C. A. Angel1 and J. C.Tucker
278
Heat Capacities and Fusion Entropies of the Tetrahydrates of Calcium Nitrate, Cadmium Nitrate, and Magnesium Acetate. Concordance of Calorimetric and Relaxational “Ideal” Glass Transition Temperatures C. A. Angell” and J. C. Tucker Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 (Received August 30, 7973) Publication costs assisted by the National Science Foundation
The latent heat of fusion and of a solid-state transition and the heat capacities of crystalline, liquid, and vitreous Ca(NO3)~.4Hz0,Cd(N03)~.4H20,and Mg(OAc)z.4HzO have been determined by differential scanning calorimetry. In Ca(N03)~.4HzOand Cd(N03)~-4HzOover 80% of the entropy of fusion has been lost before the state of the supercooled liquid is frozen in at the glass transition. By contrast, only 30% is lost in the case of Mg(OAc)z.4HzO, a substance which is unusual in several respects. The ratio T,/To, 1.07, is unusually small among glass-forming liquids for the calcium and cadmium hydrates, implying that under hypothetical equilibrium conditions the change of heat capacity cannot be much less sharp than that observed at the experimental glass transition. The T,/To ratio for the Mg(0Ac)z. 4Hz0 system, however, is approximately 1.30, a value commonly quoted for organic liquids &d polymers. The data have been used to estimate “ideal glass” temperatures, To(cal), at which S(internal1y equilibrated liquid) = S(crptal). To(ca1) is found to be 200 f 4 K for Ca(NO&.4HzO and 198 f 4 K for Cd(NO3)2.4H20, in good agreement with the respective To(transport) values of 202 f 3 K and 194 3 K salts obtained from analysis of the temperature dependence of liquid mass transport processes.
*
Introduction In recent years the physicochemical properties of molten calcium nitrate tetrahydrate have been the subject of a number of investigations. These include studies of conductance,l” shear v i ~ c o s i t y ,impurity ~ ion self-diffusion coefficient^,^ bulk v i ~ c o s i t y ,and ~ glass transition temperature.6 Much of the interest in calcium nitrate tetrahydrate has been motivated by the relative ease with which it can be cooled below its melting point of 42.5“ and by the fact that it can be studied in the vitreous state for the case of small samples rapidly cooled. The glass transition temperature, according to dta studies, falls at 217 K.6 As is commonly found in the case of easily supercooled liquids, the mass transport properties near and below the melting point do not follow the familiar Arrhenius equation, but rather are well described by the empirical VogelTammann-Fulcher (VTF) equation? W(T) = AT-112 exp( -BIT - To) (1) where W(T) is the transport property of interest, e.g., conductance or fluidity and A, B, and TOare constants. TO, which has the dimensions of temperature, replaces O”K in the Arrhenius equation as the temperature of vanishing ionic mobility and has been called an “ideal” glass transition temperature by various authors. Several theories for eq 18-11 attribute thermodynamic significance to To suggesting that if the liquid could be cooled to this temperature while in a state of internal equilibrium, the magnitude of some thermodynamic quantity which determines the state of the liquid would fall to zero. In the most plausible of such theories, due to Adam and Gibbs,lo the thermodynamic quantity which vanishes at TOis the configurational part of the total liquid entropy. In accord with such interpretations, TOfor Ca(N03)~.4Hz0has been found to have a value almost independent of the transport process studied. A TOvalue The Journal of Physical Chemistry, Voi. 78, No. 3, 1974
of 201 K is found from conductance measurements in the temperature range 0-70”, while shear viscosity studies yield a slightly higher value of 205 K U 3Either value is consistent with data on impurity ion self-diffusion coeffic i e n t ~and ~ bulk v i ~ c o s i t y . ~ Unfortunately, this appealing interpretation of eq 1 is clouded by the recent findings of Ambrus, Moynihan, and Macedo.lZ These authors demonstrated that, when similar measurements are performed a t lower temperatures and analyzed according to eq 1, not only do the best-fit values of TOfall systematically below those previously obtained, but the TOvalues for different transport processes are no longer in agreement. These discrepancies become greater the lower the temperature range of the data considered. Clearly these results raise doubt about the significance of the process-independent TOvalues obtained in the higher temperature region. The object of the present work was to determine whether the latter TOvalues could be substantiated by some direct thermodynamic measurement. For the case of congruent melting compounds such as Ca(N0&-4HzO, a calorimetric method may be used to estimate the temperature at which the excess entropy of the liquid would vanish and hence to determine directly a value of TOwhich is consistent with the Adams and Gibbs interpretation of the parameter in eq 1. The method of estimation, which is described in detail elsewhere,13 is based on Kauzmann’s original observation that liquids, by virtue of their larger heat capacities, lose more entropy than corresponding crystals when both are cooled over the same temperature interval. Since the entropy of the liquid at the equilibrium melting point only exceeds that of the solid by the entropy of fusion, A&, a liquid can evidently only be supercooled a limited number of degrees, Le., to a temperature TO,before its total entropy falls to that of the crystal. To avoid conflict with the
"Ideal" Glass Temperatures for Salt Hydrates
279 T/ K
second and third laws of thermodynamics, the heat capacity of the internally equilibrated but disordered liquid must therefore decrease to a value near that of the crystal at To if not before. To will, of necessity, fall below the experimental, cooling rate dependent, temperature Tg a t which the liquid falls out of internal equilibrium. It is for this reason that estimation of TOrequires an extrapolation of the measured equilibrium heat capacity of the supercooled liquid; for the Ca(N03)2-4HzO and Cd(N03)z. 4H20 cases the extrapolation is a very short one. The inclusion of Cd(N03)~.4H20in this study was motivated by its structural similarity to the calcium nitrate hydrate and the availability of some transport propertybased TOvalues14 with which calorimetric TOvalues could be compared. Mg(CH3C02)2.4Hz0 (hereafter abbreviated as Mg(0Ac)z .4H20) was included because of its similar stoichiometry but much higher glass transition temperature15 and its unusual appearance and rheological character near Tg.In this case, unfortunately, no mass transport measurements have yet been made.
Experimental Section Reagent grade calcium nitrate tetrahydrate, cadmium nitrate tetrahydrate, and magnesium acetate tetrahydrate (Mallinckrodt Chemical Works) were used without further purification. The water content of the calcium and cadmium hydrate salts was determined by vacuum dehydration at 140" to constant weight; the calcium nitrate tetrahydrate gave a moles of Hz0:moles of C a ( N 0 3 ) ~ ratio of 4.05 f 0.01; the cadmium nitrate tetrahydrate gave a moles of Hz0:moles of Cd(NO& ratio of 4.03 f 0.01. The water content of magnesium acetate tetrahydrate, determined by Karl Fischer titration, was found to give a moles of H2O:moles of Mg(0Ac)z ratio of 3.88 f 0.06. The specific heats of the liquid, glass, and crystal samples of Ca(N03)~.4.05HzOand Mg(OAc)z.3.88H20 were determined using a Perkin-Elmer Model DSC-lb differential scanning calorimeter. This instrument was also used to determine the heats of a solid-state transition and of fusion of Ca(NO3)-4.05€€20 and the heat of fusion of Mg( OAc)z.3.88Hz0 using, the heat of transition of dried crystalline ammonium chloride (Baker Reagent grade) and the heat of fusion of dried indium powder (Roclric 325 mesh 99.999+% pure). The heat of solid-state transition and the heat of fusion for these salts based on the two alternative standards agreed to within f l % . The stated accuracy of the instrument is f470 for heat capacity measurements and =tl% for heats of transition, which conform with our experience in its use. Using a scan speed of 10 deg min-l Ca(N03)2*4.05H20 began to fuse a t 38" and was completely melted a t 44", and the Mg(0Ac)z. 3.88H20 began to fuse at 60" and was completely melted a t 70". However, using a scan speed of 0.625 deg min-I where the samples remain in continuous internal equilibrium, the melting range was reduced to 41.5-43.5' for Ca(N03)2.4.05H~O and 62-67' for Mg( 0Ac)z3.88H20. Scan speed did not affect the values obtained for heats of transition within our accuracy limits. To confirm that small departures from stoichiometric water contents of these hydrates did not lead to data inaccuracies greater than those arising from instrumental sources, 0.05 mole of water was removed from C a ( N 0 3 ) ~ . 4.05HzO by careful vacuum dehydration, and the heat of fusion was redetermined. From our data on four separate samples, the heats of fusion of stoichiometric and
180 190 2 0 0
220
240
260
2 8 0 300 320 3 4 0 360
1"
160-
190
.
-
140-
&50;8
_E
120110-
< 100-
0" 9 0 s
220
225
230
2 3 5 .
log,,
2 4 0 T/K
245
250
255
Figure 1. Heat capacities of crystal, glass, and liquid states of (a) Ca(N03)2.4H20, (b) C d ( N 0 3 ) ~ . 4 H 2 0 ,and (c) Mg(0Ac)Z4 H 2 0 plotted vs. log T. Also shown in the figures are areas equivalent to the entropies of fusion and solid-state transition of the respective compounds, and the temperature To(cal) which satisfies the condition ASF ASt, = S ( T 0 T F ) C,(liq) C, (cryst) d log T.
+
bottle salts (Ca(N03)2.4.05HzO) are the same within f1%.laThe "extra" water content thus does not affect the values of the heats of fusion within our accuracy limits. The specific heats of the liquid, glass, and crystal samples of Cd(N03)2-4.03Hz0 were determined using the The Journal of Physical Chemistry, Vol. 78, No. 3, 1974
280
C. A. Angel1 and J. C. Tucker
TABLE I Cp, cal mol-' deg-1
T,K
140 160 180 200 220 240 260 280 300 320 340 360 380
Crystal
48.7. 55.9 59.4 67.5 72.5 71.5 75.6 77.5
Ca(N0a)z 4Hz0
Cd(N0a)z .4Hz0
Glaea
54.0 57.9 60.2 67.1
Liquid
106.3 130.8 124.2 124.2 123.4 126 .O 126 .O 126.2 126.3 126.4
crystal
Glass
55.5
55.4 59 .O 60.1
58.8 61.5 64.5 67.5 69 .O 69.1 69.4 73.5
much improved Perkin-Elmer Model DSC-2 differential scanning calorimeter. The DSC-2 is equipped with a Hewlett-Packard Model 463A precision amplifier and a Prince Applied &search Model 260 analog-digital converter, such that the data are collected on punch paper tape and the specific heat is calculated by computer. The heat of fusion of Cd(N03)2-4.03H20 was measured using the DSC-2 and an indium standard as for the Ca(N03)2. 4.05H20 and Mg(OAc)2.3.88HzO systems. The Cd(N03)2.4.03HzO sample began to fuse at 58.0" and was completely melted at 60.0" using a scan speed of 0.625 deg min-l. The temperature range of both instruments was calibrated to within k1.0" using the melting points of the following reagent grade chemicals: methanol, chloroform, mercury, water, and gallium. Throughout the specific heat measurements a scan rate of 10 deg/min was used. The specific heat values were reproducible to &2% for the DSC-2, and the accuracy of the values is expected also to be within these limits.
M g (OAc)z .4HzO
Liquid
128.6 121.6
122 .o 123.5 124.7
Crystal
63.8 67.1 67.7 69.6 76.4 76.4 74 .O
Glaas
Liquid
64.6 65.4 70.4 69.6 83.5 123.8 133.8 135.4 140.4
measurements of Ca(N03)2, and its hydrates from which a value of AHf for Ca(N03)2.4H20 of 7.705 kcal mo1-I can be calculated, 3% larger than the present value.
Discussion In Figure 1, C,(liquid) is extrapolated below Tg until (at TO)the area between the C,(supercooled liquid) and C,(crystal) curves is equal to the area representing the entropies of the first-order transitions occurring between TO and Tffor the respective salts.lS At the temperature TO,the entropies of the internally equilibrated supercooled liquid and the crystal phases would be the same; hence TO is the lowest temperature at which the liquid phase of each substance could be expected to maintain its observed heat capacity. The temperature thus defined is frequently referred to as the "ideal glass transition temperature." The value of TOdetermined in Figure 1 is 200 K for Ca(N03)2.4H20 and 198 K for Cd(N03)2.4H2O, both with an estimated uncertainty of *4 K due to uncertainties in heat capacities of &5 cal mol-l deg-l and entropy of fuResults sion of 10.5 cal mol-l deg-l. The value of TOdetermined Tg,defined as the temperature at which C, abruptly for Mg(OAc)2*4HzO is 209 K; however, the estimated uncertainty is larger (MK) due to the long extrapolation of begins to increase above the crystalline value, is found to be 217 K for Ca(N03)2.4H20, and 213 K for C d ( N 0 3 ) ~ - the heat capacity data and the unusual fusion behavior of the salt. 4H20, both values being within the *lo uncertainty For the calcium and cadmium salts the concordance of of earlier dta m e a s ~ r e m e n t s . ~Tg J ~ for Mg(OAc)2-4H20 the calorimetric value with the values obtained from is not easily assigned, owing to the rather gradual increase transport properties (see Introduction) is clearly very satin C, in the glass transition region (see Figure IC).We isfactory, encouraging belief that the TO parameter obhave chosen Tgas the initial temperature at which C, betained for transport properties measured in the first three gins to increase, yielding a Tgof -3", which is in accord orders of magnitude of change indeed does have thermowith Sare's15 dta measurement. dynamic significance. At least for the Ca(N03)2.4HzO C, values for Ca(N03)2.4H20, Cd(N03)2*4HzO, and and Cd(N03)2.4Hz0 systems, 200 and 198 K, respectiveMg(OAc)2.4HzO are given in Table I and are plotted US. ly, seem established as important reference temperatures log T i n Figure la, b, and c, respectively. for equilibrium and relaxational properties. For Ca(N03)2.4.05H20, AHtran, = 257 cal mol-l; The reason why eq 1 with these latter TO values does AStra,, = l.%cal mol-l deg-l; Ttran, = -30.5"; AHf = not accurately describe data taken at lower temperatures 7420 cal mol-1; Tf = 42.5"; ASf = 23.5 cal mol-1 deg-I; and To(ca1) = -73". and higher viscosities is not clear at this point. It is possibly due to a gradual change in character of the dominant For Cd(N0&4.03HzO, AH, = 9370 cal mol-l; A& = transport mechanism from one in which fluctuations in 28.2 cal mol-l deg-l; Tf = 59.4'; and To(ca1) = -75". configurational entropy (which determine the configuraFor Mg(OAc).3.88H20, AHf = 10.08 kcal mol-l; A& tional heat capacity) determine the relaxation time to one = 30.1 cal mol-1 deg-l; Tf = 62"; and To(ca1) = -64". in which the relaxation process is more solid-like in charThe only data in the literature with which the present acter and involves to an increasing extent successive indiresults can be compared are two heats of fusion for Cavidual ionic displacements within a temporarily rigid en(N03)2-4H20. Livingston and coworkers17 reported a vironment.20 An alternative and more specific account is value of Hf for C a ( N 0 3 ) ~of 7.91 kcal mol-l in 1907. More provided by the Environmental Relaxation model of Marecently, Ewing and coworkers18 reported heats of solution The Journal of Physical Chemistry, Vol. 78, No. 3, 1974
”Ideal”Glass Temperatures for Salt Hydrates cedo and colleagues.21 The failure of eq 1 to describe liquid transport data taken over extended viscosity ranges now appears to be rather general.22 Both Ca(N0&.4H20 and Cd(N0&-4HzO together with another ionic liquid, L ~ O A Cappear , ~ ~ unusual among substances studied to date in that T,/TO is close to unity (actual value is 1.08 for Ca(N0&.4HzO, 1.07 for Cd(N0&4H20, and 1.06 in the case of LiOAc). The Tg/To ratio for Mg(OAc)2-4H20 is 1.29, a value commonly quoted for organic molecular and polymeric liquids; indeed, near Tg this salt has peculiar, polymer-like rheological properties. When Tg/To is large, and particularly when the difference between liquid and glassy heat capacities is not very great (as in the inorganic silicate glasses), there are a variety of possible courses which the equilibrium liquid heat capacity, measured in imaginary “slow” experiments, might conceivably follow below the normal glass transition temperature. In the calcium and cadmium salt cases, however, the temperature interval between Tg and To is so small, and the necessary drop in heat capacity so great, that even the construction adopting the most gradual possible decrease in equilibrium liquid heat capacity (at temperatures below the “observational curtain” imposed by the experimental glass transition) must still involve a very rapid decrease indeed. Presumably the identification and description of a class of substances with such sharp required fall-offs in equilibrium liquid heat capacity will be helpful in the development of equilibrium theories for liquid properties and the glass transition. Finally, we note that both the Ca and Cd salt hydrates have lost more than 80% of the entropy of fusion by the time they have reached the temperature of the glass transition, making the ratio S(residual)/S(fusion) unusually small. This gives the appearance of an unusually close approach to the “ideal glass” condition,2 S(residua1) = 0. The fact that the configurational heat capacity (Cp(liquid) - C,(glass)) is very large, however, means that despite the small ( T , - TO)interval the glasses do in fact contain considerable “frozen-in’’ entropy, the magnitude
28 1
of which is given by the areas in Figure 1 (a,b) between the glass transition and the TOboundary line. Acknowledgments. The work ,was supported by grants, from the Department of the Interior, Offices of Saline Water and of Water Resources Research, and the National. Science Foundation. Referencesand Notes (1) C. A. Angell, J. Necfrochem. SOC.,112,1224 (1965). (2) C. A. Angell, J. Phys. Chem., 70,3988 (1966). (3) C.T. Moynihan, J. Phys. Chem., 70,3399 (1966). (4) C.T. Moynihan and C. A. Angeli, J. Phys. Chem., 74, 736 (1970). (5) G. S.Darbari and S. Petrucci, J. fhys. Chem., 73, 921 (1969). (6) C. A. Angell, E. J. Sare, and R. D. Bressel, J. Phys. Chem., 71, 2759 (1969). (7) (a) H. Vogei, Phys. Z.,22, 645 (1921);(b) V. G. Tammann and W. Hesse, Z. Anorg. Allg. Chem., 156, 245 (1926);(c) G. S. Fulcher, J. Amer. Ceram. SOC., 8,339 (1925). (8) . -The original VTF equation did not contain the T-”’ factor (see ref 2). (9) M. H. Cohen and D. Turnbuli. J. Chem. Phvs.. 31. 1164 119591. i o ) G. Adam and J. H. Gibbs, J. Chem. Phys.,’43; 139 (1965). 11) C. A. Angell and K. J. Rao, J. Chem. fhys., 57,470 (1972). 12) J. H. Ambrus, C. T. Moynihan. and P. B. Macedo, J. Electrochem. SOC., 119 (2),192 (1972). 13) (a) W. Kauzmann, Chem. Rev., 43, 219 (1948);(b) C. A. Angell
and C. T. Moynihan in “Molten Salts,” G: Mamantov, Ed., Marcel Dekker, New York, N. Y., 1969,p 315; (c) C. A. Angeil, J. Chem. Educ., 47, 583 (1970). 14) C. T. Moynihan, C. R. Srnalley, C. A. Angell, and E. J. Sare, J. Phys. Chem., 73,2287 (1969). 15) E. J. Sare, Ph.D. Thesis, Purdue University, 1970. 16) J. M. Berger, private communication. 17) J. Livingston, R. Morgan, and F. T. Owen, J. Amer. Chem. SOC.,
29, 1493 (1907). (18) (a) W. W. Ewing and A. N. Rogers, J. Amer. Chem. SOC., 55, 3603 (1933);,(b), W. W. Ewing, A. N. Rogers, J. Z. Miller, and E. McGovern, hid., 54, 1335 (1932). (19) There is no strong theoretical basis for the construction in which the liquid heat capacity is extrapolated at its observed equilibrium value (Figure 1). However, it is the simplest, and therefore in the absence of solid theoretical argument,to the contrary, the most appropriate construction. It also is the correct construction if consistency with the mass transport data is sought, since TO obtained from the latter is, in effect, also dependent on a linear extrapolation of higher temperature behavior. (20) F. S. Howell. R. Bose. C. T. Moynihan, and P. B. Macedo, J. Phys. Chem., in press, (21) R. Weiier, R. Bose, and P. B. Macedo, J. Chem. Phys., 53, 1258,
(1970). (22) W. T. Laughlinand D. R. Uhlmann, J. Phys. Chem., 76,2317(1972). (23) C. A. Angell and J. C. Tucker, Proceedhgs of Richardson Conference on the Physical Chemistry of Process Metallurgy, 1973
The Journal of Physical Chemistry, Vol. 78, No. 3, 1974
