets is the entropy change of the surroundings. A,S(3-4) = From eqs 11and 12 i t follows that &S(1-2) 0 and therefore
+
-
Thus it can be seen from the above result that while the entropy production occurs in the irreversible steps, 1 2 and 3 4, the heat must he lost t o the surroundings during the reversible steps of gas expansion and compression, 2 3 and4-1. Finally, in order to complete the description of hysteresis the Gibbs free energy changes for the various steps and for the cycle will be evaluated. For the two irreversible steps of hydride formation and decomposition we obtain from reversible pathways (Fig. 2)
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-
and for the complete cycle
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Journal of Chemical Education
= %(b - a ) R T in [pl/pd]
(24)
I t can be seen from eqs 14 and 15, or 21 and 22, that the entropy production, or free energy dissipation, cannot be calculated for either of the irreversible steps of hydride formation or decomposition without knowledge of the position of the equilibrium pressure (Fig. 2); however, the corresponding quantities for the complete cycle are seen to he independent of the position of equilibrium (eqs 16 and 23). Conclusions
The thermodynamic characteristics of hysteresis have been illustrated using a metal hydride as the system that exhibits hvsteresis. Hvsteresis. which is an interestine and importantphenomen&, is reproducible and the irreve&bilitv has been determined from the entroov in the . -~roduction . sjstem during hysteresis cycles. Acknowledgment
TBF wishes to thank the N.S.F. and CNP wishes to thank KOSEF for financial support of their research on metal hydrides and hysteresis. Literature Cited
1973,12.356.
1. Neurnsnn. E. Anpew. Chom. 1nfer.Ed. 2. Kstchalaky,A. 1nRiology ondthePhysimlSeienres:Dwons.S., New York, 1969; p 3. Barker. J. A ; Schreiber. E. E.; Huth, B. Everett, 0.H. Pmc. Roy.
269.
Ed.;ColurnbiaUniv.: Soc. A 1983.386.
0.: ?KT 4. Everett, D. H.; Whitton, W.I. R o c . Roy S o c A 1955,230,91. 5. Lnare,A.T.:Eyring,L.J. SolidSLofe Chem. 1975.11,383. 6. Everetf,D. H.: Whitton, W.I. Trans.Forodoy Soc. 1952,48,749. 7. Sieverts,A.;Oanz, W.Zeil.phyaL.Chem. 1937.34R. 46. S. Flsnapan,T. B.;Bowerrnm,B.S.;Biehl,G.E.Seripta Met. 1980,14,443. 9. Lewis. F. A. ThoPollodium Hydrogen System; Academic: New Yark. 1967. 10 P~ieagim.1.: Delay. R.:Evereft. D. H.,Translstor.Chsmieol Thermody~mier:Langmans Green: London. 1954.
