Statistical Thermodynamic Analysis for Isothermal Hydrogenation Performances of Mg 2-y Pr y Ni 4 Intermetallics ( y = 0 . 6 , 0 . 8 , 1 . 0 )

Isothermal hydrogenation performances of intermetallic Mg2-yPryNi4 alloys with y = 0.6, 0.8 and 1.0 reported by Terashitaet al.were analyzed on the basis of statistical thermodynamics under a simplifyinga priori assumption of constant nearest neighbourH-H interactionE(H-H) in a g iven phase at arbitrary T aiming at characterizing basic aspects of state of H atoms in the interstitial sites in H-storage alloy. To fulfill this a priori assumption, number  of available interstitial sites per metal atom was chosen by preliminary search attempt at the onset of the statistical thermodynamic analysis. Primary H solution in Mg2-yPryNi4 was analyzed by the model with  = 0.15. The chosen  value 0.15 for the model analysis was close to be 1/6 (≈ 0.167) which was half o f 1/3 (=[Mg + Pr]/[Mg + Pr + Ni])implying that about half of the (Mg + Pr) -related interstitial sites were provided as the available sites for occupation by H atoms in the primary H solution of Mg 2-yPryNi4. On the other hand, hypo-stoichiometric M4H3 type hydride of Mg2-yPryNi4 was analyzed by the model with  = 0.75 and  ' = 0.333 where ' refers to the lower limit ing composition of the phase. This model yielded situation with E(H-H) = 0 for any Mg2-yPryNi4examined. Chosen value of  ' = 0.333 appeared to imply that the filling of Ni-related interstitial sites by H atoms started after preferential full occupation of the (Mg + Pr)-related interstitial sites by H atoms in the two-phase equilib rium range at invariable p(H2) plateau during H-charging.


Introduction
Hydrogen (H) storage alloys are of strategic importance towards development of H-based energy systems with zero-CO 2 emission. In this context, many researchers have been investing efforts to discover adequate alloy compos ition to allow h igh H-storage capacity with favorable absorption/desorption performances.
In a recent publication, Terashitaet al. [1] reported isothermal hydrogenation performances of ternary intermetallic co mpounds Mg 2-y Pr y Ni 4 (0.6 ≤ y ≤ 1.4) that is considered as o n e o f can d id at e H -s t o rag e a l lo y s . In t h e ir publication, pressure-temperature -co mposition (PTC)relat io nships on H-charging/discharging cycle is presented as is otherms on P-C coordinate (log p(H 2 ) vs. x = H/M; M = Mg 2-y Pr y Ni 4 ; p(H 2 ): partial pressure of H 2 gas) for alloys with y = 0.6, 0.8 and 1.0 that maintained crystalline lattice structures during H-charging/discharging cycle (cf. Tab le 2 in Ref. [1]). Fo r the Mg 2-y Pr y Ni 4 alloys with y = 1.2 and 1.4, amorphizat ionprogressed during H-chargingand thence alloys with these compositions had to be disregarded as reversible H-storage alloyand, as such, PCisotherm for the alloys with y = 1.2 and 1.4 was not reported in Ref. [1].
In metal-hydrogen (M-H) systems, hysteretic performanceis a commonplace rather than exceptionin H-absorption/desorption cycle. In simp lify ing model for hysteresis of M-H system, hysteresis is presented for plateau levels of p(H 2 ) representing transition between primary solid solution phase of H in M and h igher hydride phase designating the equilibriu m H 2 gas pressure p(H 2 ) f for the hydride formation during ascending p(H 2 ) and the equilibriu m H 2 gas pressurep(H 2 ) d for dissociation of the hydride during descending p(H 2 ) where p(H 2 ) f >p(H 2 ) d [2]. In such simp lifying model for hysteresis of M-H system, PCis otherm in single-phase region on H-charging and that on H-discharging are considered to be comparable to one another assuming reversible nature of the H-absorption/desorption processes in single-phase regions (e.g., Fig . 1 and Fig. A1 in Ref. [2]).
Nevertheless, all the PCisotherms reported forthe Mg 2-y Pr y Ni 4 alloyswith y = 0.6, 0.8 and 1.0 by Terashitaet al. [1] show clearly the hysteretic perfo rmance in the single-phase regions as well as in the two-phase region yielding plateaus for distinguishablep(H 2 ) f and p(H 2 ) d . Thus, in the present work, PTC relationships reported forthe single-phase regions of the Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.8 and 1.0 by Terashitaet al. [1] were analyzed by statistical thermodynamics for the absorption process and for the d esorption process separately for respective isotherms aiming at identifying the possible causes leading to hysteretic performance forH d issolution in the Mg 2-y Pr y Ni 4 alloy latt ice.
Basic principles for statistical thermodynamic analysis are provided in a classical textbook authored by Fowler and Guggenheim [3] and desired thermodynamic parameter values of the calculation might be retrieved fro m NIST-JA NAF Thermochemical Tab les [4]. In statistical thermodynamic approach, PTC relationships in single-phase region are analyzed to derive ato mistic interaction parameter values in a given phase while range of composition where t wo phases are co-existing (i.e., plateau p(H 2 ) regime in isothermal plot of PC relationship) cannot be handled by statistical thermodynamics unlike by conventional thermodynamics.
Statistical thermodynamic analys es were made for extensive range of M H x and MZ z H x under standardized a priori assumption of constant nearest neighbor (n.n.)H-H interaction energy E(H-H) within a phase at respective temperature T where M might be pure metal or substitutional alloy of type A 1-y B y and Z refers to another interstitial element besides H [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. At the onset of the statistical thermodynamic analysis, number  of interstitial sites available for occupation by H atoms per M atom is chosen to fulfill the a priori assumption of constant E(H-H) within a phase by trial-and-erro r plotting ofA(x,T) ≡ RT ln {[p(H 2 )] 1/2 ·(  -x)/x} against xat an arbitrary Tto find  value yielding linear A(x,T) vs. x relationship in which slope of the plot refers to E(H-H) as explained in so me detail later in Chapter 2.
There is no firstprinciple -based justification for valid ity of the a priori assumption of constancy of E(H-H) within a phase at arbitraryT on the statistical thermodynamic modelling. In fact, in some earlier statistical thermodynamic analyses made for interstitial non-stoichiometric co mpounds MX x by other authors,  was assumed arbitrarily on the basis of crystal latt ice structure consideration and, when slope change of A(x,T) vs. x plot with co mposition x was detected, it was accepted as the inherent variation of E(X-X) with respect to composition x.Normally, E(X-X) tended to become less attractive on going from d ilute range of X to higher X concentration range in the same phase MX x in such evaluation and this trend was appreciated as the cons equence of rising elastic strain in the lattice with increasing x in the same phase. However, noting the reality that phase change even between liquid and solid is involved with enthalpy difference of up to mere 20 kJ· mo l -1 (e.g., Ref. [4] and Fig. 1 in Ref. [24]), it would be more natural and straightforward to accept that change in E(X-X) of non-stoichiometric interstitial co mpound with x at a given Twould end up with phase transformation rather than being maintained in a specified crystal lattice structure. Further, set of statistical thermodynamic interaction parameter values estimated on the basis of the simp lifying a priori assumption of constant E(H-H) for extensive range of metals and alloys appear to be self-consistent among themselves [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].
Thus, in this work, PC isotherms reported for the Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.8 and 1.0 by Terashitaet al. [1] are analy zed on the basis of statistical thermodynamics with thea priori assumption of constant E(H-H) within a phase at arbitraryT.

Statistical Thermodynamic Analysis
Noting the realitythat the statistical thermodynamic analysis procedure is not so widely known as the conventional thermodynamic analysis procedure among materials researchers, essence of the statistical thermodynamic analysis procedure is reviewed briefly in the fo llo wing.
In the statistical thermodynamics, part ition function PF for condensed phase (either solid or liquid) under consideration is composed taking into account pairwise nearest neighbor atomic interactions E(i-j) between the constituents, i and j. Then, chemical potential (i) c of the constituent element i in the condensed phase is derived through partial differentiation of PF with respect to the number n i of the constituent element i. Subsequently, (i) c in the condensed phase is put equal to (i) g of the same element i in the gas phase.
To start the statistical thermodynamic analysis using Eq.(1), the value for the parameter  must be chosen adequately to yield linear A(T) vs. x isotherms. This is to fulfill Performances of M g 2-y Pr y Ni 4 Intermetallics (y = 0.6, 0.8, 1.0) the a priori assumption of constant E(H-H) over a range of homogeneity composition x at a given T for M H x . To carry out statistical thermodynamic analysis for the MH x lattice, it was desirable to convert the graphically presented PC isotherms for the Mg 2-y Pr y Ni 4 alloys by Terashitaet al. [1]into numerical tables as summarized in Tables 1 (for y = 0.6), 2 (for y = 0.8) and 3 (for y = 1). The data were read fro m magnified PC isotherms presented in Ref. [1] as Figs. 3-5 by cutting the mesh of x (= H/M) with interval 0.025 to read p(H 2 ) value while finer mesh interval 0.0125 was taken part ially for primary H solution range in MgPrNi 4 to acquire sufficient number of data points for the analysis. With this procedure, introduction of certain extent of error in read ing the experimental data presented in graphical form is inevitable but, as p(H 2 ) enters in the calculation formu la of Eq.(1) in form of log[p(H 2 )] 1/2 , the error margin introduced by the graphical data reading into the calculation results must remain relatively small being held in acceptable level. Table 1. IsothermalPCrelationships for Mg2-yPryNi4 alloy with y = 0.6onH-charging and on H-discharging read from Fig. 3 in the publication by Terashitaet al [1].The p(H2) values are indicated with bold letters where there was no distinction between the H-charging process and the H-discharging process.Shaded data were not used for the analysis   Table 2. IsothermalPCrelationships for Mg2-yPryNi4 alloy with y = 0.8 onH-charging and on H-discharging read from Fig. 4 in the publication by Terashitaet al [1].Shaded data were not used for the analysis with θ = 0.15  Table 3. IsothermalPC relationships for Mg2-yPryNi4 alloy with y = 1.0 onH-charging and on H-discharging read from Fig. 5 in the publication by Terashitaet al [1].Shaded data were not used for the analysis with θ = 0.15 Although Terashitaet al. [1]carried outthe isothermal hydrogenation experiments for the Mg 2-y Pr y Ni 4 alloys with y = 1.2 and 1.4 as well, they did not report PCisotherms for the alloys with these compositions because of amorphization of the alloy lattice during the H-charging process and, as the consequence, reversible cyclic H-charg ing/discharging was not possible.

General Features of the PC
Following features are noticeableregarding isothermal hydrogenation performances for the Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.8 and 1.0 in Tables 1 -3; i) primary solid solubility o f H into these M lattices e xtended up to x ≈ 0.2, ii) hydride phase is with the co mposition range extending between x≈ 0.5 and x≈ 0.75, iii) PC isotherms showed hysteretic performances on H-charging and on H-d ischarging as well as for two-phase region (p(H 2 ) f >p(H 2 ) d ).
Although not analyzed in the present work, Terashitaet al. [1] reported PC isotherms for the Mg 2-y Pr y Ni 4 alloys with y = 1.0 at T = 298 K and 273 K (Fig. 6 in Ref. [1]) demonstrating presence of even higher hydride phase with x around 1(mono-hydride M H) besides the hydride phase with the co mposition range extending between x ≈ 0.5 and x ≈ 0.75 (hypo-stoichio metric M 4 H 3 ).
Noting these features of the isothermal hydrogenation performances of the Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.8 and 1.0, fo llo wing statistical thermodynamic analysis shall be made individually for the primary solid solubility range up to x ≈ 0.2 and for the M 4 H 3 type hydride phase with composition range between x ≈ 0.5 and x ≈ 0.75 distinguishing the absorption isotherm and the desorption isotherm.   (Table 1) with different choices of  parameter value.Best-fit linear relationships were calculated using all the data points plotted herein Search for  yielding linear A vs. x isothermal relationship was started fro m  = 0.25 noting that the reported PC isotherms in Fig. 3 in Ref. [1]reached a plateau at around x = Performances of M g 2-y Pr y Ni 4 Intermetallics (y = 0.6, 0.8, 1.0) 0.2. As shown in Fig. 1, slope referring to E(H-H) varied with x with the choice of  = 0.25 and 0.20 imp lying that such value of  could not be accepted for the analysis.However, when  was taken to be 0.15, linearA vs. x relationship was established for both the absorption and for the desorption. By least-mean-squares liner fitting procedure, expression for A vs. x relationship for Mg 1.4 Pr 0.6 Ni 4 at T = 323 K for absorption and that for desorption, respectively, are determined to be A(Mg 1.4 Pr 0.6 Ni 4 ; x, 323 K; abs.;  = 0.15) = 6.000 -63.884x(kJ· mo l -1 )

Analysis for Primary H Solution in
A(Mg 1.4 Pr 0.6 Ni 4 ; x, 323 K; des.;  = 0.15) = 6.086 -68.442x(kJ· mo l -1 ) As might be understood from the above attempt for d etermin ing  value for the primaryH solution the Mg 1.4 Pr 0.6 Ni 4 lattice,  for the absorption process and  for the desorption process were equally 0.15 andE(H-H) on the desorption and that on the absorption were co mparable to one another showing only slight difference between them. E(H-H) in the desorption process (Eq. (7)) was slightly mo re attractive than that in the absorption process (Eq. (6)). This order of E(H-H) on the absorption and on the desorption at a given Tappears rational suggesting that then.n. H-H interaction was more attractive in the M lattice during H-discharging than in the M lattice during H-charging not being in contradiction to the fact that p(H 2 ) f >p(H 2 ) d .
It is somewhat surprising to know that  = 0.15 appeared to be valid for analysis of primary H solution in MgPrNi 4 as well as that for Mg 1.4 Pr 0.6 Ni 4 noting that, in the earlier statistical thermodynamic analyses for A 1-y B y X x type non-stoichiometric interstitial solutions [9][10][11][12]16,19,20,22],  varied with y. As such, in the present cases for H solu-  Table 4 summarizes the thus -evaluated A vs. x relationships for primary H solutions in Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.8 and 1.0 under assumption of  = 0.15.The analysis was made for absorption and for desorption separately for H solutions in Mg 2-y Pr y Ni 4 alloys at respective T. The A vs. x relationships given in parentheses in Table 4 were derived with mere two data points and thence they are not used for the further analysis in evaluating K vs. T relationship. The A vs. x relationship was not evaluated for the H-charging in Mg 1.2 Pr 0.8 Ni 4 at T = 313 K and for H desorption in MgPrNi 4 at T = 323 K on account of unavailability of the PTC data.
Corresponding graphical presentations of A vs. x relationships are given in Figs. 3-5. Looking at Table 4, it was judged that A(x,T; = 0.15) vs. x relationships acquired during the H-d ischarging process for the primary H solutions in any examined Mg 2-y Pr y Ni 4 were insufficient for further analysis to evaluate the K vs. T relationships. Thus, further analysis for the primary H solution in Mg 2-y Pr y Ni 4 was done for the H-charg ing data for y = 0.6 and y = 1.0 alone but not for y = 0.8. Table 4. CalculatedA as a function of x at given T for primary H solution in Mg2-yPryNi4latt ice with y = 0.6, 0.8 and 1.0 for absorption and desorption processes of H reported by Terashitaet al [1] T  As such,Q value was ca. -210 kJ· mo l -1 andR ln f H was around +850 J· K -1 · mol -1 for eitherMg 1 were not estimated on account of scarcity of data points, they must be comparable to those estimated forMg 1.4 Pr 0.6 Ni 4 and MgPrNi 4 (i.e., Q ≈ -210 kJ· mo l -1 and R ln f H ≈ +850 J· K -1 · mol -1 ).

D(H2)/2 -RTC(T) A(x,T;θ= 0.15)(kJ/mol)
The virtual constancy of values of Q and R ln f H for the primary H solutions in Mg 2-y Pr y Ni 4 with respect to y in the range of y between 0.6 and 1.0 does not seem to be incompatible with the constancy of the chosen value of  to be 0.15 for these alloy lattices with varying y in the range between 0.6 and 1.0. As plotted in Fig. 7, A vs. x relationships for these isotherms were calculated with  = 0.75 and with  = 1.0 for the sake of co mparison. It looks that the data point for x = 0.45 forMg 1.4 Pr 0.6 Ni 4 at T = 323 K was out of contention and thence it was discarded for eva luation of the A vs. x relationships for Mg 1.4 Pr 0.6 Ni 4 . Figure 7. A vs. x relationships estimated for isothermal PC data for hypo-stoichiometric M4H3 alloy lattice with M = Mg1. 4Pr0.6Ni4 (Table 1) and M = MgPrNi4( Table 3) (Figs. 1 and 2).Thus, it was felt difficult to decide wh ich  value had to be chosen for further analysis. Noting that MH type mono-hydride phase would exist at least at T = 273 K and at T = 298 K for MgPrNi 4 (Fig. 6 in Ref. [1]) to which  = 1.0 might be assigned for the analysis, we decided to undertake analysis for the M 4 H 3 type hypo-stoichiometric hydride of Mg 2-y Pr y Ni 4 with the choice of  = 0.75.   (Fig. 8) appears to be greater than that of the calculated A values for the primary H solutions in Mg 2-y Pr y Ni 4 (Figs. 3-5). Thence, the error margin for the estimated K vs. T relat ionships for the M 4 H 3 type hypo-stoichiomet ric hydrides of Mg 2-y Pr y Ni 4 (Fig. 9) and that for the estimated values for Q and R ln f H for the M 4 H 3 type hypo-stoichiometric hydrides (Table 5) must be greater than the corresponding error margins for the primary H solutions in Mg 2-y Pr y Ni 4 . Table 5. Estimated values for the statistical thermodynamic parameters, Q (kJ· mol -1 ) and R ln fH (J· K -1 · mol -1 ), for primary solid solution MHx with  = 0.15 and for hypo-stoichiometric hydride M4H3 with  = 0.75 using the PTC data reported for absorption process for M = Mg2-yPryNi4 by Terashitaet al [1] primary solid solutionMH  The data set for the M 4 H 3 type hypo-stoichio metric hydrides of Mg 2-y Pr y Ni 4 listed in Table 5 appeared to be with lacking regularity.Graphical presentation of K-T relat ionships for M 4 H 3 type hypo-stoichiometric hydrides of Mg 2-y Pr y Ni 4 in Fig. 9 also does not seem to show any realistic regularity with respect to variation of y. Thence, re-consideration for the statistical modelling for the M 4 H 3 type hypo-stoichiometric hydride of Mg 2-y Pr y Ni 4 was felt desirable.

Analysis for Hypo-stoichiometric M 4 H 3 Type Hydride of
When comparing values of Q and R ln f H for Mg 1.4 Pr 0.6 Ni 4 between the primary H solution and the M 4 H 3 type hypo-stoichiometric hydride (Table 5), Q was mo re negative (i.e. E(H-M) was more attractive) in the M 4 H 3 than in the primary H solution and R ln f H was positive in both the primary H solution and the M 4 H 3 hydride. Table 6 reproduces the statistical thermodynamic analysis results for Ln Co 3 H 4x reported earlier in Ref. [16]. The analysis for LnCo 3 H 4x was done using modified basic formu la as below in place of Eq.(1) where representsthe number of interstitial sites per M atom available for occupation of H ato ms referring to the upper composition limit of the phase while ' representsthe number of interstitial sites per M ato m available for occupation of H ato ms referring to the lower co mposition limit of the phase [9,12,16].
One of remarkable aspects noticeable in Table 6 is that R ln f H for LnCo 3 H 4x in the higher hydride phase (0.75 <x< 1.1) was negative while that in the lower hydride phase Remembering co mposition procedures for the modified statistical model for the H solutions in Ln Co 3 intermetallic lattice by specifying ' referring to the lower limiting H solubility of the phase besides  referring to the higher limiting H solubility of the phase [9,12,16], introduction of ' parameter for the statistical model for the M 4 H 3 type hypo-stoichiometric hydride of Mg 2-y Pr y Ni 4 besides  = 0.75 was sought.
Noting that lattice parameter of the Mg 2-y Pr y Ni 4 Laves phase increased linearly with the composition y following the Vegard's rule (Fig. 2 Table 7. It looked that, with this modified model with  = 0.75 and ' = 0.333, slope of A' vs. x at any given T became practically 0 (i.e., E(H-H) = 0). At the bottom row in Tab le 7, averaged values of A' at given T were listed as the values of g defined in Eq. (2).The values of g calculated in the similar fashion for Mg 2-y Pr y Ni 4 at y = 0.8 and 1.0 as well as at y = 0.6 are summarized in Table 8 together with K values. At the bottom of Table 8, calculated K vs. T relat ionships are given separately for the H-absorption process and for the H-desorption process.
As seen in Table 8, the mod ified model with  = 0.75 and ' = 0.333 appeared to yield K vs. T relationships showing certain regularity with respect to variation of y in Mg 2-y Pr y Ni 4 . Graphical presentation of the drawn K vs. T relationships on the basis of this model (Fig. 10) also seem to be in better order than that drawn on the basis of a simplifying model with  = 0.75 presented in Fig. 9.  Table 8.
i) Q value tended to become more negative (that is, in-creasing extent of stabilizat ion of H ato ms in the Mg 2-y Pr y Ni 4 lattice) with the increasing y suggesting positive contribution of Pr alloying to substitute Mg towards stabilization of H in the Mg 2-y Pr y Ni 4 lattice. ii) For Mg 2-y Pr y Ni 4 with a given y, Q value for the H-discharging process was slightly more negative than that for the H-charging process.
The latter feature appears rational and acceptable noting that, on H-absorption process, H atoms are desired to be forcib ly inserted into certain interstitial sites while, on H-desorption process, H atoms in certain interstitial sites have to be pulled out of the site.

Concluding Remarks
Isothermal hydrogenation performances of Mg 2-y Pr y Ni 4 alloys with y = 0.6, 0.2 and 1.0 reported by Terashitaet al. [1] were analy zed on the basis of statistical thermodynamics under an a priori assumptionof constant E(H-H) in a given phase at arbitrary T.
Primary H solution in Mg 2-y Pr y Ni 4 was analy zed by the model with  = 0.15 to yield Q ≈ -210 kJ· mo l -1 and R ln f H = +850 J· K -1 · mo l -1 . The chosen  value 0.15 for the model was close to 1/ 6 (≈ 0.167) wh ich was half of 1/ 3 (= [Mg + Pr]/[Mg + Pr + Ni]) imp lying that about half of the (Mg + Pr)-related interstitial sites were provided as the availab le sites for occupation by H atoms in the primary H solution of Mg 2-y Pr y Ni 4 .
On the other hand, hypo-stoichiometric M 4 H 3 type hydride of Mg 2-y Pr y Ni 4 was analyzed by the model with  = 0.75 and ' = 0.333. Th is model y ielded situation with E(H-H) = 0. Chosen value of ' = 0.333 appeared to imp ly that the filling of Ni-related interstitial sites by H atoms started after preliminary fu ll occupation of the (Mg + Pr)-related interstitial sites by H ato ms in the two-phase equilibriu m range at invariable p(H 2 ) p lateauduring H-charging.