4.6 Calculated Transition Entropies for the CsHSO 4 -CsH 2 PO 4 System of
4.6.6 Summary of entropy calculations for high temperature phases
temperature phases of these compounds. For CsHSO4, CsH2PO4, and the pure cubic compounds, the total entropy of the high temperature phases was calculated using only the ideal entropy of mixing, Eq. (4-1), and applying the adjusted ice rules, Eq. (4-6). The compounds which transform to both tetragonal and cubic phases at elevated temperatures required an additional assumption to calculate their total entropy. This was that the tetragonal phase consists of pure CsHSO4. The entropy of these compounds was then straightforwardly appraised by Eq. (4-1) and Eq. (4-6). Finally, the
Cs6(H2SO4)3(H1.5PO4)4 compound obligated multiple assumptions, the most central of which was that there exist cesium vacancies in the cubic high temperature phase. This
calculation is quite speculative due to the lack of data concerning this particular high temperature structure and will need further experimental input to become more conclusive.
Table 4.5 Calculated entropies for high temperature phases
Compound Smix
(J/mol*K)
Sconfig (J/mol*K)
Stotal (J/mol*K)
CsHSO4 0 14.9 14.9
Cs3(HSO4)2.50(H2PO4)0.50 2.44 17.26 19.7 Cs3(HSO4)2.25(H2PO4)0.75 2.88 19.33 22.21
Cs3(HSO4)2(H2PO4) 5.29 21.54 26.83
Cs5(HSO4)3(H2PO4)2 5.6 20.95 26.55
Cs2(HSO4)(H2PO4) 5.76 23.76 29.52
Cs6(H2SO4)3(H1.5PO4)4 9.09 27.25 36.34
CsH2PO4 0 27.3 27.3
4.6.7 Calculated ∆Strans and comparison with experimental ∆Strans
The calculated transition entropies for these cesium sulfate-phosphate compounds are then simply the values in Table 4.4 subtracted from those of Table 4.5. We can compare these numbers to the measured entropies by dividing the experimental transition enthalpies by the mean of the various transition temperatures listed on Table 4.3. The results of this comparison are shown graphically in Figure 4.21 and listed in Table 4.6.
The first thing one should observe when viewing Figure 4.21 is the very satisfactory
agreement between the experimental and calculated transition entropies, which are quite often within error of each other.
0 20 40 60 80 100
15 20 25 30 35 40
T ransition Entropy (J/mol*K)
% PO
4∆ S
exp.∆ S
calc.Figure 4.21 Measured versus calculated transition entropies. The shape of the calculated curve closely mimics that of the experimental. Note calculated and experimental values are nearly identical for CsHSO4, for which the subjective evaluation of the room temperature entropy was not necessary.
The sometimes large errors in the experimental entropies are due mainly to the ambiguity in Tc caused by the large range over which some of the compounds transform.
From a thermodynamic perspective, one might expect that the onset temperatures, Tonset(DSC) and Tonset(σ), would tend to underestimate ∆Htrans because the compound has
not actually reached equilibrium with respect to the high temperature phase until the transition is complete. Conversely, the final temperatures, Tpeak(DSC) and Tfinal(σ), will tend to overestimate ∆Htrans as the room temperature phase stopped being the most energetically favorable phase at Tonset. For these reasons, the mean value of the transition temperatures was taken as Tc for each compound, which led to large errors in the
experimental entropies for compounds with extend transition temperature ranges.
It is interesting that the calculated and measured transition entropies for CsHSO4
are very similar. This would tend to confirm not only the hypothesis that CsHSO4 has two (rather than four) orientations in its tetragonal phase, but also justify the use of the mean transition temperature for the following reason: this compound had zero entropy in its room temperature structure and therefore the somewhat subjective entropy evaluation of its room temperature phases was avoided. Consequently, the calculated entropy for CsHSO4 should have the least amount of unaccounted for entropy. The nearly perfect match of calculated and experimental values is then very reassuring. The systematically lower values of the calculated, compared to experimental, entropies for the rest of the compounds are probably a combination of the fact that the maximum reasonable amount of entropy was assigned to the room temperature phases and that only the mixing and configurational contributions to the transition entropy were evaluated. Even the
calculated entropies for the Cs6(H2SO4)3(H1.5PO4)4 compound have the right magnitude, although this result must be taken with a large grain of salt considering the amount of speculation that went into the entropy evaluation of this compound’s cubic structure.
Table 4.6 Calculated and experimental transition entropies.
Compound Tc (mean)-
(K)
∆Hexp - (kJ/mol)
∆Sexp = ∆Hexp / Tc
(J/mol*K)
∆Scalc - (J/mol*K)
CsHSO4 419(3) 6.2(2) 14.8(6) 14.90
Cs3(HSO4)2.50(H2PO4)0.50
407(7) 7.4(2) 18.2(8) 16.82
Cs3(HSO4)2.25(H2PO4)0.75 401(11) 8.3(5) 20.7(18) 19.21 Cs3(HSO4)2(H2PO4)
395(17) 10.7(2) 27.1(17) 24.91
Cs5(HSO4)3(H2PO4)2 375(11) 9.2(7) 24.5(26) 22.22 Cs2(HSO4)(H2PO4) 364(12) 8.3(2) 22.8(13) 20.88 Cs6(H2SO4)3(H1.5PO4)4 382(14) 15.1(6) 39.6(30) 36.34 CsH2PO4
505(4) 11.3(5) 22.4(12) 21.54
Finally, it should be noted that although the investigations into the entropic driving force of these compounds were originally propelled by an apparent correlation between phosphorous content and Tc (see Figure 4.6 a)), the final results deny any such relationship. It was originally thought that the lowering of Tc with rising phosphate percentage indicated that ∆H was remaining relatively constant while ∆S increased with phosphorous content. However, as more data became available, it became clear that this was not in fact the case. With the full data set available to us now, it would seem that although there are undoubtedly very general effects to increasing the phosphorous
content, the particulars of the room temperature structures far outweigh any such effects.
This conclusion is quite evident in Figure 4.6 d), a plot of molar H-bond energy versus %PO4, where one might have guessed a priori that the energy associated with the hydrogen bonds would increase fairly linearly with phosphate, and therefore hydrogen, content. In fact, starting with just the end members CsHSO4 and CsH2PO4, such a linear relationship would have seemed justified as the hydrogen bond energy (per mole
CsHXO4) of CsH2PO4 is almost twice that of CsHSO4. The intermediate compounds, however, fall far from the line connecting the two end members and it can only be said very generally that increasing phosphate/hydrogen content correlates to higher molar hydrogen bond energies.
It is then even more pleasing that Pauling’s ice rules, adjusted to properly
describe the superprotonic phases of these cesium sulfate-phosphate compounds, produce transition entropies that compare very well with the measured values. Since these rules combine the positional disorder of the proton system with the rotational disorder of the tetrahedra, it should be applicable to any transition that involves a disordering of a hydrogen-bonded network via disorder of the hydrogen carriers. This has already been shown to be true in compounds where the hydrogen-bonded network is composed of water molecules and would now appear to be true for systems containing hydrogen- bonded tetrahedra.
4.6.8 Application of the adjusted ice rules to other superprotonic