Single Quality Factor for Enthalpy-Entropy Compensation, Isoequilibrium and Isokinetic Relationships.

Enthalpy-entropy compensation (EEC) is very often encountered in chemistry, biology and physics. Its origin is widely discussed since it would allow, for example, a very accurate tuning of the thermodynamic properties as a function of the reactants. However, EEC is often discarded as a statistical artefact, especially when only a limited temperature range is considered. We show that the likeliness of a statistical origin of an EEC can be established with a compensation quality factor (CQF) that depends only on the measured enthalpies and entropies and the experimental temperature range. This is directly derived from a comparison of the CQF with threshold values obtained from a large number of simulations with randomly generated Van 't Hoff plots. The value of CQF is furthermore a direct measure of the existence of a genuine isoequilibrium or isokinetic relationship.

Introduction

There is a longstanding debate in the literature regarding the physical basis of the so‐called enthalpy‐entropy compensation (EEC), which is observed in a wide range of fields in chemistry;1, 2, 3, 4, 5, 6, 7, 8 biology9, 10, 11, 12, 13, 14, 15 and solid‐state physics:16, 17, 18, 19 EEC describes a linear relation between two thermodynamic parameters, the enthalpy ▵H and entropy ▵S, of a series of similar reactions. Examples include the denaturation of closely related proteins, reactions in slightly varying solvents, or reactions of hydrogen with metal alloys.20 While there is a wealth of data sets available, it is unclear which ▵H‐▵S correlations are strong enough to pursue a quest for the physical basis of the related enthalpy‐entropy compensation.

Apart from its theoretical significance, in practice, the EEC effect would allow for the tuning of thermodynamic parameters, such as e. g. necessary for the design of an ideal sorption material for hydrogen storage. Such a material requires a hydrogen equilibrium pressure above 1 bar at around 300 K. EEC would help the discovery of materials combining this equilibrium pressure with an enthalpy of formation less negative than the −40 kJ/moleH2 of present storage systems. This would reduce the enormous heat load involved when fast charging a metal hydride based storage tank.

EEC is traditionally characterized by the slope of the ▵H‐▵S plot, the so‐called compensation temperature T=d(ΔH)/d(ΔS), where all compositions of a particular set of reactions have e. g. the same equilibrium pressure. Interestingly, most observed compensation effects have a T close to the harmonic mean of the experimental temperature, T:

where T with j=1,…M, is the temperature of the j‐th measurement. Hence, several authors have claimed that when T≈T, statistical and/or experimental errors are causing these compensation effects.21, 22, 23, 24, 25, 26, 27, 28, 29, 30 Indeed, whenever data are collected in a small region of the pressure‐temperature plane26 the measured Van ‘t Hoff plots of a series of reactions have the tendency to cross within the experimental region and the corresponding ▵H versus ▵S plot is approximately a straight line suggesting EEC behaviour (see Section 3 in Supporting Information).

In a more quantitative approach, Krug et al.31, 32 derived a statistical test to compare the harmonic mean experimental temperature T with the interval [T‐tσ, T+tσ]. Here, σ is the standard error in T and the Student's t‐value depends on the chosen confidence level and the number of samples, i. e. the number of (ΔH,ΔS) data pairs. For a 95 % confidence level t is close to 2 for large data sets. If T falls within this temperature interval Krug et al. consider that the EEC is probably of statistical nature at a 95 % confidence level. However, this approach does not predict the degree of coalescence of Van ‘t Hoff lines observed near T. It therefore does not provide information about the isoequilibrium relationship.

For a hypothetical, perfect EEC all related Van ‘t Hoff plots (i. e. plots of the logarithm of the pressure versus the inverse absolute temperature from which ▵H and ▵S are determined for each composition) should have a common intersection point at T. Clearly, such a unique common intersection point is never realized in practice as a result of ever‐present experimental errors. Hence, one would expect a simple correlation between the presence of a trustworthy compensation temperature and a coalescence of the Van't Hoff lines. However, Liu and Guo conclude in their much cited review article4 that the Enthalpy‐Entropy compensation effect and the isoequilibrium effect are not necessarily synonymous (see for a more detailed discussion Section 1 in Supporting Information).

At variance with this claim we show in the present article that both the nature of the EEC (statistical or non‐statistical) and the degree of coalescence of Van ‘t Hoff lines can be quantitatively characterized by a single parameter calculated from the ▵H and ▵S values derived from the measured lnP versus 1/T lines of a set of N samples. We analytically calculate the variance in lnP for the set of the N investigated samples and determine the temperature T min at which the variance of lnP reaches a minimum. The variance of lnP at T=T min is a direct measure of the degree of coalescence of the Van ‘t Hoff plots. The ratio of the variance of lnP at T=T min normalized to the largest experimentally measured lnP variance defines a Compensation Quality Factor CQF that characterizes quantitatively the extent of the crossing region of Van ‘t Hoff lines. The CQF is by definition unity for perfect compensation (T min equal to T) and tends towards zero when Van ‘t Hoff lines do not come close to a single crossing.

Remarkably, it appears that the same CQF also reveals information on the question whether the observed EEC is statistical in nature or not. For the hypothetical case of statistically independent randomly generated Van ‘t Hoff lines we define a threshold CQF value γ which depends only on the number of samples N and the chosen confidence level. For a confidence level as high as 99 % we find for example γ=0.67 for a set of N=10 samples. This means that 99 % of all CQF random simulation values are below 0.67 or, equivalently, that there is only a 1 % probability that the EEC of an experimental data set for N=10 samples with CQF>0.67 is of a statistical nature. In other words, there is a 99 % probability that the observed EEC has a genuine (i. e. physical, chemical or biological) origin.

We demonstrate our new analytical framework using experimental pressure‐composition data on the hydrogenation of two specific metal hydrides, Mg−Ti−H thin films and Pd−H nanocubes. The data set for 67 Mg−Ti alloy compositions is characterized by a T≅470 K that falls within the experimental temperature range 348Pd−H nanocubes recently published by Syrenova et al.33 have a T≅283 K that falls clearly outside their experimental temperature range 303Pd nanocubes. However, it turns out that the contrary is borne out by our analysis: a genuine ΔH‐ΔS compensation with a well‐defined crossing of Van ‘t Hoff plots and a high value for CQF≅0.94 is found for the Mg−Ti alloys while a large spread of equilibrium pressures at T and consequently a low value for CQF≅0.26 is found for the Pd‐nanocubes.

In a second step we show that our method is also applicable to kinetic studies and the determination of the isokinetic temperature. The CQF is in this case a measure of the extent of the crossing region of Arrhenius plots, i. e. plots of the logarithm of the reaction rate versus the inverse temperature. The CQF is now entirely determined by the activation energy and pre‐factor data determined from kinetic measurements (Arrhenius plots), and the temperature range of the experiments. We illustrate this case by analysing literature data on both hydrogen absorption kinetics of magnesium‐based samples28 and the thermodynamic activation data of fish myofibrillar ATPase enzyme.34

We stress that the key purpose of our work is to identify the likeliness of a non‐statistical origin of the observed EEC, which would point to a fundamental thermodynamic relation. In addition, the CQF tells experimentalists whether it is worth to aim for isoequilibrium conditions or search for new compositions based on an extrapolated ΔS versus ΔH curve.

Results and Discussion

Although the concept of the CQF could be defined in a purely formal way we choose to introduce it step‐by‐step within the context of metal‐hydrides. EEC is observed in various metal hydride systems.20, 35, 36, 37, 38, 39, 40 A typical metal hydride exists in two phases. At low hydrogen pressures, hydrogen forms a solid solution with the metal host (α‐phase). Upon increasing the hydrogen pressure, the hydrogen concentration hardly increases until a certain pressure, the plateau pressure P at which the dilute α‐phase coexists with the hydrogen‐rich β‐phase. The changes in enthalpy ▵H and entropy ▵S between the α‐ and the β‐phases are obtained by fitting the temperature dependence of P to the Van't Hoff relation:

where P 0 is the pressure at standard conditions, R the gas constant, and ▵H and ▵S are expressed per mole H2. Note, that the determination of the entropy requires extrapolation over a large temperature range. In general hydrogen absorption in metals is measured in one sample at the time. The study of an EEC requires a relatively large number of samples which induces statistical errors related to the temperature and H2 pressures of each experiment. In order to avoid this source of statistical errors we choose to measure a large number of samples simultaneously by means of Hydrogenography (see the Experimental Section). The results obtained with this technique for the hydrogen absorption in 67 Mg−Ti thin films are given in Section 2.1. These results are contrasted with those of a recently published study on Pd‐nanocubes in Section 2.2.

Enthalpy‐Entropy Compensation in Mg−Ti Hydrides

To simultaneously investigate a large range of thin film alloy compositions, we measure the hydrogen absorption process by means of Hydrogenography.20 In this optical method we obtain the logarithm of the applied hydrogen pressure as a function of the hydrogen concentration for hundreds of compositions in a single experiment. In these compositional gradient thin‐film samples, the change in optical transmission T is a measure of the hydrogen concentration. We thus obtain Pressure‐optical Transmission‐Isotherms (PTIs), similar to the more commonly obtained Pressure‐Composition‐Isotherms (PCIs).41

The PTIs for 67 different MgyTi1–y alloys with 0.62hydrogenography (upon 1st hydrogenation) are shown in Figure 1. For Mg‐rich alloys there are clear plateaus indicating the formation of transparent MgH2 dihydrides. The isotherms get gradually more sloping when y approaches 0.62. This is due to an increasing admixture of Ti in the Mg‐rich domains .42 For each composition y the transmission ln(T/T o) is normalized to its maximum value [ln(T/To)]max and all the equilibrium pressures used to generate Van ‘t Hoff plots are taken at [ln(T/T o)]/ [ln(T/T o)]max=0.5. The enthalpy ▵H and entropy ▵S obtained from a linear fit to the individual Van ‘t Hoff plots are shown in Figure 2a. Both ▵H and ▵S increase as a function of the Mg/Ti fraction y. A linear relation between ▵H and ▵S is observed for 0.62MgH2 crystallizes in the rutile instead of the fcc phase, leading to a deviation from linearity.

Figure 1

Pressure‐optical Transmission‐Isotherms PTIs for 67 MgyTi1–y alloys with 0.62

Figure 2

a) Enthalpy (filled symbols) and entropy (open circles) for the 67 MgyTi1–y alloys shown in Figure 1. b) Enthalpy versus Entropy plot from the ΔH and ΔS values of the left panel. The compensation temperature is T=470.1±1.3 K. The coefficient of determination of the fit is R=0.9995 and T=404.8 K. The colours of the data points are the same as in Figure 1. c) Van ‘t Hoff plots constructed from the ΔH and ΔS values in Figure 2a. For clarity at each temperature the average < lnP > taken over all y is subtracted from the lnP . The colours of the data points are the same as in Figure 1. d) Temperature dependence of the spread of lnP values calculated by means of Eq. (6). The minimum spread occurs at T min=470.3 K. The experimental temperature range 348 to 473 K is indicated as blue rectangle.

Pressure‐optical Transmission‐Isotherms PTIs for 67 MgyTi1–y alloys with 0.62Hydrogenography. The colours correspond to the Mg concentration increasing in steps of Δy=0.00288 from 0.62 (blue) to 0.81 (red). The preparation of the MgyTi1−y (0.62≤y≤0.81) thin film gradient samples is described in the Experimental Section 4. The logarithm of the optical transmission change ln(T/T 0) is linearly related to the hydrogen concentration in the MgyTi1–y gradient films. The transmission in the metallic state of the film is T0. The horizontal x‐axes of all the PT Is are normalized to 1. The equilibrium pressures used for the Van ‘t Hoff plots in Figure 2 are evaluated at x=0.5. The harmonic mean temperature is T 404.8 K.

The slope of the linear regression line defines the so‐called compensation temperature:

We obtain T=470.1±1.3 K for the data in Figure 2b. While T differs from the harmonic average temperature T=404.8 K, it still falls within the experimental range [348, 473 K] of temperatures. While this suggest a statistical origin of the ΔH‐ΔS compensation, the large number of investigated samples (N=67) leads to a small standard error σ=1.3 K for T. The Krug‐Hunter‐Grieger criterion for an EEC of statistical origin26

is clearly not satisfied as ≅2. Accordingly, a non‐statistical origin of the ΔH‐ΔS compensation in Figure 2 is thus highly probable (at a 95 % confidence level).

The very high value of the coefficient of determination R=0.9995 of the experimental data in Figure 2 suggests that there should be a well‐defined crossing of the 67 Van ‘t Hoff plots at T=470.10 K. Indeed, at the level of the PTIs in Figure 1 there is a clear indication that a genuine crossing of Van ‘t Hoff plots does exist: at temperatures 348, 383 and 438 K the equilibrium pressure for MgyTi1–y alloys increases with increasing Mg concentration while at 473 K there is an opposite trend. This implies that somewhere in the interval between 438 and 473 K the equilibrium pressures should be essentially independent of the alloy composition, or, equivalently, that the spread of Van ‘t Hoff plots should have a minimum at a temperature T min such that 438

with h=ΔH /R and s=ΔS /R. The index i runs from 1 to 67 and labels the Mg−Ti alloy compositions. For clarity at each temperature the average taken over all y is subtracted from the lnP.

To quantify the coalescence of the Van ‘t Hoff plots we define LnP‐Spread(T), the temperature dependent spread of Van ‘t Hoff plots, as follows:

where N is the number of measured samples and the < > bracket indicates the average taken over all samples. Although LnP‐Spread(T) can directly be evaluated numerically from the Van ‘t Hoff plots (see Figure 2d), we derive in Section 2.3 of the Supporting Information the following analytic expression:

which shows explicitly that LnP‐Spread(T) is solely determined by the enthalpy and entropy variances and their enthalpy‐entropy covariance:

and

From Eq. (7) follows immediately that the smallest lnP spread occurs at the temperature

and is given by

The experimental data in Figure 2 lead to VAR(ΔH)=14.70×106 J/(moleH2), VAR(ΔS)=66.496 J/(K.moleH2) and COVAR (ΔH, ΔS)=30.793×103 (J/moleH2)2/K and consequently to VAR(h)=212700 K2, VAR(s)=0.9620 and COVAR(h,s)=445.483 K. With these values we obtain T min=470.33 K and in excellent agreement with the minimum in the calculated curve based on Eq. 6 shown in Figure 2d. As a result of the very high value of the coefficient of determination R=0.9995 of the ΔH versus ΔS fit, the difference between T 470.1 K and T min=470.33 K is very small. One has to realize, however, that these two temperatures are essentially different as T is given by (see Section 2.2 of the Supporting Information):

This expression that is clearly different from that in Eq. (11). This is explicitly shown in the next Section using the thermodynamic data derived from pressure composition isotherms measured by Syrenova et al33 on Palladium nanocubes of various sizes.

Enthalpy‐Entropy Compensation in Pd−H Nanocubes

From the enthalpy and entropy values for the hydrogenation of Pd nanocubes33 of sizes between 17 and 63 nm measured between 303 and 333 K, we obtain a linear ▵H and ▵S plot with R=0.9793 (see Figure 3). The compensation temperature T=282.7 K is slightly lower than the lowest experimental temperature, 303 K. Classically, together with the relatively high value of the coefficient of determination R, this is an indication for a non‐statistical behaviour. In addition, the Krug‐Hunter‐Grieger criterion26 for a genuine, non‐statistical EEC is met, since T 317.6 K falls just outside the statistical range [Tcomp‐2.2σ, Tcomp+2.2σ]=[255, 310 K]. However, the large scatter in ▵H and ▵S data lead Syrenova et al. to conclude that the compensation effect is probably due to statistical effects.33 Hence, according to the present understanding, it is difficult to decide whether a further analysis of a physical reason for the compensation effect is called for.

Figure 3

a) Enthalpy (filled symbols) and entropy (open circles) for Pd nanocubes of sizes between 17 and 63 nm determined from pressure‐composition isotherms between 303 and 333 K in Ref.33. b) Corresponding ΔH versus ΔS plot. The compensation temperature is T=282.7±12.4 K and lies, therefore outside the temperature range (303–333 K) of the measurements with average temperature T=317.6 K. The R is 0.97926. c) Van ‘t Hoff plots constructed from the ΔH and ΔS values in Figure 3a. For clarity at each temperature the average taken over all nanocube sizes is subtracted from the individual lnP. The colours of the data points and lines in panels (a), (b) and (c) indicate the nanocube sizes. d) Temperature dependence of the LnP‐Spread calculated by means of Eq. (6). The minimum spread occurs at T=288.6 K. This value is in excellent agreement with the result predicted by the analytic Eq. (11). The experimental temperature range 303 to 333 K is indicated as coloured rectangle.

A much clearer picture emerges from our analysis shown in Figure 3c, obtained by generating Van ‘t Hoff plots from the enthalpy and entropy values in Figure 3a. We find that there is no well‐defined crossing region. The LnP‐Spread(T) varies by only 25 % within the experimental range 303–333 K. Quantitatively, our analysis of the Pd nanocube data yields VAR(ΔH)=4.601×106 J/(moleH2), VAR(ΔS)=56.389 J/(K.moleH2) and COVAR (ΔH, ΔS)=14.713×103 (J/moleH2)2/K. The corresponding values VAR(h)=66559 K2, VAR(s)=0.8158 and COVAR(h,s)=212.85 K lead to T min=288.65 K and in excellent agreement with the calculated curve in Figure 3d.

In the next two Sections we show that both the Mg−Ti alloy and the Pd‐nanocube data lead naturally to the definition of a single parameter that 1) describes quantitatively the degree of coalescence of Van ‘t Hoff plots near T min (or T without generating the Van ‘t Hoff plots from the ▵H and ▵S data, and 2) allows to characterize the likeliness of the statistical origin of the EEC.

Compensation Quality Factor for the Isoequilibrium Region

Although the R values for the thin film MgyTi1–y alloys (R=0.9995) and the Pd nanocubes (R=0.97926) are both close to unity it is evident from Figure 2d and Figure 3d that the crossing region of the Van ‘t Hoff plots is much better defined for the MgyTi1–y alloys. For the MgyTi1–y alloys the LnP‐Spread varies by more than one order of magnitude within the experimental temperature range. For Pd nanocubes this spread varies only by 25 %. This means that the minimum spread is only slightly smaller than the largest LnP‐Spread actually measured for the nanocubes. To quantify the essential difference between the MgyTi1–y alloy and Pd nanocube data we introduce a Compensation Quality Factor (CQF) defined as:

where

is the minimum of the lnP variance and

is the largest variance of lnP actually measured in the experiment. As the variance is a quadratic function of (1/T) the largest variance occurs either at T*=T if 1/T min is closer to 1/T or at T*=T if 1/T min is closer to 1/T. The CQF is equal to 1 for a perfect compensation with all Van ‘t Hoff plots intersecting at a common isoequilibrium point in the pressure‐temperature plane. In an experimental situation we always have to deal with experimental errors. The isoequilibrium point is then replaced by an isoequilibrium region characterized by a finite spread in lnP at T min. Speaking of an isoequilibrium region is only meaningful if the LnP‐Spread at T min is clearly smaller than the largest spread observed in the experiment. Hence, we choose as isoequilibrium criterion:

which means that LnP‐Spread at T min needs to be 10 times smaller than at the temperature T*. As from Eq. (18):

we find then with T*(MgTi)=348 K and T* (Pd)=333 K, and . From Eq. (14) and the minimum variances indicated above we obtain:

and

This means that the isoequilibrium criterion Eq. (17) for a well‐defined crossing region of Van ‘t Hoff plots is only satisfied for the Mg−Ti alloys and not at all for the Pd‐nanocubes. The definition of CQF in Eq. (14) is solely based on the measured Van ‘t Hoff plots and does not depend on whether T min falls within or outside the range of measurements.

In literature many articles do not show the Van ‘t Hoff plots but list the derived enthalpies and entropies. It is therefore useful to express the CQF directly in terms of ▵H‐▵S data. This is readily obtained by noting that the coefficient of determination R of a ▵H versus ▵S plot (such as those in Figure 2 and Figure 3) is given by Eq. (S21) in the Supporting Information:

From Eqs. (11), (13) and (21) follows then:

Equation (22) is especially interesting as it expresses CQF in terms of the parameters R and T that are directly obtained from a linear fit to enthalpy and entropy data for a given set of experiments. It is well suited to understand how the Compensation Quality Factor depends on the quality of the linear fit of the ΔH versus ΔS plot, which is usually taken as the value of the R parameter. Equation (22) shows explicitly that values of R very close to 1 are needed to satisfy the isoequilibrium criterion in Eq. (17). This key result is easily illustrated by using in (22) the same characteristic temperatures T=432.9 K and T*=348 K as for the Mg−Ti alloys. In order to satisfy the criterion in Eq. (17) we need R>0.99875. Similarly for the nanocubes with T 432.9 K and T*=333 K, R >0.99977 is needed. In other words R is not an adequate indicator of a well‐defined isoequilibrium point. Linear ΔH versus ΔS plots are easily obtained even for sets of Van ‘t Hoff plots with a low CQF. This is demonstrated in Section 4 of the Supplementary information where we show that an extremely high R=0.99987 can still lead to a rather low CQF if the average experimental temperature is close to T and the denominator in Eq. (22) is small.

CQF Parameter as Discriminator for a Non‐Statistical Origin of Enthalpy‐Entropy Compensation

As described in the introduction a fundamental question about EEC is whether or not the linear relation is due to statistical errors or due to a genuine physical/chemical/biological compensation. We show here that the CQF can indeed provide a way to answer this question quantitatively. Specifically we show that it is extremely unlikely that data satisfying the criterion CQF>0.9 are the result of independent statistical errors.

For this we ran a large number of simulations in which we randomly generated Van ‘t Hoff plots for N=4, 8, 16 and 32 samples in the temperature interval [T T]. The only constraint is that the random pressures are drawn independently within a chosen interval [lnP min(T), lnP max(T)] and [lnP min(T), lnP max(T)]. More details about the simulations are given in Section 3 of the Supplementary Information. For each simulation we calculate T, R min and CQF. The distribution of simulated CQF values are shown as histograms in Figure 4.

Figure 4

Histograms and cumulative percents for the CQF parameter obtained from 105 simulations with randomly generated van ‘t Hoff plots for N=4, 8, 16 and 32 samples. Details of the simulations are given in Section 3 of the Supporting Information. The threshold CQF values γ [N; CL%] for 95 % (black dashed line), 99 % (red line) and 99.5 % (dashed blue line) confidence levels indicated in the central panel are well described by the simple algebraic relations given in Eqs. (23), (24) and (25), respectively. If the CQF falls within the red‐dotted region the simulations indicate that the EEC is of statistical origin at a 99 % confidence level.

For N=4 the values of CQF are spread between 0 and almost 1, whereas with increasing number of samples the CQF distribution is compressed towards increasingly lower values. Clearly, CQF values close to one become less likely with increasing number of samples N, as shown by the cumulative percentage curves in blue. Defining a threshold value γ [N; CL%] as a function of the number of samples N and the confidence level CL%, we observe that 99 % of the simulated CQF values are smaller than γ=0.935, γ=0.728, γ=0.586 and γ=0.487 for N=4, 8, 16 and 32 samples, respectively. These threshold values depend of course on the choice of the confidence level CL% and are well described by:

These lines are plotted in the middle panel of Figure 4: If the CQF falls within the red‐dotted region the simulations indicate that the EEC is of statistical origin at a 99 % confidence level. The asymptotic value is given in Eq. (S49).

For a 99 % confidence level Eq.(24) implies that γ [N=5; 99 %]≅0.853 and there is only a 1 % chance that a CQF larger than this value is due to statistical effects. As γ[N; 99 %] is a decreasing function of N the following three cases may occur when N≥5:

When CQF≥0.9>γ[N; 99 %] the isoequilibrium criterion is satisfied and the Van't Hoff plots coalesce in a small region of the pressure‐temperature plane. There is a 99 % chance that the EEC is of non‐statistical (i. e. physical, chemical or biological) origin.

When 0.9>CQF≥γ[N; 99 %] the EEC is of non‐statistical (i. e. physical, chemical or biological) origin. The crossing of the Van ‘t Hoff lines occurs over a wider region of the p‐T plane.

When γ[N; 99 %]>CQF the EEC is of statistical origin and there is essentially no coalescence of Van ‘t Hoff lines. For example, for N=10 samples, γ[N; 99 %]=0.67. A CQF smaller than 0.67 implies that LnP‐Spread at T min is only 3 times smaller than the largest measured LnP‐Spread.

We have now all the ingredients to draw conclusions about the Mg−Ti alloys and Pd‐nanocube data for which

and

The large CQF value for the 67 Mg−Ti alloys, which satisfies the isoequilibrium criterion Eq.(17), is much higher than the threshold γ [N=67; 99 %] and a statistical origin of the observed ▵H‐▵S compensation in Figure 2b can safely be discarded. For Pd‐nanocubes, however, the linear ▵H versus ▵S plot is probably due to statistical errors since CQF is much smaller than γ [N=13; 99 %]. This conclusion is confirmed by the analysis of hydrogenation data of Pd nanocubes reported by various research groups in Ref. [40].

The method described so far can easily be used to analyze any (published) ▵H‐▵S data. In the Supplementary Information we apply it to the largest ▵H‐▵S data set ever published. The latter data20 are obtained for hydrogen absorption obtained in parallel from 3859 MgyNizTi1–y–z thin film alloys and provide an additional insight about ▵H‐▵S compensation and correlation as well as isoequilibrium temperature in systems with very large numbers of samples. For example, 668 MgyNizTi1–y–z samples with a high Mg‐content (0.79ellipse. There is therefore a high degree of correlation but the coefficient of determination R=0.84 and CQF=0.29 are relatively small and there is no EEC.

Quality Factor for Isokinetic Relationship

The analysis developed in the previous sections can be applied straightforwardly to kinetic studies. The rate constant k of thermally activated processes is usually well described by an Arrhenius expression:43

where E is the apparent activation energy. Equation (28) implies that a plot of ln k measured for sample versus the reciprocal temperature is a straight line with slope E and intercept lnA. With these fitted values one can generate lnk by means of:

From a comparison of Eq. (29) with Eq. (2) we see the following correspondence between the enthalpy and entropy for equilibrium measurements and the activation energy and prefactor of kinetic measurements:

All the relations derived for the equilibrium measurements can therefore be adapted to the analysis of kinetic data by using the substitutions in Eq. (30). To illustrate this we analyse the kinetic data of the Mg‐based hydrides reported by Andreasen et al..28 The values of activation energies and prefactors obtained from a linear fit to rate constants measured during dehydrogenation of 8 samples are shown in Figure 5.

Figure 5

a) Modified Cremer–Constable plot of the kinetic dehydrogenation data of the 8 Mg based hydrides given in Table 1 of reference 28. The isokinetic temperature is T=678.4 K and the coefficient of determination R is 0.973. b) Arrhenius plots constructed from the E and RlnA values in panel (a). For clarity at each temperature the average < lnk > taken over the 8 hydrides is subtracted from the individual lnk. The colours of the lines are the same as in a). c) Temperature dependence of the spread of lnk values calculated by means of Equation (29). The minimum spread occurs at T min=697.3 K. This value is in excellent agreement with the result predicted by the analytic Equation (35). The experimental temperature range 629 to 662 K is indicated as yellow rectangle. Within this range the spread of lnk varies only little. This leads to the small CQF=0.162.

The slope of this modified Cremer‐Constable plot,43 the so‐called isokinetic temperature T is the analog of the compensation temperature T introduced in Eq. (31):

The Compensation Quality Factor for kinetic data is given by the same expression as Eq. (22) with T replaced by T:

with, as shown in Section 2.4 in the Supporting Information:

For the dehydrogenation of the 8 Mg‐based hydrides, the experimental data in Figure 5 Eq. (32) together with T*=629 K leads to:

Although R=0.973 is close to unity the low value of the CQF indicates that there is no well‐defined isokinetic region. This is mainly due to the narrow range of experimental temperatures, 629 to 662 K, which is only 5 % of the average temperature. Furthermore, as:

we conclude within a 99 % confidence level that the E versus lnA compensation is of statistical origin. In the Supplementary Information we apply it to the thermodynamic activation parameters of fish myofibrillar ATPase enzyme.44, 45

Conclusions

The Compensation Quality Factor CQF plays a pivotal role in the characterization of an Enthalpy‐Entropy Compensation (EEC). Before summarizing its main properties we provide a quick recipe for its determination. For this we assume that the enthalpy h=ΔH /R and entropy s=ΔS /R for a series of N samples (i=1,…,N) have been determined from measurements within the temperature range [T]. The existence of a genuine isoequilibrium state and the likeliness of a non‐statistical origin of the EEC are then easily evaluated following the steps:

Using standard software (for example Excel) calculate the variances VAR(s), VAR(h) and covariance COVAR(h,s)

Calculate with Eq. 11, T min =(N‐1)VAR(h)/N/COVAR(h,s)

Calculate with Eq. 12, VAR min =VAR(s)‐[N/(N‐1)COVAR(h,s)]2 /VAR(h)

Determine T* from the condition: If|1/ T min−1/T|<|1/T min−1/T|then T*=T otherwise T*=T

Calculate with Eq.18, VAR max =(1/T*)2 VAR(h)‐2 N/(N‐1)(1/T*)COVAR(h,s)+VAR(s)

The Compensation Quality Factor CQF is then simply CQF=1‐(VAR min /VAR max)1/2

The power of the single CQF value is to characterize quantitatively both the degree of coalescence of Van ‘t Hoff plots and the probability of the statistical origin of the enthalpy‐entropy compensation (EEC):

A genuine isoequilibrium state exists if CQF >0.9. This condition guarantees that the minimum spread in the N Van ‘t Hoff plots is one order of magnitude smaller than the largest measured spread.

At a 99 % confidence level the observed EEC is a real effect, i. e. not due to a statistical artefact if CQF>γ [N;99 %]=0.29+1.41/N 0.57

With this procedure we have established that for the 67 thin Mg−Ti−H films, CQF=0.937 while CQF=0.262 for the 13 Pd−H nanocubes. We conclude therefore that a true coalescence of Van ‘t Hoff plots occurs only for the Mg−Ti alloys.

Furthermore, from CQF (67 Mg−Ti alloys)=0.937>γ [N=67; 99 %] and CQF (13 Pd nanocubes)=0.262<γ [N=13; 99 %] it follows that at a 99 % confidence level the EEC for the Mg−Ti−H films is a genuine compensation effect, while it is of statistical origin for the Pd−H nanocubes.

Although experimental data for metal‐hydrides have been used in this work, any results obtained from equilibrium measurements based on Van ‘t Hoff plots as well as from kinetic measurements based on Arrhenius plots can be analyzed with the method described in this work. This is explicitly demonstrated for the dehydrogenation kinetics of 8 Mg‐based hydrides and the thermodynamic activation parameters of fish myofibrillar ATPase enzyme. The CQF values CQF (8 Mg samples)=0.162 and CQF (7 Fishes)=0.746 are too low for a clear isokinetic state. As γ [N=8; 99 %]≅0.72 the EEC observed for the 8 Mg samples is of statistical origin. For the 7 fishes the situation is borderline as γ [N=7; 99 %]≅0.755 is very close to the CQF value. An attractive feature of our analysis is that CQF depends only on enthalpy and entropy values and the experimental temperature range. This is a great advantage for the evaluation of published data for which very often Van ‘t Hoff data are not available.

Finally, it is important to mention that the very large data set for hydrogen absorption in 3859 MgyNizTi1–y–z alloys exhibit a high degree of enthalpy‐entropy correlation that is not always linear. However, for some large families of alloys (for example the 668 Mg rich alloys) EEC does occur with CQF much larger than γ [N=668; 99 %] implying a non‐statistical origin of the EEC.

Experimental Section

Mg−Ti Sample Preparation

MgyTi1–y thin films with a compositional gradient are prepared in a 7‐gun ultra‐high‐vacuum dc/rf magnetron co‐sputtering system (AJA Int., base pressure 10−7 Pa) at room temperature and in 3 μbar of Ar on 50×5 mm2 quartz substrates. Mg and Ti are facing each other in tilted off‐axis sputtering guns. By adjusting the power applied to each gun the desired region of the binary phase diagram is obtained. The Mg fraction of the 50 nm MgyTi1–y films along the length of the sample varies between 0.62≤y≤0.81. The Mg films is sandwiched between two 10 nm Fe layers and capped by 10 nm Pd. The iron layers serve to minimize the interaction of the Mg with the substrate and the Pd layer. The latter promotes H2 dissociation and prevents oxidation of the underlying film. While the Fe and Pd are deposited on a rotating substrate resulting in uniform thicknesses, the MgyTi1–y compositional gradient layer is obtained by co‐sputtering from Mg and Ti sources tilted towards a stationary substrate. The composition of the films is gauged by measuring the thickness gradients of single element films. The same sputtering system is used for ternary MgyNizTi1–y–z gradient thin films described in the Supplementary information. In this case, the Mg, Ti and Ni off‐axis sputter guns are positioned every 120° on a circle and 3‐inch diameter sapphire substrates are used.20

Hydrogenography

After deposition, metallic films are transferred into an optical cell to monitor their optical transmission during hydrogenation.20 The whole cell is placed in an oven to control temperature up to 300 °C. The complete thermal equilibration of the setup is checked by comparing the output of two PT100 resistors placed at different locations in the oven, one of them being in contact with the sample holder. A 150 W diffuse white light source illuminates the sample from the substrate side, and a 3‐channel (RGB) SONY XC‐003 charged‐coupled device (CCD) camera continuously monitors the transmitted light as a function of hydrogen pressure. The 3‐channel transmission intensities are added, resulting in a 1.1 to 3.3 eV photon energy bandwidth. The transmission intensity is integrated over the width of the sample as no significant change of composition or thickness occurs across this direction. The gas pressure increase is controlled by a MKS 248/250 forward Proportional‐Integral‐Differential (PID) system that regulates both inlet and outlet gas flows. 0.1 % to 100 % hydrogen in argon mixtures are used to achieve hydrogen (partial) pressures of between 10−1H2)<106 Pa. Typical pressure sweeps have a duration of 8.6×104 s.

The Pressure‐Transmission‐Isotherms (PTIs) for the Mg−Ti samples are obtained by measuring the optical transmission of every spot (expressed in pixels) of the gradient sample as recorded by the 3CCD camera. The transmission data are averaged in a direction perpendicular to the composition gradient. The background pressure of the cell is 10−2 Pa. The hydrogen pressure is gradually increased exponentially from 4 to 300 kPa, depending on the temperature. For this purpose, a 4 % H2/Ar gas mixture and pure (100 %) H2 gas are used with a gas flow set to 20 sccm. Thus, in the gradient MgyTi1–y thin films the hydrogenation is optically monitored for all compositions simultaneously, which minimizes systematic experimental errors, the hydrogen gas pressure and the temperature being exactly the same for all samples. More details about hydrogenography of 3859 Mg−Ti−Ni samples are given in Refs.46 and47.

Author Contributions

C. Boelsma. and R. Griessen developed the Enthalpy‐Entropy Compensation verification method. C. Boelsma, R. Gremaud. and Ch.P. Broedersz designed and performed the experiments and analyzed data. H. Schreuders prepared the samples and provided technical support. R. Griessen and B. Dam wrote and edited the manuscript and supervised the project.

Conflict of interest

The authors declare no conflict of interest.

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Supplementary

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