Proton jumps in crystalline hydronium perchlorate

Physica

72 (1974) 168-I 78

PROTON

0

North-Holland

Publishing

Co.

JUMPS IN CRYSTALLINE

HYDRONIUM

PERCHLORATE J. M. JANIK Institute

and M. RACHWALSKA

of Chemistry

of the JagieiIonian

Krakdw,

University,

Poland

and J.A. JANIK Institute

of Nuclear

Krakdw,

Physics,

Poland

Received 8 August 1973

Synopsis Calorimetric measurements have been made for solid hydronium perchlorate (H30C104). The existence of the first-order phase transition at 248.4 K hes been proved and the thermodynamic parameters of this transition have been evaluated. Cuasielastic ceutron-scattering measurements have been made for the same substance below and above the phase trarsiticn by applying the method of differentiation of the beryllium edge. In the high-tcmperatme phase a lorentzian component of the quasi-elastic peak has been obtained. From its width the average time between proton jumps in a double-well potential of the hydrogen bond has been evaluated cu. 3.5 x lo-” s.

1. Introduction. In earlier works the existence of two crystalline phases in perchloric acid monohydrate (hydronium perchlorate) was established’-‘). It was discovered that phase II, which exists below the phase transition point at 248 K, corresponds to a monoclinic structure (P2&) whereas phase I (above the transition temperature and below the melting point at 323 K) corresponds to an orthorhombic structure (Pnmu). In both phases the existence of torsional motions of the hydronium groups was established on the basis of both NMR2p4) and neutron inelastic-scattering measurements 5-7), the rotational activation barriers being 4.8 kcal/mole for phase II and 4.2 kcal/mole for phase I. As the thermodynamic phase-transition parameters at 248 K are still lacking we decided to perform the specific-heat measurement for the hydronium perchlorate crystal. Also, the neutron inelastic-scattering measurements5) indicate an interesting difference between phase I and phase II, when observing the neutron168

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JUMPS

IN CRYSTALLINE

HYDRONIUM

PERCHLORATE

169

scattering region. Whereas in phase II the beryllium cutoff edge is very sharp, corresponding indeed to elastic scattering, in phase I there appears a distinct broadening of the beryllium edge suggesting the existence of stochastic proton jumps. Taking this fact into consideration, we decided to analyse quantitatively the elastic region of the neutron data obtained in ref. 5. Section 2 describes some details of the applied adiabatic-calorimetry technique and the results of specific-heat measurements. Section 3 describes the method of analysing the neutron elastic-scattering region by differentiation of the (broadened) step function characteristic of the beryllium filter, and the results of such an analysis obtained on the basis of neutron data from ref. 5. Section 4 presents a discussion. 2. Specific-heat measurements. Specific-heat measurements of the hydronium perchlorate crystal were performed with the adiabatic calorimeter, which was described in detail in ref. 8. The measurements were made in the temperature range from cu. 100 K to ca. 300 K. The calorimeter vessel was made of brass,

Fig. 1. Specific heat of solid hydronium

perchlorate.

170

J. M. JANIK,

M. RACHWALSKA

AND

J. A. JANIK

TABLE I

Specific-heat values of solid hydronium perchlorate Temperature

Specific heat

Temperature

Specific heat

(K)

(Cal/mole K)

(K)

(Cal/mole K)

96.71

13.84

187.95

22.94

101.59

189.28

23.33

107.02

14.19 14.77

191.51

23.45

108.90

14.97

193.25

23.44

112.60

15.34

195.19

23.87

113.84

15.56

196.84

24.05

117.94

15.99

198.49

23.96

118.02

16.10

200.75

24.57

121.96

16.07

202.14

24.51

123.28

16.62

204.10

24.59

126.21

16.96

205.04

24.80

128.44

16.98

205.97

25.00

130.91

17.16

207.52

25.12

133.45

18.47

209.22

25.04

135.85

17.71

209.99

25.81

136.18

17.76

211.16

25.38

140.89 141.44

18.07

211.38

25.75

18.11

212.94

25.21

144.00

18.41

214.39

25.32

146.14

18.72

215.13

26.11

149.29

18.96

216.24

26.35

151.49

19.11

216.29

25.79

151.63

19.27

218.38

26.46

154.57

19.56

219.48

26.59

156.75

19.65

221.31

26.88

159.75

20.45

222.22

161.63

20.23

223.62

26.58 27.14

161.92

20.17

224.40

27.29

164.85

20.76

225.19

27.71

166.53

21.02

226.28

27.73

166.99

227.74

27.72

170.25

20.65 21.20

228.85

28.06

170.80 171.98 174.46

21.40 21.19 21.81

229.30 230.10

27.87 28.41

233.16

175.30

21.88

28.46 29.42

177.18

21.73 22.17

234.40 234.62

28.68

237.51

29.46

22.32

238.73

29.60

183.56

22.37 22.59

239.22 239.75

29.56 30.15

186.11

23.07

240.32

25.99

178.48 180.65 182.59

PROTON JUMPS IN CRYSTALLINE TABLE 1

HYDRONIUM

171

(cont.)

Temperature (K)

Specific heat (Cal/moleK)

Temperature

242.07 243.17 243.90 244.06 244.86 244.91 245.28 246.88 247.05 247.18 247.47 248.13 248.19 248.31 248.37 248.38 248.39 248.40 248.43 248.45 249.31 249.98 250.07 250.41

30.33 34.10 45.52 45.46 70.51 57.32 161.7 74.66 77.27 143.3 168.3 378.7 5957 1925 co co 00 co co 1730 91.88 38.14 60.85 71.07

251.71 253.00 254.80 255.58 255.60 256.52 258.95 260.33 261.79 264.41 264.45 267.59 269.01 273.13 274.32 275.88 278.62 279.92 281.56 284.05 285.81 287.32 289.61 291.87

covered electrolytically

PERCHLORATE

(K)

Specificheat @al/moleK) 34.89 35.34 35.37 35.48 35.17 35.67 35.49 35.04 35.27 35.68 35.87 35.83 36.52 36.39 36.97 37.27 37.23 37.08 37.82 37.85 38.42 38.71 38.98 39.56

by a’ layer of gold in order to protect the brass against

the chemical action of perchloric acid monohydrate. The specific-heat values obtained are listed in table I and plotted in fig. 1. The maximum

deviation

experimental

of points from a smooth curve does not exceed 2%. The

points were obtained in five experimental runs. In the first two runs

the whole temperature range was investigated, in the next two the phase-transition region was additionally

investigated.

In the fifth run the phase-transition

region

was once more studied after the substance had been melted and resolidified.

In

this way the possible influence of impurities was expected to appear. Here it should be stressed that all five runs gave consistent results, thus showing negligible

in-

fluence of possible impurities. The heat-capacity

effect of the empty calorimeter

vessel was, of course, also

measured, so that the specific-heat data of table I and fig. 1 for the hydronium perchlorate

crystal are the result of the subtraction of the heat capacities of the

calorimeter vessel with the substance and the empty calorimeter

vessel.

J. M. JANIK, M. RACHWALSKA AND J. A. JANIK

172

The amount of hydronium perchlorate was 88.83 g, the mass of the calorimeter vessel amounted to 249.23 g. 3. Neutron incoherent quasielastic-scattering results. An analysis of the elastic part of the neutron spectra obtained after scattering from hydronium perchlorate was made on the basis of data obtained in ref. 5. In the same paper there is also a description of the experimental conditions and detailed information concerning the sample. The data are presented in fig. 1 of ref. 5. The elastic region of these data, being in the form of a beryllium edge, was transformed to the form proportional to the scattering law by using a generalization of the method described, for instance, in Turchin’s book9). A similar generalization was also applied by Larsson and Dahlborg”). The principle of the method is as follows. If neutrons, after having been scattered by the sample, are transmitted through the beryllium filter, the observed number of counts (No,_) is a convolution of the beryllium filter edge function (N) and a function @ which contains both the instrumental and the sample scattering-law contributions. Nob&) =

7N(t’)

CD(t’ - t) dt’.

6

t stands here for the neutron time of flight. We replace function N by a step function N(t’) =

F

for

t’ > rB,,

for

t’ <

tBe,

where tBe is the neutron time of flight corresponding to the beryllium edge. Hence No&)

= No

T@ (t’ -

dN,,, dt

= No@ (tB,

t) dt’ = No h-r

TBe

-

7@(Odt,

t)*

In this way we have shown that by differentiation of the elastic region of the data obtained by the beryllium-filter method we obtain directly the scattering law broadened by the instrumental resolution. We had at our disposal the four experimental runs of ref. 5. Two of these runs corresponded to hydronium perchlorate in phase II at temperatures 123 K and

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HYDRONIUM

PERCHLORATE

173

223 K, and two corresponded

to phase I at temperatures 258 K and 294 K. In addition to the procedure described above, we introduced several corrections. First of all, the spectrum of incident neutrons has an intensity variation following the maxwellian velocity distribution. In order to correct for this, all experimental points in the beryllium-edge region were multiplied by a factor L5, where 1 is the incident neutron wavelength. Moreover, it is convenient to express the scattering law not in the time of flight scale but in the energy scale; therefore a transformation @(E) = di [t(E)]

F

= CD[t(E)]

E - 3’2,

was needed. Finally the neutron cross section and the neutron scattering law are connected by the formula -=

d20

d0 do

A5

e-fio/k,T

s

(KY WI,

k,

where k. and k’ are the incident and scattered neutron wave vectors, respectively; ~CO= E,, - E’, EO and E’ are the incident and scattered neutron energies, respectively; and K is the neutron wave-vector transfer. Thus, in order to pass from the neutron cross section to the scattering law, we must divide all experimental points of the beryllium-edge region by (EJE)* exp (EBe - E)/k,T. After all these corrections we could expect that the curve obtained by differentiation would be a convolution of the scattering law and the gaussian resolution. The differentiation procedure was made in the time of flight scale by taking equal time of flight intervals corresponding to the sequence of the channel numbers. We were able to apply this procedure only to the left half of the quasielastic peak, because the right side was distorted by the appearance of some small Bragg peaks arising from the aluminium sample holder and the sample itself. Hence after obtaining half of the quasi-elastic peak, we plotted, for the sake of presentation, its mirror reflection on the right ride. It should be stressed here that this mirror reflection is justified only if our processed data are in the form proportional to the scattering law, which, according to all existing theories, must be symmetric. We should also point out that the differentiation procedure was made on the basis of the smoothed-out data of paper 5 and after the subtraction of the empty sampleholder background. The thus obtained neutron quasielastic-scattering results for hydronium perchlorate are presented in fig. 2. 4. Discussion. The calorimetric data give direct evidence of the phase transition in the hydronium perchlorate crystal. It is evident from fig. 1 that it is a first-order transition and occurs at temperature 248.4 K. A presentation of the occurrence

J.M. JANIK,

M. RACHWALSKA

AND J.A. JANIK

INCIDENT NEUTRON ENERGY w km-‘1

Fig. 2. QNS spectra of solid hydronium perchlorate. Inelastic background subtracted. Curves a and b: experimental points corresponding to temperatures - 150°C and - 50°C respectively. Solid curves are gaussian fits to the points. Curves c and d: experimental points corresponding to temperatures - 15 “C and + 21 “C, respectively. Solid curves are (6 -t GE) convoluted by gausSian fits to the points. (6 is the Dirac function, a is a fittable coefficient, L is the Lorentz function with fittable width.)

of the first-order transition is also made in fig. 3, where the sample temperature US. heat transfer to the sample is shown. The temperature of transition (248.4 + 0.1 K) is in good agreement with the value 248.25 K obtained by Rosolovskii and Zinovevl). The latent heat of phase transition obtained by us is AH = 1.30 4 0.03 kcal/mole and the corresponding entropy change at the transition is AS = AH/T = 5.25 + 0.12 Cal/mole K.

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JUMPS

Fig. 3. Sample-temperature

IN CRYSTALLINE

us. heat-transfer

HYDRONIUM

dependence

PERCHLORATE

for solid hydronium

175

perchlorate.

If we assume that the only difference between phase II and phase I lies in the fact that protons of an H30+ in phase II have well-determined positions, whereas in phase I each proton may choose at random one of two equilibrium positions in the double-well potential of the hydrogen bond, we shall find in phase I eight possible configurations of one H,O+ group as shown in fig. 4. The corresponding entropy change when passing from phase II to phase I should be AS = R In 8 = 4.13 Cal/mole K, which is to be compared with our experimental values 5.25 cal/ mole K. We should remember that some entropy change must occur due to the

Fig. 4. Eight configurations

of the H30+ group with protons jumping in a double-well potential.

176

J. M. JANIK, M. RACHWALSKA AND J. A. JANIK

crystal-structure change and to the frequency-spectrum change which are probably responsible for the difference. The neutron quasielastic-scattering results shown in fig. 2 lead to the following conclusions. In the low-temperature phase the peak represents an elastic-scattering process. Its form is nearly a gaussian representing the instrumental resolution. One may see that the differential method, as applied to the neutron spectra of the pulsed reactor at Dubna, leads to a very good resolution; the full width at half maximum (FWHM) of the gaussian is cu. 0.6 cm- ’ and AE/E = 1.4 %. It is also evident that the intensity of the elastic peak in phase II is lower at higher temperature (it has, however, the same width), in accordance with the fact that it is ruled by the Debye-Waller factor. In phase I the quasielastic scattering appears and the peak has a complicated form. According to existing theories of neutron quasielastic scattering by molecular crystals 11*12), the peak should consist of two components: the Lorentz one, broadened by the resolution, and the elastic one (6 function), also resolution-broadened. The result of a Lorentz theoretical fit to our experimental data is shown as a solid line in fig. 2. The FWHM parameter of the Lorentz functions for the two temperatures of phase I are J’zssK=2.68cm-’ and r,,,, = 3.48 cm-l. From these parameters it is possible to obtain the time between proton jumps from the formula t = 2fi/I’. This time at the two temperatures is tZS8k = 4.0 x lo-l2 s and tZg4k = 3.0 x 10-l* s. We now come to the point at which this time should be interpreted. At first one is tempted to consider it as the time between orientational jumps of the H,O+ group. If, however, one takes into consideration the relatively high barrier to rotation of this group, being, according to ref. 4 4.8 kcal/mole in phase II and 4.2 kcal/mole in phase I, in agreement with the neutron inelastic-scattering measurements5), one must come to the conclusion that proton jumps over this barrier must occur much less frequently. As a matter of fact, if we apply the Bauer formula13) t = (x1/2k,T)*

exp (AH/RT)

exp (-AS/R)

and use for estimation the following parameters: I, the moment of inertia of the H,O+ group with respect to the threefold symmetry axis, equal to 5.12 x 10-40gcm2, /lH, the activation enthalpy, equal to the barrier to rotation, i.e., 4.2 kcal/mole; and AS, the orientational activation entropy, equal to zero, we obtain tZS8k = 5.4 x 10-r’ s, i.e., two orders of magnitude more than our neutron quasielasticscattering experiments. Moreover, we must note that barriers to rotation of H30+ groups in phases I and II quoted above are not very different, so that the striking difference between fig. 2a, b and fig. 2c,d cannot be explained on the basis of change of orientational rate even if we put aside the estimation based upon the Bauer formula. Hence, the only conclusion we can come to is that an additional motion is

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JUMPS IN CRYSTALLINE

HYDRONIUM

PERCHLORATE

177

switched on when passing from phase II to phase I and we believe that this motion consists of proton jumps along each hydrogen bond, which are due to the existence of a double-well potential. An estimation of the barrier separating the two minima of this potential is possible on the basis of the Eyring formula14) t = (i&T)

exp (&!/RT)

exp (-AS/R).

Using for t the value 4.0 x lo-l2 s, corresponding to T = 258 K (i.e., - lS”C), and assuming dS = 0, we obtain for dH a value of 1.6 kcal/mole. This estimation should be considered as a preliminary one. It would be useful to study the neutron momentum-transfer dependence of the quasielastic peak in both phases. This study is under consideration in our laboratory. 5. Conclusions. 1) Calorimetric experiments made for solid hydronium perchlorate have shown that a phase transition of the first order occurs for this substance at 248.4 K. 2) The entropy change obtained in the calorimetric measurements at the phase transition point is 5.25 cal/mole K. 3) The elastic region of neutron scattering shows a single gaussian peak in phase II and the appearance of a Lorentz component in phase I. 4) It is proposed that the Lorentz component corresponds to proton jumps in a double-well potential characteristic for the hydrogen bond in phase I. The average time between these jumps is ca. 3.5 x lo- l2 s. 5) From the last statement it follows that in phase I there is a dynamical equilibrium: H,0C104 + H20*HC104. Acknowledgement. Our thanks are due to Dr. T Waluga, Mr. J.Mayer, and Miss T.Grzybek for their help in the calorimetric measurements. We also thank Dr. P. Goyal, Dr. W. Nawrocik, Dr.G. Pytasz, and Mr. K. Rosciszewski for helpful discussions.

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

Zinovev, A. A. and Rosolovskii, V. Ya., Zh. neorg. Khimii 3 (1958) 2382. Hennel, J. W. and Pollak-Stachura, M., Acta Phys. Polon. 35 (1969) 239. Nordman, C.E., Acta Cryst. 15 (1962) 18. O’Reilly, D.E., Peterson, E. M. and Williams, J. M., J. them. Phys. 54 (1971) 96. Janik, J.M., Pytasz, G., Rachwalska, M., Janik, J.A., Natkaniec, I. and Nawrocik, W., Acta Phys. Polon. A43 (1973) 3, 419. Janik, J.M., Janik, J.A., Bajorek, A. and Parlinski, K., Phys. Status solidi 9 (1965) 905. Janik, J.M., Acta Phys. Polon. 27 (1965) 491. Mayer, J. and Waluga, T., Report INP, No 75O/PI, Krakow (1971). Turchin, W.F., Myedlennye nejtrony, Moskva (1963). Dahlborg, U., Gr&Jund, C. and Larsson, K.E., Physica 59 (1972) 672.

178 11) 12) 13) 14)

J.M. JANIK,

M. RACHWALSKA

AND J. A. JANIK

Sktild, K., J. them. Phys. 49 (1968) 2443. Larsson, K.E., Phys. Rev. A3 (1971) 1006; Larsson, K.E., to be published. Bauer, E., Cahiers Phys. 20 (1944) 1. Glasstone, S., Laidler, K. J. and Eyring, H., The Theory of Rate Processes, (New York, 1941).

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