1. Introduction

Hydrogen (H) storage alloys are of strategic importance towards development of H-based energy systems with zero-CO₂ emission. In this context, many researchers have been investing efforts to discover adequate alloy composition to allow high H-storage capacity with favorable absorption/desorption performances.

In a recent publication, Terashitaet al.[1] reported isothermal hydrogenation performances of ternary intermetallic compounds Mg_2-yPr_yNi₄ (0.6 ≤ y ≤ 1.4) that is considered as one of candidate H-storage alloys. In their publication, pressure-temperature-composition(PTC)relationships on H-charging/discharging cycle is presented as isotherms on P-C coordinate (log p(H₂) vs. x = H/M; M = Mg_2-yPr_yNi₄; p(H₂): partial pressure of H₂ gas) for alloys with y = 0.6, 0.8 and 1.0 that maintained crystalline lattice structures during H-charging/discharging cycle (cf. Table 2in Ref.[1]). For the Mg_2-yPr_yNi₄ alloys with y = 1.2 and 1.4, amorphizationprogressed 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 simplifying model for hysteresis of M-H system, hysteresis is presented for plateau levels of p(H₂) representing transition between primary solid solution phase of H in M and higher hydride phase designating the equilibrium H₂ gas pressure p(H₂)^f for the hydride formation during ascending p(H₂) and the equilibrium H₂ gas pressurep(H₂)^d for dissociation of the hydride during descending p(H₂) where p(H₂)^f>p(H₂)^d[2]. In such simplifying model for hysteresis of M-H system, PCisotherm 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-yPr_yNi₄ alloyswith y = 0.6, 0.8 and 1.0 by Terashitaet al.[1] show clearly the hysteretic performance in the single-phase regions as well as in the two-phase region yielding plateaus for distinguishablep(H₂)^f and p(H₂)^d. Thus, in the present work, PTC relationships reported forthe single-phase regions of the Mg_2-yPr_yNi₄ 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 desorption process separately for respective isotherms aiming at identifying the possible causes leading to hysteretic performance forH dissolution in the Mg_2-yPr_yNi₄ alloy lattice.

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 from NIST-JANAF Thermochemical Tables[4]. In statistical thermodynamic approach, PTC relationships in single-phase region are analyzed to derive atomistic interaction parameter values in a given phase while range of composition where two phases are co-existing (i.e., plateau p(H₂) regime in isothermal plot of PC relationship) cannot be handled by statistical thermodynamics unlike by conventional thermodynamics.

Statistical thermodynamic analyses were made for extensive range of MH_x and MZ_zH_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-yB_y and Z refers to another interstitial element besides H[5-23]. At the onset of the statistical thermodynamic analysis, number θ 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-error plotting ofA(x,T) ≡ RT ln {[p(H₂)]^1/2·( - x)/x} against xat an arbitrary Tto find θ value yielding linear A(x,T) vs. x relationship in which slope of the plot refers to E(H-H) as explained in some detail later in Chapter 2.

There is no firstprinciple-based justification for validity 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 compounds MX_x by other authors, was assumed arbitrarily on the basis of crystal lattice structure consideration and, when slope change of A(x,T) vs. x plot with composition 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 dilute range of X to higher X concentration range in the same phase MX_xin such evaluation and this trend was appreciated as the consequence 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·mol^-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 compound 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 simplifying a priori assumption of constant E(H-H) for extensive range of metals and alloys appear to be self-consistent among themselves[5-24].

Thus, in this work, PC isotherms reported for the Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.8 and 1.0 by Terashitaet al.[1] are analyzed on the basis of statistical thermodynamics with thea priori assumption of constant E(H-H) within a phase at arbitraryT.

2. 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 following.

In the statistical thermodynamics, partition 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 μ (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, μ (i)^c in the condensed phase is put equal to μ (i)^g of the same element i in the gas phase.

The expression for μ (X)^gof ideal diatomic gas X₂ is readily made available in the classical textbook authored by Fowler and Guggenheim[3]. The detailed derivation procedure of μ (X)^c for the condensed phase MX_x might be referred to elsewhere[7,11].Anyway, the statistical thermodynamic equilibrium condition is eventually reduced to the following Eq.(1) for the purpose of analyzing H solution under consideration[5-23]

(1)

(2)

(3)

(4)

(5)

wherevalues for the dissociation energy D(H₂) (kJ∙mol^-1) of H₂ molecule per mole, characteristic temperature Θ_r (= 85.4 K) for rotation of H₂ molecule and characteristic temperature Θ_v (= 6100 K) for vibration of H₂ molecule might be taken from available thermodynamic table[4]. Definitions of symbols used throughout the text are listed in the APPENDIX.

To start the statistical thermodynamic analysis using Eq.(1), the value for the parameter θ must be chosen adequately to yield linear A(T) vs. x isotherms. This is to fulfill the a priori assumption of constant E(H-H) over a range of homogeneity composition x at a given T for MH_x.

3. General Features of the PC Isotherms for H Solutions in Mg_2-yPr_yNi₄Alloy Lattice (y = 0.6, 0.8 and 1.0)

To carry out statistical thermodynamic analysis for the MH_xlattice, it was desirable to convert the graphically presented PC isotherms for the Mg_2-yPr_yNi₄ 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 from 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(H₂) value while finer mesh interval 0.0125 was taken partially for primary H solution range in MgPrNi₄ to acquire sufficient number of data points for the analysis. With this procedure, introduction of certain extent of error in reading the experimental data presented in graphical form is inevitable but, as p(H₂) enters in the calculation formula of Eq.(1) in form of log[p(H₂)]^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-yPr_yNi₄ 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-charging/discharging was not possible.

Following features are noticeableregarding isothermal hydrogenation performances for the Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.8 and 1.0 in Tables 1 - 3;

i) primary solid solubility of H into these M lattices extended up to x ≈ 0.2,

ii) hydride phase is with the composition range extending between x≈ 0.5 and x≈ 0.75,

iii) PC isotherms showed hysteretic performances on H-charging and on H-discharging as well as for two-phase region (p(H₂)^f>p(H₂)^d).

Although not analyzed in the present work, Terashitaet al.[1] reported PC isotherms for the Mg_2-yPr_yNi₄ 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 MH) besides the hydride phase with the composition range extending between x ≈ 0.5 and x ≈ 0.75 (hypo-stoichiometric M₄H₃).

Noting these features of the isothermal hydrogenation performances of the Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.8 and 1.0, following statistical thermodynamic analysis shall be made individually for the primary solid solubility range up to x ≈ 0.2 and for the M₄H₃type hydride phase with composition range between x ≈ 0.5 and x ≈ 0.75 distinguishing the absorption isotherm and the desorption isotherm.

4. Analysis for Primary H Solution in Mg_2-yPr_yNi₄Lattice (x< 0.2; y = 0.6, 0.8 and 1.0)

To commence statistical thermodynamic analysis process, adequate value for θ must be chosen first of all. A trial-and-error test for evaluating A vs. x relationships defined by Eq.(1) was done using sets of data for Mg_1.4Pr_0.6Ni₄ at T = 323 K in the range ofx< 0.2 listed in Table 1. This set of data showed the smallest extent of hysteresis between the H absorption and the H desorption processes.

Figure 1. A vs. x relationships estimated for isothermal PC data at T = 323 K forprimary H solution in Mg1.4Pr0.6Ni4 alloy lattice on H-absorption and on H-desorption (Table 1) with different choices of θ parameter value.Best-fit linear relationships were calculated using all the data points plotted herein

Search for θ yielding linear A vs. x isothermal relationship was started from θ = 0.25 noting that the reported PC isotherms in Fig. 3 in Ref.[1]reached a plateau at around x = 0.2. As shown in Fig. 1, slope referring to E(H-H) varied with x with the choice of θ = 0.25 and 0.20 implying that such value of  could not be accepted for the analysis.However, when θ 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_1.4Pr_0.6Ni₄ at T = 323 K for absorption and that for desorption, respectively, are determined to be

(6)

(7)

As might be understood from the above attempt for determining θ value for the primaryH solution the Mg_1.4Pr_0.6Ni₄ lattice, θ for the absorption process and θ for the desorption process were equally 0.15 and E(H-H) on the desorption and that on the absorption were comparable to one another showing only slight difference between them. E(H-H) in the desorption process (Eq.(7)) was slightly more 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(H₂)^f>p(H₂)^d.

Figure 2. A vs. x relationships estimated for isothermal PC data at T = 353 K forprimary H solution in MgPrNi4 alloy lattice on H-absorption and on H-desorption (Table 3) with different choices of θ parameter value, Best-fit linear relationships were calculated excluding the data points at x = 0.05

Similar attempt of search for θ parameter value was made for primary H solution in MgPrNi₄ lattice using PC isotherm at T = 353 K. As plotted in Fig. 2, θ = 0.15 appears to be the adequate choice for the θ value like for H solution in Mg_1.4Pr_0.6Ni₄ to yield linearA vs. x relationships

(8)

(9)

Like for the Mg_1.4Pr_0.6Ni₄ lattice, E(H-H) during H-discharging (Eq.(9)) was more attractive than that during H-charging (Eq.(8)) for the MgPrNi₄.

It is somewhat surprising to know that θ = 0.15 appeared to be valid for analysis of primary H solution in MgPrNi₄ as well as that for Mg_1.4Pr_0.6Ni₄noting that, in the earlier statistical thermodynamic analyses for A_1-yB_yX_x type non-stoichiometric interstitial solutions[9-12,16,19,20,22], θ varied with y. As such, in the present cases for H solutions in Mg_1.4Pr_0.6Ni₄ and MgPrNi₄, θ = 0.15 was evaluated to yield equally the linearA vs. x relationships for y = 0.6 and 1.0. Thus, the analysis for the primary H solution in Mg_1.2Pr_0.8Ni₄ was decided to be done also with θ = 0.15. Reason for why θ for primary H solution in Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.8 and 1.0 held constant with y is not clear. The reason for this might be speculated with reference to the outer shell electronic configurations of elemental states of Mg, Pr and Ni which are 2p⁶3S², 4f³5p²6s² and 3p⁶3d⁸4s², respectively, where incompletely filled electron shell, 4f in Pr and 3d in Ni, are presented with bold letters to distinguish from completely filled outermost p and s electron shells. That is, Mg and Pr distributed over the same metal sub-lattice positions on account of similarity of the outermost electron shells although with scarcely populated 4f shell of Pr. On the other hand, Ni distributed over the other group of metal sub-lattice positions on account of existence of nearly filled3d shell. The above cited electron configurations for Mg, Pr and Ni are the electron configurations in the elemental state and thence they must be certainly modified in the Mg_2-yPr_yNi₄ intermetallic alloy lattice. Thus, reason for why the θ parameter value for the analysis of the primary H solution in Mg_2-yPr_yNi₄ alloys with varying y in the range between 0.6 and 1.0 was estimated to hold constant at 0.15 must be elucidated somehow by further investigation.

Table 4 summarizes the thus-evaluated A vs. x relationships for primary H solutions in Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.8 and 1.0 under assumption of θ = 0.15.The analysis was made for absorption and for desorption separately for H solutions in Mg_2-yPr_yNi₄ 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_1.2Pr_0.8Ni₄ at T = 313 K and for H desorption in MgPrNi₄ 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; θ = 0.15) vs. x relationships acquired during the H-discharging process for the primary H solutions in any examined Mg_2-yPr_yNi₄ were insufficient for further analysis to evaluate the K vs. T relationships. Thus, further analysis for the primary H solution in Mg_2-yPr_yNi₄was done for the H-charging 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-yPryNi4lattice with y = 0.6, 0.8 and 1.0 for absorption and desorption processes of H reported by Terashitaet al[1]

Figure 3. A vs. x relationships estimated for isothermal PC data at different T forprimary H solution in Mg1.4Pr0.6Ni4 alloy lattice on H-absorption and on H-desorption (Table 1) with θ =0.15.Best-fit linear relationship at T = 323 K on H-absorption was calculated using the all data points plotted herein whereas best-fit linear relationships at T = 313 K and 293 K on H-absorption were calculated excluding the data points at x = 0.05

Figure 4. A vs. x relationships estimated for isothermal PC data at different T forprimary H solution in Mg1.2Pr0.8Ni4 alloy lattice on H-absorption and on H-desorption (Table 2) with θ =0.15.Best-fit linear relationships at T = 353 K and 333 K on H-absorption were calculated using the all data points plotted herein

Figure 5. A vs. x relationships estimated for isothermal PC data at different T forprimary H solution in MgPrNi4alloy lattice on H-absorption and on H-desorption (Table 3) with θ =0.15.Best-fit linear relationships at T = 373 K and 353 K on H-absorption were calculated excluding the data points at x< 0.05

Figure 6. K vs. T relationships estimated for primary H solutions in Mg1.4Pr0.6Ni4 and MgPrNi4 alloy lattices on H-absorption with θ =0.15

From these estimation results, K vs. T relationships were determined as exhibited in Fig. 6 and, as such, values of Q and R ln f_H are evaluated to be

(10)

(11)

(12)

(13)

As such,Q value was ca. -210 kJ·mol^-1andR ln f_H was around +850 J·K^-1·mol^-1for eitherMg_1.4Pr_0.6Ni₄ or MgPrNi₄. Thus, although values of Q and R ln f_H for Mg_1.2Pr_0.8Ni₄ were not estimated on account of scarcity of data points, they must be comparable to those estimated forMg_1.4Pr_0.6Ni₄ and MgPrNi₄ (i.e., Q ≈ -210 kJ·mol^-1 and R ln f_H ≈ +850 J·K^-1·mol^-1).

The virtual constancy of values of Q and R ln f_H for the primary H solutions in Mg_2-yPr_yNi₄ 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 θ to be 0.15 for these alloy lattices with varying y in the range between 0.6 and 1.0.

5. Analysis for Hypo-stoichiometric M₄H₃Type Hydride of Mg_2-yPr_yNi₄ (0.45 <x< 0.75; y = 0.6, 0.8 and 1.0)

Search of θ for analysis of hypo-stoichiometric M₄H₃ type hydride of Mg_2-yPr_yNi₄ (0.45 <x< 0.75; y = 0.6, 0.8 and 1.0) was done in the same fashion as that for the primary H solutions described above in Chapter 4. For this, the analysis was done for the PCisotherms for H-absorption and H-desorption at T = 323 K for Mg_1.4Pr_0.6Ni₄ and at T = 353 K for MgPrNi₄.

As plotted in Fig. 7, A vs. x relationships for these isotherms were calculated with θ = 0.75 and with θ = 1.0 for the sake of comparison. It looks that the data point for x = 0.45 forMg_1.4Pr_0.6Ni₄ at T = 323 K was out of contention and thence it was discarded for evaluation of the A vs. x relationships for Mg_1.4Pr_0.6Ni₄.

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) with choices of θ = 0.75 and 1.0.Best-fit linear relationships for Mg1.4Pr0.6Ni4on H-absorption and on H-desorption were calculated excluding the data points at x< 0.50 while those for MgPrNi4 on H-absorption and on H-desorptionwere calculated using all the data points plotted herein

At either choice of θ = 0.75 or 1.0, linear A vs. x relationships were drawn for the H solutions in hypo-stoichiometric M₄H₃ type hydride of Mg_2-yPr_yNi₄ yielding positive E(H-H) (repulsive n.n. H-H interaction) in contrast to negative E(H-H) for the primary H solutions in Mg_2-yPr_yNi₄ (Figs. 1 and 2).Thus, it was felt difficult to decide which θ 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 MgPrNi₄ (Fig. 6 in Ref.[1]) to which θ = 1.0 might be assigned for the analysis, we decided to undertake analysis for the M₄H₃ type hypo-stoichiometric hydride of Mg_2-yPr_yNi₄ with the choice of θ = 0.75.

Figure 8. A vs. x relationships estimated for isothermal PC data at different T forhypo-stoichiometric M4H3 alloy lattices with M = Mg1.4Pr0.6Ni4and MgPrNi4on H-absorption withθ = 0.75

Figure 8 plots the calculated A vs. x relationships for M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄ at y = 0.6, 0.8 and 1.0 with the choice of θ = 0.75. In most cases, E(H-H) appears to be positive (repulsive) while E(H-H) for Mg_1.2Pr_0.8Ni₄ at T = 313 K was estimated to be slightly negative (attractive).

Figure 9. K vs. T relationships estimated for hypo-stoichiometric M4H3 alloy lattice with M =Mg1.4Pr0.6Ni4, Mg1.2Pr0.2Ni4 and MgPrNi4withθ = 0.75

Extent of scatter of calculated A values from the least-mean-squares lines for the M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄(Fig. 8) appears to be greater than that of the calculated A values for the primary H solutions in Mg_2-yPr_yNi₄ (Figs. 3-5). Thence, the error margin for the estimated K vs. T relationships for the M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄ (Fig. 9) and that for the estimated values for Q and R ln f_Hfor the M₄H₃ type hypo-stoichiometric hydrides (Table 5) must be greater than the corresponding error margins for the primary H solutions in Mg_2-yPr_yNi₄.

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 solutionMHx hypo-stoichiometric hydride M4H3 M T (K) (θ= 0.15) (θ= 0.75) Q (kJ/mol) R ln fH (J/K/mol) Q (kJ/mol) R ln fH (J/K/mol) Mg1.4Pr0.6Ni4 293-323 -208.753 +852 -224.842 +829 Mg1.2Pr0.8Ni4 313-353 ------- ------- -183.232 +951 MgPrNi4 323-373 -216.321 +841 -275.092 +696

Table 6. Valuesof statistical thermodynamic parameters, Q and R ln fH, estimated for LnCo3H4x type intermetallic hydride (reproduced from Table 5 in Ref.[16])

The data set for the M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄ listed in Table 5 appeared to be with lacking regularity.Graphical presentation of K-T relationships for M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄ 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 M₄H₃ type hypo-stoichiometric hydride of Mg_2-yPr_yNi₄ was felt desirable.

When comparing values of Q and R ln f_H for Mg_1.4Pr_0.6Ni₄ between the primary H solution and the M₄H₃ type hypo-stoichiometric hydride (Table 5), Q was more negative (i.e. E(H-M) was more attractive) in the M₄H₃ than in the primary H solution and R ln f_H was positive in both the primary H solution and the M₄H₃ hydride.

Table 6 reproduces the statistical thermodynamic analysis results for LnCo₃H_4x reported earlier in Ref.[16]. The analysis for LnCo₃H_4x was done using modified basic formula as below in place of Eq.(1)

whereθ representsthe number of interstitial sites per M atom available for occupation of H atoms referring to the upper composition limit of the phase while θ' representsthe number of interstitial sites per M atom available for occupation of H atoms referring to the lower composition limit of the phase[9,12,16].

One of remarkable aspects noticeable in Table 6 is that R ln f_H for LnCo₃H_4x in the higher hydride phase (0.75 <x< 1.1) was negative while that in the lower hydride phase (0.25 <x< 0.5) was positive. This was interpreted as suggesting drastic modification of electronic surroundings around H atom from the former to the latter. It might be that, in the higher hydride phase with x>0.75, filling of Ln-related sites ([Ln]/[Ln + Co] = 1/4]) might have proceeded with Co-related sites ([Co]/[Ln + Co] = 3/4) being fully occupied preferentially during isothermal increase of p(H₂) judging from the values of θ and θ' chosenfor the model(θ= 1.05 and θ' ≈ 0.75±0.025; Table 6) whereas, in the lower hydride phase of LnCo₃, partial (probably, in regular fashion) filling of the Co-related sites with H atoms have been in progress (θ≈ 0.5 and θ' ≈ 0.25).

Remembering composition procedures for the modified statistical model for the H solutions in LnCo₃ intermetallic lattice by specifying θ' 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 θ' parameter for the statistical model for the M₄H₃ type hypo-stoichiometric hydride of Mg_2-yPr_yNi₄ besides θ= 0.75 was sought.

Noting that lattice parameter of the Mg_2-yPr_yNi₄ Laves phase increased linearly with the composition y following the Vegard's rule (Fig. 2 in Ref.[1]), Mg and Pr substituted one type of metal sub-lattice positions randomly while Ni occupied another type of metal sub-lattice positions. Ratio of[Mg + Pr]/[Mg + Pr + Ni] in the Mg_2-yPr_yNi₄lattice was 1/3 (≈ 0.333). Thus, modified model calculation was attempted with the choice of θ= 0.75 and θ' = 0.333.

Table 7. Estimated values of A'(x,T; θ= 0.75, θ' = 0.333) using Eq.(14) for hypo-stoichiometric M4H3 type hydride of Mg1.4Pr0.6Ni4and calculated values of gby simple arithmetic averagingof A' values under assumption of E(H-H) = 0 x A'(x,T; θ = 0.75, θ' = 0.333) for Mg1.4Pr0.6Ni4(kJ/mol) 323 Kabs 323 Kdes 313 Kabs 313 Kdes 293 Kabs 293 Kdes 0.45 5.234 5.042 ----- ----- ----- ----- 0.475 4.649 4.481 ----- ----- ----- ----- 0.50 4.420 4.420 2.917 ----- ----- ----- 0.525 4.380 4.380 2.558 1.926 1.168 ----- 0.55 4.303 4.303 2.409 1.829 0.868 0.023 0.575 4.409 4.409 2.390 1.651 0.771 -0.008 0.60 ----- ----- 2.521 1.733 0.778 0.012 0.625 ----- ----- 3.006 2.592 1.057 0.303 0.65 ----- ----- ----- ----- 1.332 0.838 g(kJ/mol) 4.566 4.506 2.634 1.946 0.996 0.234

Table 8. Calculatedvalues of gand K for thehypo-stoichiometric M4H3 type hydrides ofMg2-yPryNi4 with y = 0.6, 0.8 and 1.0 for absorption and desorption processes of H reported by Terashitaet al[1]according to a modified statistical model with the choice ofθ= 0.75 and θ'= 0.333 that led to E(H-H) = 0

The model calculation results made for the non-stoichiometric M₄H₃ type hydride of Mg_1.4Pr_0.6Ni₄with θ= 0.75 and θ' = 0.333 are summarized in Table 7. It looked that, with this modified model with θ= 0.75 and θ' = 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 Table 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-yPr_yNi₄ 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 relationships are given separately for the H-absorption process and for the H-desorption process.

As seen in Table 8, the modified model with θ= 0.75 and θ' = 0.333 appeared to yield K vs. T relationships showing certain regularity with respect to variation of y in Mg_2-yPr_yNi₄. 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 θ= 0.75 presented in Fig. 9.

Figure 10. K vs. T relationships estimated for hypo-stoichiometric M4H3 alloy lattice with M =Mg1.4Pr0.6Ni4, Mg1.2Pr0.2Ni4 and MgPrNi4with θ=0.75 and θ' = 0.333

Following trends are noticeable for the M₄H₃ type hypo-stoichiometric hydrides of Mg_2-yPr_yNi₄ in Table 8.

i) Q value tended to become more negative (that is, increasing extent of stabilization of H atoms in the Mg_2-yPr_yNi₄ lattice) with the increasing y suggesting positive contribution of Pr alloying to substitute Mg towards stabilization of H in the Mg_2-yPr_yNi₄ lattice.

ii) For Mg_2-yPr_yNi₄with 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 forcibly inserted into certain interstitial sites while, on H-desorption process, H atoms in certain interstitial sites have to be pulled out of the site.

6. Concluding Remarks

Isothermal hydrogenation performances of Mg_2-yPr_yNi₄ alloys with y = 0.6, 0.2 and 1.0 reported by Terashitaet al.[1] were analyzed 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-yPr_yNi₄was analyzed by the model with θ= 0.15 to yield Q ≈ -210 kJ·mol^-1 and R ln f_H = +850 J·K^-1·mol^-1. The chosen θ value 0.15 for the model was close to 1/6 (≈ 0.167) which was half of 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-yPr_yNi₄.

On the other hand, hypo-stoichiometric M₄H₃ type hydride of Mg_2-yPr_yNi₄ was analyzed by the model with θ = 0.75 and θ' = 0.333. This model yielded situation with E(H-H) = 0. Chosen value of θ' = 0.333 appeared to imply that the filling of Ni-related interstitial sites by H atoms started after preliminary full occupation of the (Mg + Pr)-related interstitial sites by H atoms in the two-phase equilibrium range at invariable p(H₂) plateauduring H-charging.

APPENDIX / List of Symbols

A(x,T): ≡ RT ln {[p(H₂)]^1/2·(θ - x)/x} (kJ∙mol^-1); calculated from experimentally determinedvalues of p(H₂), T and x for specified value of θusing Eq.(1)

C(T): defined by eqn.(3) to represent contributions of translational, rotational andvibrational motions of H₂ molecule

D(H₂): dissociation energy of H₂ molecule per mole (kJ∙mol^-1)

E: lattice energy (kJ∙mol^-1)

E(i-j): nearest neighbor pair-wise interaction energy between iand j atoms in MH_x lattice

f_H(T): partition function of H in MH_x lattice at temperature T

g: parameter determined as the intercept of the A(T) vs. x plot at x = 0 using Eq.(1)

g(v): distribution function as a function of vibrational frequency νof H atom in MH_x lattice

h: Planck constant

k: Boltzmann constant

K: parameter calculated from g using Eq.(2)

m_H: mass of H atom

n_H: number of H atoms in the MH_x lattice

n_V: number of M atoms in the MH_x lattice

p(H₂): partial pressure of ideal H₂ gas molecule (atm)

P-T-C: pressure-temperature-composition

Q: degree of stabilization of H atom in MH_x lattice with reference to isolated H atom in vacuum

R: universal gasconstant (= 0.0083145 kJ∙mol^-1·K^-1)

T: absolute temperature (K)

x: atom fraction of H against M in MH_x

β: geometrical factor determined from crystal structure consideration

θ: number of available interstitial sites for occupation by H atom per metal atom in MH_x

Θ_r: characteristic temperature for rotation of H₂ molecule (= 85.4 K)

Θ_v: characteristic temperature for vibration of H₂ molecule (= 6100 K)

μ(H)^c: chemical potential of H atom in the condensed phase MH_x

μ(H)^g: chemical potential of H atom in the ideal diatomic H₂ gas molecule

ν:vibrational frequency of H atom in MH_x lattice

ρ: nuclear spin weight (= 2 for H while 3 for D)

υ₀: statistical weight of tightly bound electrons around H in MH_x lattice

υ₀*: statistical weight of electrons in H₂ molecule in normal state (= 1)