Edgar F. Westrum, Jr. University of Michigan Ann Arbor, Michigan
Cryogenic Calorimetric Contributions to Chemical Thermodynamics
The present scientific era is characterized by a broad coverage of many facets of cryogenic endeavor as well as by rapid applications of the results of such research t o commercial and military uses. Many areas of engineered cryogenic devices, for example, masers, superconducting computers, and high-field magnets; cryogenic fuels; cryogenic isotopic separation of deuterium from hydrogen, and of SHe from 'He by superfluid osmosis; and "cryopumping" of high vacua, are public knowledge. Chemical application of low temperatures has been less spectacular but perhaps equally fundamental. It is not always appreciated that in the great period at the eve of the present century which saw the birth of cryogenics and in which Kammerlingh Onnes a t Leiden practically singlehandedly kept the low temperature flag flying, the research problems awaiting solution by such studies were largely those of interest to the physical chemist. Moreover, it was from the physico-chemical area that most of the fundamental contributions were made. Walter Nernst's practical objective, which ultimately led t o the important third law of thermodynamics, was that of finding a reference point for the numerical equations of chemical equilibrium. The concept of absolute zero as the absence of kinetic motion remained unchallenged in this era, and it appeared strange when Nernst concluded that the entropy rather than the energy was the variable which vanished with the absolute temperature. Although the true significance of the entropy function was initially veiled in mystery, the development of quanta1 mechanics and the realization that phenomena observed at low temperatures were due to energy quantization shifted the focus of research from the physico-chemical area to that of chemical physics. Hence, problems of the validity of the third law and its applicability to systems with "frozen in" equilibria have become of historical interest and have been replaced by the quantitative study of the entropy increments predicted by it. The solid state researcher so often has occasion to acknowledge the extraordinary wealth of information that can be obtained about substances from lowtemperature heat capacity determinations that one is tempted to attribute to such measurements a more fundamental significance than to any other unique method of investigation. Certainly, applied to high temperatures such a statement would appear rather exaggerPresented as part of the Symposium on Recent Advances Thermodynamics before the Division of Physical Chemistry the 140th ACS Meeting, Chicago, September, 1961. The author and editor gratefully acknowledge permission reproduce figures, granted by authors of books and editors journals in which certain of the figures were first printed.
in at
to of
ated; it may be suspect even for ambient temperaturesbut the deeper one delves into the cryogenic region be; yond the "thermal chaos," the more true it becomes. It is the purpose of this presentation to review some areas of cryogenic calorimetry in which progress is currently being made. The treatment is typical rather than exhaustive, and most of the subjects treated are those with which the author has had recent experimental familiarity. This inherent subjectivity has been tempered and reinforced by related endeavors wherever practicable within the limitations of space. Thermodynamic Interlude
I n contrast t o some work in current cryophysics, a prime motivation of chemical cryogenic studies has been the provision of definitive data for thermodynamic purposes. The heat capacity, besides being a sensitive and revealing parameter of the energetics of substances and one of the most powerful tools available for the study of condensed phases, provides the raw material for chemical thermodynamic functions useful in science and technology. Measurements of the heat capacity at constant pressure, C,, of the various phases (or, ?, . . . , 7 ) involved together with the enthalpy increments of such phase transitions, AH, occurring reversibly at, temperatures, T,,permits evaluation of the standard enthalpy increment of a substance by the equation:
and the standard entropy by the relat,ed equation:
These functions may be combined to yield the standard Gibbs energy function for the suhstance:
Symbolizing a generalized chemical reaction as in which S, indicates a mole of the ith substance involved and vi is the stoichiometric coefficient (positive for EDITOR'SNOTE: The use of the tern "Gibbs energy" follows the proposals now before the IUPAC which will probably beeornc recommendations soon. The quantity designated is the "Gihbs function (see (IUPAC report in J. Am. C h m . Sac., 82, 55'20 (1960), the "Gibhs free energy," or the Lewis and Randell "frcc energyM(F).
(r)"
Volvmc
39, Number 9, September 1962
!443
products and negative for reactants), the Gibbs energy increment of the reaction, AGO, may be found by summation of the Gibbs energy functions and the enthalpies of formation, AH,",for the substances involved: aGor = X V ~ ( G ~=TZvi )~
[COT
Hoo]i
- ZY~(AH~&
From AGOT the equilibrium constant of the reaction, K, may be readily evaluated from the familiar relation: AG'T = -RT In K
R here represents the gas constant in appropriate u ~ t s . Although the chemical thermodynamicist's utopia will have a t his fingertips complete thermodynamic functions on all interesting chemical substances) is far removed, systematic experimental and theoretical research endeavors of the Petroleum Thermodynamics Laboratory of the U. S. Bureau of Mines a t Bartlesville, Oklahoma, for example, deserve recognition for providing extensive data on homologous series of hydrocarbons and organic sulfur compounds (1). Although less highly organized, the by-product contribution of chemical thermodynamic data from other definitive cryogenic calorimetric studies made over a sufficiently extensive range of temperature represents a real contribution from such studies whab ever their ,prime motivation may be.l However, strict assiduity regarding such matters as high purity and adequate characterization of the sample studied, high precision, and accuracy of measurement is mandatory for definitive studies. (in which he
Cryostat and Calorimeter
Accurate heat capacity data are typically procured in an instrument depicted in Figure 1, which is a modern development of a Nernst adiabatic, vacuum calorimetric cryostat. This assembly provides a way of maintaining a sample of chemical substance within a calorimeter vessel at any desired temperature under such conditions that no heat is exchanged with the environment except that introduced by an electrical heater during the measurement. The calorimeter vessel is suspended in isolation by a braided silk line and is surrounded by an adiabatic shield. The calorimeter and adiabatic shield may he mechanically brought into direct thermal contact with a refrigerant tank and be thus cooled to the desired operating temperature and again isolated by lowering it. Two chromium-plated copper refrigerant tanks provide thermal sinks. For studies above 90°K, liquid nitrogen is used as refrigerant in both tanks; temperatures as low as 50°K may be achieved by evacuating the lower tank and solidifying the nitrogen. For the range 4 ' to 50°K, liquid helium is used as refrigerant in the lower tank. Temperatures approaching l°K may be achieved by pumping on the liquid helium. Suitable modification in the fabrication of the instrument will permit its operation up to 60O0K. Chromium-plated copper radiation shields not only conserve refrigerants but generate zones of uniform and oroeressivelv lower tem~eratures. Thermal isolation
' An appreciation of the total magnitude of the internationd endeavor is gained from the recent listing ( B ) of the existence of unpublished cryogenic data suitable far entropy evalustions at 298°K on 210 suhstanees (of which 197 are from the United States)
444
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Journal of Chemical Education
and achievement of adiabatic conditions is facilitated by the high vacuum within the apparatus and the elimination of gas conduction. Thermal conduction along the electrical leads is minimized by firmly anchoring them to the refrigerant tanks and to an "economizer" which serves as a heat exchanger by using the cold effluent helium gas to absorb much of the heat conducted d o m the leads, and thus conserve liquid helium. The temperature of the lead bundle is tempered by a ring and finally adjusted to the temperature of the calorimeter by the adiabatic shield which is maintained within +0.002°K of the calorimeter temperature. Copper-constantan thermocouples monitor the difference in temperature between calorimeter and shield and between shield and ring and actuate three separate channels of recording electronic circuitry provided with proportional, rate, and reset control actions for establishment of adiabaticity. The equilibrium temperature of the sample is determined with a platinum-encapsulated, platinumresistance thermometer mounted within the calorimeter vessel, as shown in Figure 2. As the temperature of the calorimeter is increased by electrical energy input to the heating element installed within the vessel, that of the shield is made to follow it so closely that no unmeasured energy enters or leaves the calorimeter. After the input the new equilibrium temperature is determined. Under normal conditions for typical substances several such runs may be made per hour. Especially during phase and other transformations, equilibration periods of many hours may occasionally be required. Here the precise automatic control of the
CM.
Figure I. Schemalic cross section of modern cryostot for low tempemlure adiabatic calorimetry: A, liquid nitrogen tank; B, liquid helium tonk; C, D, and E, radiation shields; F, adimbatis shield; G, O-ring gasket sealing brass vocvvm jockel; H, effluent helium vapor exchanger ("economizer"); I,helium exit tube; 1, ring for ~ d p ~ t i ntemperolure g of leads; K, calorimeter arrembly; 1, platinum-reridonce thermometer; and M, connection to vocvvm diffusion pump.
adiabatic shield is indispensable in making reliable heat capacity determinations. The heat capacity of the sample may then be calculated from its mass, the observed temperature increment, the measured energy input, and the previously-determined heat capacity of the empty calorimeter-heater-thermometer system. A precision of the order of a few hundredths of a per cent may be obtained over most of the temperature range. Accuracy is assured by ultimately referring all determinations of mass, time, temperature, resistance, and potential t o calibrations performed by the National Bureau of Standards and by the measurement of heat capacity standards provided by the Calorimetry Conference (8). Even Further Beyond the Thermol Chaos
An important aspect of even lower temperature calorimetric investigation is the achievement and measurement of temperatures and temperature increments in suitably well-defined systems. Evaporation of 4Heand 3He will permit an extension of the working range t o approximately 0.9 and 0.3"K, respectively. Adiabatic demagnetization may also be utilized to yield temperatures several orders of magnitude lower; relatively little calorimetry, however, is in process a t these temperatures. Although vapor pressure thermometry and gas thermometry are of prime concern to the calorimetrist as a means of establishing temperature scales, there is good prospect for the emergence of an acoustical (soundvelocity) thermometer as a primary standard in the near future. For the region down to about 4'K, platinum-resistance thermometers and copper-constantan thermoelectric thermometers have been most frequently utilized as wcrking thermometers. Below 15°K ordinary carbon resistors have been used extensively in many studies, but are gradually being displaced by germanium resistance thermometers because of ability of the latter t o withstand cycling t o room temperature without significant shift in calibration. Magnetic thermometry based upon the measurement of the susceptibility of a paramagnetic substance has heen
widely applied to thermodynamical measurements below 1°K. For the determination of very low temperatures the direct application of Kelvin's definition of the absolute temperature scale in terms of the relation dS = dQ/T (in which dS is the infinitesimal change in entropy associated with an infinitesimal and reversible heat input dQ) is made with gamma-ray heating. Two excellent technical treatments of the problems of experimental cryophysics have appeared recently (4'5) and one concerned primarily .with calorimetric aspects (6) is in preparation. Heat Capacities of Crystalline LaWices
As a prelude to discussing the anomalous behavior of the heat capacity, it is perhaps worthwhile to consider the characteristic sigrnoid-shape curves generated by those solids whose energetic spectra are composed largely of the vibrational motion of their constituent atoms. Examination of the heat capacities of the four Group IV tetrafluorides (7-1 1) in Figure 3 reveals the behavior of a rather typical inorganic family. It is evident that the heat capacities of these substances tend t o vanish as O0K is approached and that they tend toward the empirical ambient temperature limit of Dulong and Petit, coupled with the Kopp-Neumann rule, thereby aggregating 30 cal mole-'OK-'. For convenience, only the curve for TiF4 includes the original experimental points. I t should probably be noticed, incidentally, that TiF4is not isostructural with the other three substances ( l l ) , which fact accounts (at least in part) for the inconsistent trend of its heat capacity relative t o that of the other substances. ZrR, CeFn, HfR, ThF4, and the other actinide tetrafluorides, however, form an unusually instructive isostructural series with wide variation in 1at.tice dimensions, in masses, and consequently in force constants. The zirconium and hafnium tetrafluorides, for example, have nearly identical lattice constants because of the lanthanide contraction; and yet they have cation masses differing by almost a factor of two. This series poses a very interesting ground for the test of theories of corresponding states for crystalline solids. The development of such a relationship beyond its present embryonic state (12, 0
I
Figure 2. Cross-sectional diagram of nickel calorimeter for tronritionelement herofluorides A, cone far thermal codact; 8, demountable valve plug; C, vane, sample space; D, thermocouple well; E, heater sleeve; F, platinum-resistance thermometer; G, spool to thermally equilibrate leodr with calorimeter.
TEMPERATURE, 'K 100 ZOO 300 I I //!
I
.. TEMPERATURE. 'K Figure3. Heat capacities of four Group IV tetrafluorides; titonivm tetrafluoride ( I I ) , zirconium tetroflvoride (101, thorium tetrafluoride 191, and uranium tetrafluoride (7,81.
Volume 39, Number
9, September 1962
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13) would he a real desideratum if for no other reason than t o aid in the resolution of magnetic, electronic, or other contributions to the heat capacity of isostructural crystalline substances from that of the lattice. Although the general trend of heat capacity with t,emperature has been semiquantitatively explained in terms of the Dehye theory and more accurately in terms of the recent developments in lattice dynamics, it is worth noting that the model used by Debye was that of a monatomic, structureless, isotropic solid. Among the predictions of Debye theory (which have been verified more generously even than deserved) is the lowtemperature limiting law that the heat capacity at constant volume be proportional to the cube of the temperature. Many crystalline substances, however, are st,rongly anisotropic and in extreme cases may be described as being either essentially lamellar (planar) or fibrous (linear-in which long chhins of atoms occur) crystals. Prime examples of lamellar crystals are the lattices of graphite, of hexagonal boron nitride, or of a slightly more complex structure, the "sandwich-layerlattice" of molybdenite (IIIoSz). Forces between atoms within a plane of the lamellae are strong, but intraplanar forces are relatively weak. Crystalline polyethylene or silver cyanide serve as examples of essent,ially fibrous lattices. Theoretical Debye-like heat capacity equat,ions have been developed and applied by Tarasov in an extensive series of publications ( I d ) , by Lifshits (15), by Gurney (16), and by others (17). The Tarasov limiting law, for example, predicts a heat capacity proportional to T 2 for a lamellar crystal and proportional to T' for a fibrous crystal. Although heat capacity studies published by others on both graphite (18,19) and on hexagonal horon nitride (20) were held t o provide confirmation of the Tarasov limiting law, it may be contended on the basis of more precise measurements on hexagonal boron nitride (21) shown in Figure 4 that the very low temperature behavior follows the T3 relationship of Dehye. This may he seen most clearly in the logarithmic plot of the heat capacity versus t,he logarithm of the temperature as depicted in Figure 5. Here the slope of tangent or chord, x,may he interpreted as the exponent of temperature in the relationship , C = aTz. It is immediately evident that the data may he best interpreted as involving a continuously changing exponent from a value of 3 at the lowest temperatures of measurement,
TEMPERATURE. 'K Figure 4.
The heot capacities of boron nitride 1211 and of graphite 118).
446 / Journd of Chemical Educafion
gradually decreasing, and probably approaching zero at, sufficiently high temperatures. No significant region over which x is exactly 2 appears. Moreover, it is extremely probable that the same situation obtains for graphite (qf. 22) although here the situation is romplicated by the presenre of electronic contributions to the heat capacity as evidenced by the electrical conductivity of graphite. Unfortunately, sufficiently macroscopic crystals of hexagonal horon nitride have not been available to eliminate all concern about the surface contribution to the heat capacity of the finely divided boron nitride used for the calorimetric determinations. 5
10
TEMPERATURE, 'K 25 50
100
- 0.5
- 1.0
boron nitride
P
&
" - 1.5 0 0 J
-2.0 -2.5
- 3.0 0.8
10 .
1.2
1.4
1.6
18 .
2.0
LOG T Figure 5. Logorithmt plot of the heat ),NCl(s)
+ HCl(g)
derived from direct equilibrium constant measurement,^ was compared with the entropy increment from the third law using low-temperature heat-capacity measurements on tetramethylammonium chloride and on tetramethylammonium hydrogen dichloride, together with statistically calculated entropy values for gaseous hydrogen chloride based upon spectroscopic data (54). Although certain assumptions were necessarily involved here, the conclusion that the proton is symmetrically situated is consistent with the great similarity of the infrared spectra, of the [Cl-H-C1]-ion and the [F-H-FJ-ion reported by Waddington. The st,rength of the bond has been attested to by McDaniel et al. (55). Anornolies in Thermal Properties
Thus far, only that energy which is involved in the excitation of thermal vibrations of the crystalline lattice or the vitreous solid has been considered. If other energy states of the molecule are occupied, this occurrence will be revealed sensitively in the heat capacity. The temperature at which an excess heat capacity occasioned by the occupation of the energy state appears is of the order @/k, in which AE is the excitation energy of the process per molecule and k is the Boltzmann constant. The form of the resultant heat, capacky anomaly is dependent upon the nature, mechanism, and energetics of the process involved. In the course of cryogenic measurements, phase tra!lsformations such as crystalline polymorphic transitions, fusion, and vaporization are likely to be encountered. On the basis of thermodynamic theory these have often been designated as first-order transformat i o n ~ . ~So-called higher-order transitions are also encountered in cryogenic studies. For example, if the probability of transition is indcpendent of the degree to which the excited state is occupied, each atom thus excited facilitates excitation for its neighbors, and the entire process becomes a cascading one. For this circumstance the excess heat capacity will accordingly rise with increasing steepness until the transition is completed, giving an excess heat capacity resembling the shape of the Greek letter lambda, thereby occasioning the designation of such a phenomenon as a lambda-type anomaly.
d
300
TEMPERATURE. 'K Figure 6. The deviation, AC,,, of the heat capacities of vitreous silica relative to the smoothed heat capacity curve for quartz (101. 0 represents experimental points for vitreous riiico onneoied a t 1070°C (24); doto on similar materid annealed at 1300°C (241 is reprerented by 0 ; the determinotionr of Simcn and Longe (25) by A; the data of Nernrt end the doto of Simon (271 by 0. The deviation curve of (26) by cristobaiite is represented b y ---,that of neutron irradiated quartz ond after after on integrated damaging Rux of 2.5 X 101envtb y 7 . 7 X IO'*nrtby
+;
---.
dered structure a t low temperatures and hence that the IF-H-F]- ion is symmetrical. Subsequent interpretation of infrared and Raman spectra, dielectric constant, neutron diffraction, nuclear magnetic resonance, and other data has confirmed this conclusion. Examination of the infrared spectrum of tetramethylammonium hydrogen dichloride suggested to Waddmgton (55) that its [Cl-H-CIJ- ion might prove t o be another example of symmetrical bonding. The entropy increment of the reaction
-.-.-,
Transition in Magnetic Materials
A lambda-type anomaly occasioned by the alignment of electronic spins a t the NBel point in hauerite (57) (mineralogically occurring manganese disulfide with pyrite structure) is shown in Figure 7. This transition Although discussion of the classification oi phase transformations is not s, part of the present presentation, attention is directed to a phenomenological classification scheme by McCollough (38). Volume 39, Number 9, September 1962
/
447
a t 47.g°K delineates the disappearance of the ordered antiferromagnetic state with opposed spins and no resultant magnetic moments (58). The entropy associated with this transformation approximates 0.71 cal mole-' OK-' and, like that of other similar transitions, will be seen to comprise only a fraction of the total entropy associated even with the spin-only contribution from the magnetic moment. It is possible to treat the decrease in order of these spins with rising temperature by thermodynamic theory and to relate a second parameter (e.g., the saturation magnetizati0n)with the heat capacity 2 by the relation,
C
= dg2/dt
and thereby to tie in thermal measurements with magnetic determinations. In some instances it is even possible to obtain data pertinent to the magnetic behavior of complex magnetic substances such as the technologically important ferrospinels (59), perovskites, garnets, etc., more successfully by heat capacity measurements than by more direct magnetic studies. For example, the presence of a small contamination of metallic iron in a ferrimagnetic substance mould seriously affect susceptibility determinations but would affect heat capacity data to an extent only roughly equal to that of the fractional concentration of this contaminant. The wide occurrence of ferro-, ferri-, and antiferromagnetism among the compounds of transition, lanthanide, and actinide elements provides many other examples of thermal contributions from transitions between magnetic states.
One of the interesting revelations of the past decade involves the discovery of the widespread manifestations of ferri- and ferroelectric behavior in both organic and inorganic substances. Since energy must he supplied to a ferroelectric crystal in order to destroy the spontaneous polarization, the heat capacity may be expected to be anomalous in those regions where the spontaneous polarization is changing with the temperature. Although many classifications of ferroelectric substances are possible, a phenomenological classification based on their thermodynamics has been
30
40
50
60
T E M P E R A T U R E . 'K occorioned b y the Figvre 7. The lambda-type anomaly in houerite (Mnh) Irmrformation from the antiferromagnetic state to the higher temperotvre ferrimagnetic ifate (37).
448
/
'
100
200
300
TEMPERATURE, ' K Figvre 8. The heat capocity of the ferroelectric anomaly in ammonium d f o t e ~howingd t a the doto d Shomote ( 4 1 ) and of Horhinoef ol. (42).
Ferroelectricity in Crystals
20
made (40). In the fist class are substances with a transitional ent,ropy increment of the order of magnitude of several cal mole-' OK-' , irrespective of the Curie temperature. Hence these materials are considered to undergo order-disorder type transformations. Among common substances, ammonium sulfate has a ferroelectric Curie temperature of 223.ioK with a typical discontinuous phase hnsition and an associated ent,ropy increment of 3.9 cal mole-' OK-'. A recent redetermination of its low temperature thermal properties (10) confirms the earlier finding of Shomate (41) and contrasts with the claim of Hoshino et al. (48), based on conduction calorimetry, that the transition consists of two maxima as shown in Figure 8. I n the second class are those suhstances with transition entropies of a few hundredths of a calorie. Because
Journal o f Chemical Education
the entropy increments are so small it is highly probable that the transitions in this class of material involve only enhanced libration of dipoles with increasing temperature instead of the complete reorientation of dipoles characteristic of the first class. The recent discovery of ferroelectricity in thiourea [SC(NH,),] by Goldsmith and White (45) was claimed to be unique in that its ferroelectric behavior is associated with the relative displacement of entire molecules rather than with the motion of ions within the crystal. Three distinct, hut very small, heat capacity anomalies were observed (44) as shown in Figure 9 and correlated in Figure 10 with the interpretation of the changes in molecularstructure of Calvo (45). The lowest temperature anomaly, occurring a t 169.3'K, has an excess entropy increment of only 0.04 cal mole-' OK-'. Closely following this at 171.Z°K occurs an even smaller peak with an entropy increment less than 0.01 cal mole-' OK-'. At 200°K still another gradual and possibly more energetic transition is terminated by a sudden rise and sharp decrease in heat capacity. The total effect associated with this third transition is approximately 0.17 cal mole-' OK-', provided the unusually long pretransitional rise on the low temperature side is included. The small magnitudes of the entropy effects associated with these transitions are especially noteworthy, but are consistent with the small rotational alteration in the ordering of the molecules required to account for the change in crystallographic and electrical properties.
J 5
0
TEMPERATURE.
20
25
O K
TEMPERATURE. OK 150
15
TEMPERATURE. .K
Figure 9. The heat capacity of thicurea 1441 in the region of the dielectric on0ma1ieS.
100
10
200
Figure 11. Total heat capacity and resolved electronic heat capacity of uranium tetrafluwide 181 with indication of orsumed energy level scheme.
degenerate ground level and no other low-lying levels. A comparison of the heat capacities of dineodymium and dilanthanum trioxides (47) in Figure 12 provides immediate evidence for the existence of contributions t o the heat capacity of dineodymium trioxide other than those associated with the vibration of its lattice.
250
TEMPERATURE. 'K
I
I
Figure 10. Dielectric conrtmh (431, pyroeledric coefficients 1431, heat copocity moximo and osrocioted entropies of transition (441. The moleculor configuration changes in the regions identified ore (451: 1, disordering ferrielectric; 11, discontinuity in disorder; Ill, disordering compensated ferrielectrici IV, disordered ferrielectnc statistically antiferroelectric; and V, ercitntion of molecular orcillotion.
Schottky Effects in Crystals
-
A characteristic non-cooperative process involving electronic excitation to higher levels has been treated by Schottky (46). I n the simplest case of a single excited electronic state separated from the ground level by an energy increment, AE, it follows that at T = 0 only the ground state will be occupied, a t T = = the two states will be equally populated, and a t the intermediate temperature T = 0.417 AE/k the excitation will be a maximum. The characteristic skewed bell-shaped curve reaches a maximum of 0.872 cal mole-I OK-', provided the degeneracies of both levels are equal. The presence of additional excited levels or of nonequal degeneracies will, of course, influence the shape of the Schottky curve. Recent studies at the Argonne National Laboratory by Burns et al. (8) reveal the presence of a Schottky-type anomaly a t 6.4'K, which accounts for the relatively high heat capacity in this compound shown in Figure 3 in the vicinity of 10°K. The heat capacity associated with the Schottky anomaly has been resolved from the total measured heat capacities and is depicted in Figure 11, together with a theoretical curve based on the assumptions that onethird of the uranium ions have a nou-degenerate ground level and a non-degenerate level a t U / l c 15.4'K and that the other two-thirds have a non-
-
TEMPERATURE. 'K Figure 12. (39.1
Heot capacities o f dineodymium and dilonthonum trioxides
These isostructural lattices are composed of cations of nearly identical mass and similar chemical interactions and consequently would be expected to have virtually identical lattice contributions to the heat capacity. The excess heat capacity of dineodymium trioxide, thus resolved and shown in Figure 13, is to be attributed to a splitting of the electronic states of its 4f shells by the crystalline electrical field. The shape of the Schottky anomaly is less characteristic here because of the complication introduced by the presence of three sets of excited energy levels in addition to the ground level, all making contributions a t temperatures within the range of measurement. Similar effects are found in other paramagnetic lanthanide oxides (47) and have also been used to delineate low-lying electronic energy levels of these elements since this information has not been available from spectroscopic studies. Inelastic neutron scattering, neutron diffraction, and magnetic susceptibility data reveal the presence of other, possibly cooperative, anomalies in Volume 39, Number 9, September 7962
/
449
rare earth chalcogenides below 6°K. Accurate thermal data on these materials to lower temperatures will be required to define accurately the thermodynamic properties of these oxides for high temperature use. 1.2
z 2
"
1.0
0.8
- 0.6 Y 0 4 0
0.4
i
0
0 0.2
0
0
20
40
60 80 TEMPERATURE.
100
200
300
O K
Figure 13. The Schottky anomaly in dineodymium trioxide with indication of t h e derived energy leveircheme 1391.
Confirming the report hy Penningt,on el al. (48), the heat capacity of 2-methylthiophene crystals was found (49) to increase abruptly by 0.7 cal mole-' OK-' near 163OK as shown in Figure 14. Moreover, the results of direct enthalpy determinations across t,his
crystalline state; high triple point pressure and temcubic or hexagonal-symperature; high-usually metry; and transparency, tackiness, and easy deformability. That the orientation of globular molecules a t latt,ice sites is highly disordered is evidenced by the microscopic interpretation of the small entropy of fusion. It seems likely that about ten distinguishable orientations of molecules at lattice sites represent the minimum degree of disorder to be found in substances normally considered to be plastic crystals. I n this light, "rotation" in plastic crystals can he adequately described in terms of orientational disorder with the molecules "flipping" into discrete, distinguishable positions in the crystalline lattice with such high frequency that X-ray and KMR measurements cannot be expected to distinguish between this movement and free rotation. A stochastic treatment of the transition entropies has been achieved by Guthrie and McCullough (51). As might be expected, many tetrahedral molecules such as methane, pentaerythritol [C(CHd3H)4],and pentaerythrityl fluoride [C(CH2F)n] manifest characteristic plastically cryst,alline behavior. A somewhat more complex molecule, adamantane [CmHlel, depicted in Figure 1.5, is the simplest saturated, polycyclir hydrocarbon that has its carbon atoms arranged in a cage-like skeleton identical with t,hat of the so-called "characteristic cell" of the diamond lattice. A sharp transition to the plastically crystalline state with apparent heat capacities in excess of 4000 cal mole-' OK-' was observed in adamantane (52) at 208.G°K, as shown in Figure 16, with an associated entropy of transition of 3.87 cal mole-' OK-'. Proton magnetic
Hexomethyfenetetromlne
Adomontone TEMPERATURE. 'K F , * I , ~ ,l
Figure 14. The h e a t capacity in the region of the 2-methyl-thicphene transformation. D o t a of Pennington et 01. ( 4 8 ) ore shown by 0 ; thore of Carison 1491 by 0.
region revealed no isothermal absorption of heat (i.e., no latent, heat). Occasioned by the onset of internal rotation of the methyl group, this anomaly which resembles a "glass" transformation is paralleled by similar effects in 3-methylthiophene and toluene. Molecular Disorder-The
Plastically Crystalline State
The term "plastic crystal" was used first by Timmermans (50) to characterize a phase found helow the meking temperature in crystals made up of "molecules globnlaire." Such crystalline substances may be dist,inguished by the following characteristics: an entropy of fusion of less than 5 cal mole-' OK-' compensated by one or more energetic transitions in the 450
/
Journal o f Chemical Education
Figure 15. Structure m01e~"le~
of
adamantane
and
heromethylenetetromine
esonance studies on adamantane by McCall and Donglass (55) have confirmed the interpretat,ion of this transition. Although normal decane would have a melting point about -31°C, the phase properties of adamantane have been so profoundly affected that. this substance does not melt until a temperature of 2 7 4 T is reached. Substitution of four nitrogen atoms for the CH groups at the bridgehead positions in adamantane yields another diamondoid molecule of identical symmetry with that of adamantane (cf.Fig. l j ) , hut one in which the plastically crystalline phase apparently does not appear. Figure 16 evidences its absence up to 350°K, as does thermal analysis to the
meking point. Becka and Cruickshank (54) have calculated the entropy of hexamethylenetetramine in a realistic approach based on Debye terms for the translat,ional modes and Einstein terms for the fundament.al vibrational frequencies and torsional oscillations
Table
Commund
1.
Thermodynamics of Some Hexotluorides
Atomic weight
Tt
AS,
Transition-Metal
Tt.
AS,
Data of Cady and Hargreaves (56). " M a of Brady, Myers, and Clauss (57)
a
Methanol [CH,OH] is among the familiar smaller molecules classified by Timmermans as forming plastic crystals (58). Although many investigators have examined the heat capacity of this substance (.5941), a re-examinat,ion (49) on sampl~sof both high and reduced purity provides definitive data and establishes Figure 16. The heat capacities of adomantme ond heiamethylenetetramine showing the transition in odammtone 1521.
of molecules a t lattice sites. Analysis of the elastic scattering of X-rays was used to evaluate the amplitudes for the first two contributions. The hexafluorides of the third group of transition metals, WF6, ReFs, OsF6, IrF6, and PtF6 provide an nnusually finely-graduated set of closely-packed molecular crystals of orthorhombic symmetry, which undergo transition to the cnbic plastically-crystalline phase just prior t,o fusion as shown in Figure 17 for three members of the series. The nuclear framework of these interesting molecules is that of a regular octahedron in the vapor which undergoes a small tetragons1 distortion in the crystalline phase. Removal of the electronic degeneracy in ReFs and OsFs gives rise to Schottky anomalies a t very low temperatures. Low temperature optical spectra provide quantitative details concerning the split,ting of the vibrational electronic degeneracies in these compounds arising from the molecular distortion and permit an unusually thorough discussion of the energy spectrum and the dahn-Teller effect in these crystals. Regularities in the entropies of fusion and transit,ion may be noted in Table 1. 0
TEMPERATURE. 'K 100 200
300
40
-a
-
30
0
n m
, "
20
i
U
TEMPERATURE. 'K
.
Figure 18. The heat capacity of methanol 1491 in the fusion region. Data for a high purity sample are represented b y 0 and those for o l e s pure sample by O 1491. Values of Stoveley .and Gupto ( 6 0 ) and of Kelley 1611 oreindicotedby .and e,respedively.
the existence of a fist-order transition between the orthorhombic crystal I and monoclinic crystal I1 phases a t 155'K with an entropy increment of 0.966 cal mole-' OK-'. As shown in Figure 18, the range of the crystal I phase is small, for fusion occurs with an entropy increment of 4.37 cal mole-' OK-' a t 17A.fi°K. Although the entropy of fusion is within the limit prescribed by Timmermans, the entropy of transition is considerably smaller than would be anticipated for a plastically crystalline transition. Comparison of the entropy of the liquid methanol with that of other relatively small molecules containing methyl groups indicates that the small entropy of fusion should perhaps be attributed rather to the ordering occasioned by the hydrogen bonding in the liquid state than to the formation of a plastically crystalline phase. An interesting example of the profound influence of
to 0
0
10
20
TEMPERATURE. 'K Figure 17. The heat capocities of three platinum-meto1 hexduorides 1551 showing the low temperotvre orthorhombic phases, the cubic pladicdly crystalline phase and the liquid phase.
in heat capacity of the crystal I phase on two samples with mole fractional purities of 0.9998 and 0.9988, as well as with the results of previous investigators. Both samples readily undercooled through the transition, making it possible to measure the heat capacities of the metastable crystal I phase below its normal Volume 39, Number 9, September 1962
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451
the fact that a rather significant contribution to the measured heat capacity a t constant pressure arises from the term
,I BL 0
1
I
I,
10 20 TEMPERATURE. .K
Figure 19. The heat copacity of the tetrachloromethane for mole fraction of CClr
system
tetramethylmethone-
= 0.666 (64).
t,ransformation temperature. I t was exceedingly difficult to obtain the conversion of crystal I1 to crystal I in t,he purer sample; the relatively impure sample, however, readily underwent this transition. Application lo Phase Diagram Determination
Although non-equilibrium methods such as different,ial thermal analysis may he adequate for the study of metalh~rgical systems a t high temperatures, the increased t,ime required for thermal equilibration a t low temperatures in many systems makes the equilibrium methodof delineatingphase behavior a real desideratum. Kot only is the sensitivity of the calorimetric approach greatly superior to that of thermal analysis, but quantitative enthalpy increments of the transformation may be obtained. This method is one of such power that it,s use should certainly not be restricted to the cryogenic area. Although the existence of solid solut,ions in which ice is the major component has long been a controversial subject, the solubility of isostmctural, similarly hydrogen-bonded ammonium fluoride in ice to the extent of about 7 mole % at the eutectic temperature has been clearly established (63) by calorimetric determination and by Schreinemakers' method of wet residues. Moreover, the existence of the compound NHIF.HZO, which undergoes peritectic melting a t 246.0°K, was also demonstrated and its thermodynamic properties derived. At the present time, ammonium fluoride is the only substance known to form solid solutions involving ice as a major component. Thermal studies on the system N H 9 - H F (63) revealed the existence of a solid solution range a t composit,ionNH4F3HF.
(in which ar and 6 represent, respectively, the coefficients of expansion and compressibility) and by the lack of an adequate understanding of the heat capacity of complex crystals. I n some cases, however, considerable insight can be obtained by diluting the lattice of a compound which undergoes a phase transition with a second kind of molecule or ion andobserving the effect on the nature and magnitude of the transition and its thermodynamic properties. If the original substance is a globular molecule, the diluents may produce a more uniform potential field, resulting in less inhihition to reorientational-rotation of the molecule. Although usually the added component has been one without transitions, the phenomena are naturally more mmplw 1rht.n holid solntiol~itire mu&. of wmlkmrnti which both rshihir nol\~nimnhictransitions. \lorrowr the determination of thermal data may be used to extend reliable thermodynamic functions of mixing taken in the liquid phase to those pertaining to the solid solution phase. I n order to be assured of relat,ively rapid achievement of equilibrium, a solid solution composed of two plastically crystalline substances was selected for study, and heat capacity measurements were made on a number of compositions of the system tetramethylmethane-tetrachloromethane (64). Enthalpies of mixing in the crystalline phase provided from the existing dat,a of Englert-Chwoles (65) in the liquid region have been extended to the plastically crystalline range (and to temperatures as low as 5%) by integrating heat capacities and enthalpies of fusion of several compositions of the system and combining these values with enthalpy data on the pure components. The enthalpy of mixing of the equimolar mixture is 281 cal (mole solution)-' at 230°K. This quantity involves the mixing of the plastically cryst,alline compounds to yield that of the plastically crystalline solid solution lattice. The heat capacity curve for the solution of mole fraction tetrachloromethane = 0.666 is shown in Figure 19. Of the four clearly discernible transformations, that a t 142OK is associated with the reorient,ation of the symmetrical tetramethylmethane molecules in a phase which is practically pure tetramethylmethane, that at 192'K is due to eutectoid transition, that a t 201°K is related to the onset of the reorientation of t,he tetrachloromethane molecule in the solid solution, and fusion a t 236°K is accompanied by an entropy increment of only 2.36 cal (mole solution OK)-'. Hence it may be concluded with certitude that the solid existing below this temperature is plastically crystalline indeed.
. " .~~
~
~~
~~
The Future of Cryogenic Calorimetry Excess Functions for Mixing o f CrystoNine Solids
Although in principle the heat capacity curve will provide considerable information about the manner in which the absorption of energy by the crystalline latt,ice depends upon temperature and hence on the mechanism of the phase transitions which occur therein, in practice information thus gained is limited by 452 / Journol o f Chemical Education
Diverse applications to chemical and other problems, such as the evaluation of barriers to internal rotation (66) and the molecular freedom and disorder of ions in crystals [e.g., (NH,)+ (10, 67, 68)], might be noted. I n determining the structure of organic molecules (69), statistico-mechanical thermodynamic properties are computed on the basis of an assumed model and
then are compared with measured thermal data. A novel determination (70) of the kinetics of the orderdisorder transitions in the MgCd, system by cryogenic calorimetry introduces time as a variable. Other actively developing fields include determinations of purity (71); crystal-glass comparisons (e.g., BZ03) (10); the nature of the liquid state; and the properties of free radicals in crystals or frozen into matrices (10, 72), of atomic exchange disorder (73), of the motion of molecules trapped in Pquinol clathrates (74, 75), of biochemically important systems (76), of adsorption phenomena (77), and of ferromagnetic superconductors (78). Cryogenic calorimetry is a viable area with opportunities for many talents such as penetrating and intuitive insight, capable application of mathematics and recent developments in quanta1 theory, and precise, skilled, and ingenious experimental technique to provide further progress. Improvements in calorimetric t,echnique, in thermometry, and in new techniques of measurement are needed to extract the wealth of energetic and thermodynamic data from substances. Internationally acceptable temperature scales and reliable heat capacity standards for the very low temperatures are urgently needed. I t should be emphasized that the current interest in very high temperature chemistry provides an additional stimulus for the determination of high precision thermodynamic data for many substances over the low temperature range as well as in the higher regions of temperature. From the sum total of such data, eventually it may be possible to create complete tables of thermodynamic functions, from which it then should be possible to calculate and thereby predict chemical behavior generally. However, for the preutopian era a t least, the thermodynamicist will be obliged to rely in many instances on estimated quantities shrewdly devised from the carefully thorough study of a selected group of materials, which will permit the formulation of rules and generalizations for predicting the energetic behavior of many other substances whose sheer number will preclude experimental investigation in the foreseeable future.
"Experimental
mi.
Cryophysics," Butterwortha, London,
MCCULLOUGH, J . P., Ed., "Experimental Thermodynamics," (in preparation). E. F., JR., A N D LOHR,H. R., OSBORNE, D. W., WESTRUM, J . Am. Chem. Soc., 77, 2737 (1955). BURNS,J . H., OSBORNE, D. W., AND WESTRUM, E. F., JR., J . C h m . Phys., 33, 387 (1960). LOHR,H. R., OSBORNE, D. W., A N D WESTRUM, E. F., JH., J . Am. C h m . Soe., 76, 3837 (1954). WESTRUM,E. F., JR. (unpublished data). E. F.,JR., J . Phys. Chen~., EULER,R. D., A N D WESTRUM, 65, 132 (1961). STOUT,J. W., AND CATALANO, E., J . Chem. Phys., 23, 2013 i\."--,. lQR51 HOFMANN, J. A,, ET AL., J . P h y ~C h m . Solids, 1, 45 (1956). TAHASOV, V. V., Dok. Akad. Nauk., 54, 803 (1946); Dok. Akad. Nauk., 58, 577 (1947); Dok. A h d . Nauk., 100, 307 (1955); Z h w . Fiz. Khim., 24, 111 (1950); Zhur. Fiz. Khim., 27, 1430 (1953). Lmsnrrs, I. M., Zhur. Ekspll. Theowl. Fiz., 22, 471 (195'2); Zhicr. Ekspll. l'heoret. Piz., 22, 475 (1952). GURNEY, R. W., Phys. Rev., 88, 465 (1952). KLEMENS,P. G., Auslmlian J . Phys., 6 , 405 (1953); KRUMHANSL, J., A N D BROOKS,H., J. Chem. Phys., 21, H . B., J . Chem. Phys., 21, 1663 (1953); ROSENSTOCK, T., J . 2064 (1953); KOMATSU,I
