This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 131.94.195.200

This content was downloaded on 24/01/2017 at 18:37

Please note that terms and conditions apply.

Thermodynamic Properties of Krypton. Vibrational and Other Properties of Solid Argon and

Solid Krypton

View the table of contents for this issue, or go to the journal homepage for more

1961 Proc. Phys. Soc. 78 1462

(http://iopscience.iop.org/0370-1328/78/6/347)

Home Search Collections Journals About Contact us My IOPscience

You may also be interested in:

A Low Temperature Adiabatic Calorimeter for Condensed Substances. Thermodynamic Properties of Argon

P Flubacher, A J Leadbetter and J A Morrison

Theory and properties of solid argon

E R Dobbs and G O Jones

Thermoelastic properties of some cubic close-packed lattices

T H K Barron and M L Klein

On the Statistical Mechanics of the Ideal Inert Gas Solids

G K Horton and J W Leech

Thermodynamics of solid argon at high temperatures

R K Crawford, W F Lewis and W B Daniels

Longitudinal wave velocity in solid argon

D J Lawrence and F E Neale

Anharmonic effects in the thermodynamic properties of solids II. Analysis of data for lead and

aluminium

A J Leadbetter

Anharmonic effects in the thermodynamic properties of solids IV. The heat capacities of NaCl, KCl

and KBr between 30 and 500 °C

A J Leadbetter and G R Settatree

Anharmonic effects in inert gas solids. II. Method applied to Ne, Ar, Kr, Xe

J W Leech and J A Reissland

1462

Thermodynamic Properties of Krypton. Vibrational and Other Properties of Solid Argon and Solid Krypton

BY R. H. BEAUMONTtf, H. CHIHARAj-8 AND J. A. MORRISOE Division of Pure Chemistry, National Research Council, Ottawa, Canada

Communacated by M . Blackman; MS. recezved 25th May 1961

Abstract. The calorimetric methods described in the preceding paper have been used to measure thermodynamic properties of solid and liquid krypton. Besides heat capacities and vapour pressures over a range of temperatures, the following were obtained : heat of fusion = 392.0 2.3 cal mole-l, T (tr. pt.) = 115.77,'~~ p(tr.pt.)= 548.7+ 0.1 mm, heat ofvaporizationat 116*85"~=2179*2+ 0.9 calmole-' and at the normal boiling point =2162 i: 1 calmole-l.

A number of properties of solid argon and solid krypton has been derired, in particular, the apparent Debye characteristic temperatures at O'K and as a function of temperature, the heats of sublimation at 0 OK, the static lattice energies and the zero point energies. The results of the thermal measurements on the solids have been correlated with expansivity and compressibility results and certain inconsistencies resolved. The calorimetric results are shown to be internally consistent.

The shapes of curves of OD( T) against temperature indicate that anharmonic contributions to the vibrational properties of solid argon and solid krypton are appreciable, particularly in the region T > @,/lo.

In the region below the melting points, Cp for both argon and krypton increases rather rapidly with temperature. This is interpreted as an effect of formation of vacancies in the solids. Enthalpies of formation are found to be 1280 + 130 cal mole-1 for argon and 1770 These are about two-thirds of values estimated from theory. ,4n effect of vacancy formation may also be seen in the vapour pressures.

200 cal mole-1 for krypton.

INTRODUCTION

N the first part of this paper we give the results of measurements of the The results are I analogous to those for argon reported in the preceding paper (Flubacher,

Leadbetter and Morrison 1961 a). Since the experimental methods used were exactly the same, they require no further description.

thermodynamic properties of solid and liquid krypton.

f National Research Council Postdoctorate Research Fellow. $ Now at Dunlop Rubber Company, Birmingham. I Now at Department of Chemistry, Osaka University, Osaka, Japan.

Thermodynamic Properties of Krypton 1463

The second and longer part of the paper is concerned with vibrational and &er properties of solid argon and solid krypton. The experimental results given here and in the preceding paper are analysed and correlated with other information on these solids so as to obtain derived quantities such as the apparent Debye characteristic temperature, @,( T ) , at O'K and as a function of temperature and the zero point properties, namely, heats of sublimation at O'K, zero point energies and static lattice energies. A rather long section ($2.3) is devoted to the correlation of the thermal properties with existing data on the expansivity and compressibility of solid argon and solid krypton. Here, there are inconsistencies &lch we have tried to resolve. I t would be valuable if our conclusions could be checked by further measurements of expansivities and compressibilities. Because of uncertainties which remain we cannot say very much about the detailed temperature dependence of O , ( T ) above 1 5 " ~ , other than that @,(T) definitely increases with increasing temperature for both argon and krypton, and that OD( T ) for argon seems to increase more rapidly. Clearly, this behaviour should be ascribed to anharmonicity, but to what extent it is due to zero point energy or to thermal energy cannot be established at present. The fact that the effect seems greater for argon, for which the ratio of zero point energy to lattice energy is greater, suggests that zero point energy is an important factor. The comparison of the experimental properties with those derived from theoretical models corresponding to the inert gas solids is largely left to a theoretical paper by Horton and Leech (1962, to be published).

The heat capacities in the region below the melting point have been analysed so as to obtain estimates of the enthalpies required to form vacancies in solid argon and solid krypton. The enthalpies found from experiment are only about two-thirds of those computed from models. .A further investigation of the discrepancy using expansivity data suggests that the relaxation of surrounding atoms into the vacancy is very much larger than has been estimated by theory.

The results are discussed in $2.5.

PART 1.

The following three sections contain the experimental results obtained for krypton and brief accounts of relevant corrections. The krypton used was of a grade similar to that of argon and it also was obtained from the Linde Company. The purity was checked during the determination of the triple point ($1.3).

THERMODYNAMIC PROPERTIES OF SOLID AND LIQUID KRYPTON

1 .l. Heat Capacities of Solid and Liquid Krypton

The measured values of the heat capacity are listed in the Appendix and values for the solid at rounded temperatures in Table 1. The accuracies to be assigned to the results are essentially the same as those applicable to the results for argon. There is, however, a slightly greater uncertainty in the results at higher temperatures. The second virial coefficient of gaseous krypton has not been determined experimentally below room temperature. It was therefore necessary to estimate its value at low temperatures by extrapolation of high temperature results (Beattie, Barriault and Brierly 1952, Whalley and Schneider 1955). The extrapolation was made down to 7 6 " ~ by a method due to Hirschfdder,

1464 R . H. Beaumont, El. Chihara and J. A. Morrison

T K ) 2 3 4 5 6 8

10 12 15 20 25 30 35 40 45

Table 1. The Heat Capacity of Solid Krypton at Rounded Values of the Temperature

Cp (cal mole-l deg-') T('K) c p (cal mole+ deg-1) 0 01050 50 5.978 0.0375, 55 5.140 0.09638 60 6.296 0.2039 65 6.45 1 0.3721 70 6 569 0.8572 75 6 702 1.41 8 80 6.824 1.999 85 6 974 2.798 90 7 146 3.817 95 7.338 4.516 100 7.585 4.990 105 7.841 5.345 110 8.139 5.612 115 8.552 5.802

Curtiss and Bird (1954).t Fortunately, the correction to the heat capacity on account of gas imperfection is rather small, so that the virial coefficients do not need to be known with high accuracy. An uncertainty of 10% in the virial coefficient leads to an uncertainty in the heat capacity of 0.4% at worst. The densities of solid krypton used were those obtained by Figgins and Smith (1960). In correcting the heat capacities of the liquid a single value of the density (2-43 g ~ m - ~ ) was used.

o Present results 4 Clusius (1936) 'I =. A Clusius,Kruis and Konnertz (1938)

I

80 I LO -1

- 4 ~ 40 T(OK)

Fig. 1. A comparison of heat capacity results for solid krypton.

x 100 Cp(obs) - Cp(smooth)

Cp(smooth) 4%) =

where Cp(smooth) is given in Table 1.

?The quantum-mec hanical correction (de Boer and Michels 1938) was small, W O W t i @ to only 0.23% at 1 1 6 " ~ .

Thermodynamic Properties of Krypton 1465

The measured heat capacities of solid krypton in the region 10" < T < T(tr. pt.) are compared with earlier results (Clusius 1936, Clusius, Kruis and Konnertz 1938) in Fig. 1. The results are plotted as differences from a smooth curve corresponding to the values given in Table 1. The average deviation of the present results from the smooth curve is less than 0.2%. T h e other results deviate more strongly, particularly at the lower temperatures.

T(OK)

83.282 86.319 89 266 89 273 92.603 95 831 98 959

101 984 104 691 104 906 107.651 108 125 110.495 110 836 11 3.228 113 447 115 112 115 626 115.661 115.758

116.045 116.129 11 6,240 11 6.845 117.024 117.045 117.880 117.937 118.923

- Table 2. Vapour Pressures of Solid and Liquid Krypton

P ( r m Hd Solid

6.07 10.70 17.81 17.89 31.25 50.76 78.42

116.44 162.76 167.42 230.65 243.66 316.81 328.86 423.30 433.27 513.49 539.91 542.04 548.75

560 82 564.68 570.33 598.98 608.15 609.57 652.39 655.73 709.70

Liquid

- -0 03 + 0.12 + O 04 -0 14 - + O 03 - 0 34 - 0.12 - 0.21 -0.13 -0.12 +0.16 + 0.69 + 0.20 + 0.43

-0 05 -0.18 + 0.20 -0 49 -0 24 + O 14 -0.11 + O 25 + 0.14

The heat capacities of solid argon and solid krypton have also been measured between 1.2" and 20°K by Anderson (1960). A comparison of Anderson's results? with ours can be summarized as follows. Over a large part of the temperature region, in particular between 3" and 4°K and above 9"K, the two sets for both argon and lrrypton agree within their probable accuracies. Between 4" and 9 ' ~ Anderson's heat capacities for argon are larger by as much as 4y0, and for krypton smaller by as much as 5%. These differences are larger than the estimated accuracy of the present measurements (see 8 3 ( a ) of Flubacher, Leadbetter and Morrison 1961 a). At the lowest temperatures ( T < 3 ° K ) Anderson's results are consistently higher than ours by up to 3%.

t We should like to thank Dr. Anderson for sending us the tables from his thesis.

1466 R. H . Beaumont, H . Chihara and J. A. Morrison

1.2. Vapour Pressures of the Solid and Liquid The vapour pressures of solid and liquid krypton, determined in conjunction

with the measurements of the heat capacity, are listed in Table 2. The pressures are of comparable accuracy to those obtained for argon. The results were fitted to the following equations by the method of least squares:

solid krypton : loglop (mm) = 7.70741 - - 575:67 , 1

......( 1) 489.70 . T ' liquid krypton : loglop (mm) = 6.96880 - -

the deviations of the actual pressures from the equations are given in the last column of Table 2. The differences, of course, simply reflect the fact that the heats of sublimation and vaporization are temperature dependent. The constants of Eqns (1) are significantly different from those found by Freeman and Halsey (1956), namely 7.7447 and 579.6 for solid krypton, and 6.9861 and 491.9 for liquid krypton. The difference between the two sets of measurements can also be seen in the estimates of the triple point temperature and pressure of krypton (see Table 4).

1.3. The Triple Point of Krypton and the Heats of Fusion and Vaporization

Three determinations of the heat of fusion of krypton were made, and the results are given in Table 3. During the first, the fusion was stopped at intervals for the determination of the triple point pressure and temperature. For this reason the first result may be the least accurate, but on the other hand the amount melted was 40% larger, which would tend to improve the accuracy. I n obtaining the average value the three results were given equal weight. The average value agrees well with the result found by Clusius (1936).

Table 3. The Heat of Fusion of Krypton

Amount melted (moles) Temperature range (OK) 0.1550 11 5.758-116.240 388.6 0.1105 115.661-116.037 393.5 0.1 105 115'626-116'129 394.0

Clusius (1936) 390.7

Heat of fusion (cal mole-')

average 392.0 k 2.3

The triple point temperature corresponding to pure krypton was estimated from a plot of the equilibrium temperatures against the reciprocal of the fraction melted. The relevant data are given in Table 4. The amount of impurity was also estimated and the result was 4 e 3 parts per million. The triple point temperature found here, 115.776"~, lies in between the values given by others. Little can be said about this because the different temperature scales are involved in an absolute way. The triple point pressure does not involve the temperature

Thermodynamic Properties of Krypton 1467

Table 4. The Triple Point of Krypton

Fraction melted 0.20 0.40 0.60 0.80

TQ (pule krypton) Freeman and Halsey (1956) clusius, Kruis and Konnertz (1938) clusius (1936) Keesom, Mazur and Meihuizen (1935)

T W 115.7748 11 5.7756 11 5.7756 115.7761

115.776 115.6 115-97+0*05 116.0 f0.05 115.95 f0.03

average

P (mmHg) 548.90 548.64 548.60 548.66 548.70 ? 0.10

538-1 548.7 f1.5 549.5 k1.0 549 k1.5

scale and it is interesting to see that our result agrees so well with the values found by Clusius and his colleagues (1936, 1938) and by Keesom, Mazur and Meihuizen (1935). Freeman and Halsey's result is much lower but was probably calculated from a vapour pressure equation.

Table 5. The Heat of Vaporization of Krypton at T= 116.85"~

(moles) (cal mole-l) Amount evaporated Heat of vaporization

7.946 x 2178.6 8.167 x 2178.4 5.365 x 10-4 2180.5

Average 2179.2 + 0.9 Calculated from vapour pressures 2186 +9 Heat of vaporization at normal boiling point 2162 f 1 Clusius, Kruis and Konnertz (1938) 2158 f 3

The results of the direct measurement of the heat of vaporization of krypton are given in Table 5, and the average value agrees with that calculated from the vapour pressures to well within the assigned uncertainties. It is also in accord with the average of much more extensive determinations by Clusius, Kruis and Konnertz (1938).

PART 2. VIBRATIONAL AND OTHER PROPERTIES OF SOLID ARGON AND SOLID KRYPTON

2.1. General Remarks In this second part of the paper our object is to analyse the results of thermal

and other measurements on solid argon and solid krypton to see what can be learned about their vibrational properties. For instance, we should like to know t o what extent the properties can be described in terms of harmonic vibrations. This is a matter which becomes very important when the experimental results are compared with properties corresponding to models, such as those calculated bY Horton and Leech (1962, to be published). Before conclusions about the

PROC. PHYS. SOC. LXXVIIJ, 6 4 F

1468 R. H . Beaumont, H . Chihara and J. A . Morrison

interatomic forces can be reached from the agreement or lack of agreement between theory and experiment, effects due to anharmonicity must be clearly Understood, While some general conclusions about effects of anharmonicity can be drawn, uncertainties in some of the auxiliary data required are just too great, so that a complete detailed analysis of the thermal measurements cannot be justified at present.

2.2. Zero Point Properties 2.2.1. The characteristic temperature at O'K. An investigation of the low temperature heat capacities of a number of examples has shown (Barron and Morrison 1957) that the expansion

C = a T 3 + b T 5 + c T 7 + . . . may be expected to describe experimental results in the temperature region T < @,/25. The apparent Debye characteristic temperature at O@K, @,, may be calculated from the first coefficient, i.e.

. . . . . . (2)

..... 464.5 , where a has the units calmole-l deg-4. Figs 2 and 3, which are graphs of C,/T3

r I I I I 7 0 20 60 80

Fig. 2. A graph of Cv/T3 against T 2 for argon.

against T2 for the heat capacity results below ~ O K , illustrate the estimation of 0

for argon and krypton. The temperature @,/25 is approximately 4" for argon and 3" for krypton. Since the lowest temperature of the experiments is 2% and since the experimental accuracy is least in this region, it is obvious that we cannot conclude from the experimental results whether or not the expansion (Eqn (1) ) applies. If, however, we accept the principle of the expansion, we are led to suggest that the dashed lines in the figures represent reasonable limits on the extrapolation of the results to O'K. The coefficients a so derived correspond

Thermodynamic Properties of Krypton 1469

to @,=93*3 f 0.6" for argon and Oo= 71.7 k 0.7" for krypton. No attempt hag been made to estimate the coefficients b and c. Anderson's heat capacity results (Anderson 1960) yield values of Oo which are about lg% lower.

I I

I- . 5 0 x

0

I

0 20 40 60 a0 T2 (T inOK)

Fig. 3. A graph of CV/T8 against T 2 for krypton.

2.2.2. The heat of sublimation at O'K. The thermodynamic properties given i n this and in the preceding paper may be used to calculate the heats of sublimation of argon and krypton at O'K. The relevant thermodynamic expression is a standard one and requires no detailed discussion. It is

A X , , ( 0 " ~ ) = A i Y ~ , , ( T ) - ~ ~ C ~ ( g a ~ ) d T + ~ ~ C ~ ( s o l i d ) d T + 0 RPT2 ( d d 3 . . . . . . (4)

The last term takes account of gas imperfection ; the sources of numerical values of the second virial coefficient B p have been indicated ($ 1.1 above, and $3.1 of the preceding paper). The results are summarized in Table 6. The estimated uncertainties are also given.

Table 6 . Calculation of the Heat of Sublimation at O"K (in units of cal mole-1)

Argon 1861 + 5 Heat of sublimation at T= T, 1" C,(S)dT

- IT& Cp(G)dT

0

0

R P T ~ (s) at T = T~~

391 + I

,416

1 0 + 1 . .

Heat of sublimation at T = O O K

Dobbs and Jones (1957) Whalley and Schneider (1955)

1846+7 1850 f 12 1818 +40

Krypton 2579 + 3

648 2

- 575

1 4 + 2 -

2666 f 7

2589 + 50 -

1470

2.2.3. Recently, Salter (1962) has shown how vapour pressure data for monatomic solids may be analysed to obtain values of the static lattice energy E, and the geometric mean frequency vg of the lattice frequency distribution. In his paper he uses the vapour pressures of argon and krypton given here to illustrate the method and we shall summarize his results. The numerical values provide a useful test for the consistency of the thermodynamic information. The zero point energy E, may be obtained as the difference between E,, computed from the vapour pressures and the heat of sublimation at OOK, Alternatively E,, which is equivalent to the first moment of the lattice frequency distribution, may be estimated through use of the v D ( n ) function (Barron, Berg and Morrison 1957)

R. H. Beaumont, H. Chihara and J. A. Morrison

The static lattice energy.

vD(n) ={i(n+ 3)/-h)l’m, . . . .. . ( 5 )

where pVL is the nth moment, provided that the general shape of the curve of v,(n) against n can be established. Three values of vD(n), in particular vD( - 3), vD(o) and ~ , ( 2 ) , can be obtained from e,, vg and 0, respectively, and since the vD(n) curve should be smooth, interpolation to find ~ ~ ( 1 ) is not difficult.

Two sets of results calculated from the vapour pressures of argon and krypton are given in Table 7. The uncorrected values refer to direct use of the vapour

Table 7. The Static Lattice Energy and Geometric Mean Frequency Computed from the Vapour Pressures

Eo (cal mole-l) 10-l* x vg(s-1)

Uncorrected Corrected Uncorrected Corrected Argon 1970 ? 10 2005 ? 10 1.06 ? 0.01 1.15+0.01 Krypton 2740 ? 15 2790k15 0.81 ? 0.01 0.88 + 0.01

pressure data as found experimentally. These give Ez (argon) = 124 & 17 and E, (krypton) = 74 f 22 cal mole-1 from E,, which are to be compared with 158 k 5 and 117 f 5 cal mole-1 from vD(n) curves (not illustrated) based on estimates of 0, given in $2.4. The differences between the independent estimates of E,, which are clearly outside the limits shown, turn out to be due to an effect of the presence of vacancies in the lattices at higher temperatures. When an allowance is made for the free energy of formation of vacancies (see $2.5), the corrected results in Table 7 are obtained. These lead to E, (argon) = 159 k 17 and E, (krypton) = 124 f 22 cal mole-1 from E, and 171 5 and 134 & 6 cal mole-’ from vD(n) curves. The agreement is now quite satisfactory.

2.3. The Estimation of C, - C, Before the apparent Debye characteristic temperature, OD( T ) , at higher

temperatures can be calculated it is necessary to know C, - C,. Estimates of this quantity as a function of temperature have been made for both solid argon (Clusius 1936, Dobbs and Jones 1957) and solid krypton (Clusius 1936), but a

Thermodynamic Properties of Krypton 1471

in the light of more recent experimental and theoretical information turns out to be worth while. The standard expression is

* * (6)

\"here U is the coefficient of cubical expansion, p the density and xT is the isothermal compressibility. 2.3.1, Argon. Barker and Dobbs (1955) have measured directly the velocities of transverse (wt) and longitudinal (yi) lattice waves in solid argon in the temperature range 65" to ~ O " K , have extrapolated their results to O'K, and so have obtained estimates of the adiabatic compressibility, xs, for the range 0" to 8 0 " ~ . A check on their extrapolation can be made by using their values of vt and we at O O K to calculate 0 (elastic), which, assuming harmonic vibrations should be equal to eo, deduced from the heat capacity data. Barker and Dobbs's results give 0 (elastic) = 80.5 OK, which is much smaller than 0, = 93.3 0 . 6 " ~ ( 0 2.2.1). hharmonicity introduced by zero point energy, which would seem to be the only reasonable cause of a difference between 0, and 0 (elastic), is believed to make 0 (elastic) greater than 0, (Ludwig 1958), and so we conclude that the extrapolation made by Barker and Dobbs cannot be correct.

Velocities of the lattice waves at O'K have been calculated from theory by Bernades (1960) and Horton and Leech (1962, to be published). The results (ot=0*94 x lo4 cm sec-1 ; we = 1.61 x lo4 cm sec-I) lead to 0 (elastic)= 92.6"~, which is in better agreement with 0,. We have therefore calculated xS at O'K from the theoretical wave velocities and have used it with the experimental compressibilities derived by Barker and Dobbs to obtain xs as a function of temperature. Fortuitously, the values of xs at intermediate temperatures are rather close to those tabulated by Dobbs and Jones (1957) (Table I11 in the appendix to their paper) ; the maximum difference is 2%.

The density of solid argon has been measured between 20°K and the melting point by Dobbs et al. (1956) using x-ray diffraction and volumometer techniques- More recently, Henshaw (1958) has reported a single value of the density at 4 . 2 " ~ , obtained from neutron diffraction experiments, which is about 3 yo higher than that given by an extrapolation of the results of Dobbs et al. (1956). The difference is disturbingly large, and since there do not appear to be any obvious errors in either the x-ray or neutron diffraction experiments, it is necessary to distinguish between their results using other information.

Theoretical work on the inert gas solids (Barron 1955, Horton and Leech 1962, to be published) suggests that the Gruneisen parameter y should be only slightly temperature dependent. In particular, the indications are that y for argon should decrease by probably not more than 10 to 20% when the temperature falls below 50°KK. y is given by

U .... Y= -* P XSCP

We may therefore conveniently test the internal consistency of the quantities on the right-hand side by plotting u/p against xsCp. Such a graph is shown in Fig. 4. The points represent smoothed values of u/p given by Dobbs et al. (1957) at the temperatures 20, 30, 40, 50 and 6 0 " ~ . The corresponding values of xs Were taken from the (xS, T ) relation established as indicated above, and the values Of c, from Table 1 of the preceding paper. The light dashed lines represent

1472 R. H. Beaumont, H . Chihara and J. A. Morrison

7=3-0 and y=2-7. The points at 20" and 3 0 " ~ lie well below the dashed lines but the others come nicely in between. If we rely on the theoretical deductions about y, we are obliged to discard the points at 20" and 3 0 " ~ and to say that the solid curve in Fig. 4 is what should be used to fix u/p below 4 O " ~ t . When this is done one finds that the resulting change in the density below 4 0 " ~ is small. F~~ example, the density at O'K becomes 1.776 g ~ m - ~ compared with 1,77Ogcm-~ given by Dobbs et al. (1956). I t seems impossible to make the density value found by Henshaw (1958) consistent with the other data.

Fig. 4. A graph of alp against xSC, for argon (see 2.3.2 of the text).

A summary of the values of U, p, xs and xT used in computing Cp - C, is given in Table 8. xp was calculated from xs using

x T = X s ( l f u T y ) * . . . . . . (8)

A final check may be made by comparing xT so derived at 65" and 7 7 ° K with values determined directly by Stewart (1956) using a piston displacement method. T h e agreement at 6 5 " ~ is good ( ~ 2 % ) but Stewart's result at 7 7 " ~ is much higher ( >20yo). A complicating factor at this higher temperature is that the concentration of thermally created vacancies is appreciable (see 9 2.5). The general effect of the vacancies would be to enhance the compressibility as determined by displacement under pressure. It is difficult, however, to construct a quantitative argument. Stewart (1956) has considered other possible reasons, such as errors in density measurements, and shown that they are insufficient to account for the observed difference in xT. 2.3.2. Krypton. Figgins and Smith (1960) have measured the density of solid krypton between 20" and 9 0 " ~ using x-ray diffraction and between 70" and the melting point using a volumometer technique. From the results they have com- puted the expansivity for the temperature range 20" to 9 0 " ~ . Very much less information is available for the compressibility of solid krypton. In fact, there

One is based on Cp as given in the Table, and the other, which is closer to the ordinate, On cP corrected for vacancy formation (see 5 2.5).

t It will be noted that the measurements at 60"~ are represented by two points.

~( '4

Argon 0 5

10 15 20 25 30 40 50

Krypton 0 5

10 15 20 25 30 40 50

Thermodynamic Properties of Krypton 1473

Table 8. Adopted Values for Calculating C, - Cv

1.776 0 1.776 0.04, 1.775 1.5,

1.768 6*10 1.773 3.9,

1.762 7.91 1.755 9.35 1.736 11.70 1-715 13.5

3.093 0 3-093 0.1, 3.092 1-68 3.088 3*49 3.081 4.90 3.073 ' 5.99 3.063 6.7, 3.041 7.88 3.01 5 8-60

1010 xs (cmz dyn-l)

0.39, 0.40, 0 ~ 4 0 ~ 0.404 0.40, 0*411 0.41, 0.42, 0.44,

0.388 0*3g9 0.39, 0.3g1 0*392 0*393 0.39, 0.40, 0.41,

0-398 0.40, 04O4 0.41, 0.42, 0.43s

0.48, 0.44,

0.52,

0.388 0 ~ 3 8 ~ 0*391 0.396 o4Oo 0406 0.414 0*43* 0.45,

- -

0.003 0.030 0.096 0.196 0.318 0.623 0.960

- -

0.005 0.030 0.078 0.144 0.217 0.378 0.549

Fig 5. A graph of alp against xsCp for krypton (see 2.3.2 of the text).

appears to be but a single experimental value of xT at 77°K obtained by Stewart (1955). In order to obtain the compressibility of krypton as a function of tem- perature our only recourse has been to do a type of corresponding states treatment based on the results for argon and using xs at O"K obtained from theory (Horton and Leech 1962, to be published). (The velocities were equivalent to @(elastic)=72.9"~.) The particular method chosen was to use a graph of X,(T)/X~(O"K) against T/T,, where T, is the triple point temperature. It is "possible to fix the accuracy of this procedure but it is useful to note that xT at 77°K derived using it agreed with Stewart's result to about 3 yo.

1474 R. H. Beaumont, N. Chihara and r. A. Morrison

The test for the consistency of the different properties of solid krypton is shown in Fig, 5, which is a graph of a l p against xsCp corresponding to the temperatures 20,25, 30,35,40, 50,60,70 and 80"~. In contrast to the example of argon, none of the values of a / p needs to be eliminated. A11 of the points lie close to the solid CUme. The only troubling feature is that the magnitude of y is smaller than is suggested by the calculations from models (Horton and Leech 1962, to be pub. lished). This may, of course, simply be caused by a consistent error in the estimation of the compressibility, but on the other hand it could represent a significant departure of experiment from theory. The second part of Table 8 contains the values of Cp- Cv for krypton and of the quantities used in their calculation.

2.4. The Temperature Dependence of OD(T) Figures 6 and 7 are graphs of OD( T ) as a function of temperature for argon and

In the calculation of the plotted points, Cp- Cv as given in Table 8 kryptont.

I 95 t -0- V f (T)

----- VsV(OoK) 0'

80 I I I I

0 IO 30 40

Fig. 6. The apparent Debye characteristic temperature of argon.

-0- V = f (T)

V- V (0" K) -----

I I I I I

IO 30 40 0

Fig. 7. The apparent Debye characteristic temperature of krypton

t A graph of O,(T) against T for argon is contained in a preliminary report (Flubacher, It Leadbetter and Morrison 1961 b) and its shape is a little different from that of Fig. 6.

was, however, based on a different estimate of C, - C,.

Thermodynamic Properties of Krypton 1475

I I I

was used, and therefore O,(T) corresponds to a volume which is varying with temperature. What is required for comparison with theory is OD( T ) for a fixed volume. Correction of OD( T ) to the volume of the crystal at O'K has been made using the expression (Barron, Berg and Morrison, 1957)

. . . . . . (9 )

and the results are indicated by the dashed curves in Figs 6 and 7. We do not show OD( T ) abova 4 0 " ~ for two reasons. First, above 4 0 " ~ , Cp - Cv is becoming alarge fraction of C p (> 11% for argon, > 7% for krypton), and at best we only b o w its value to 5%. Second, a non-vibrational contribution to the heat capacity, namely that due to vacancy formation, begins to come in.

While the general shape of the OD(T) curves in Figs 6 and 7 is rather similar, there are differences in detail. The curve for krypton shows a more pronounced minimum in the vicinity of 8"K, and rises less rapidly at the higher temperatures. The latter feature is displayed more clearly in Fig. 8, where the two curves are compared on a reduced basis using@, (argon) = 93.3"~ and 0, (krypton) = 71.7"~. The difference between the two curves in the region T/Oo < 0.15 is of interest in connection with a comparison with the calculated curves of Horton and Leech (1962, to be published). It turns out that the curve for krypton agrees most

closely with one for a central force model including all neighbour interactions, while that for argon agrees better with a model including only first neighbour interactions. Since it is difficult to accept that the nature of the interatomic forces in the two crystals is different in any fundamental way, one is almost obliged to ascribe the difference in the reduced OD( T ) curves to anharmonic effects. Here It IS relevant to note that the ratio of zero point energy to the static lattice energy for argon is nearly twice that for krypton, i.e. 0.085 compared with 0.048.

The rise in @,(T) at the higher temperatures must also be attributed to an- harmonicity, and again it appears significant that the effect is much greater for argon. At ~ O " K , the increase of O D ( T ) for argon above the minimum value corresponds to a change of about 4% of C,, which is outside the combined un- certainties of the experimental heat capacities, of the estimation of Cp - Cv and ofthe use of Eqn (9). The effect is less certain for krypton where the increase in % ( T ) to 4 0 " ~ amounts to only 1% of Cv, but the chances are great that it is

I .

1476

real. This being so, it is interesting to note that the behaviour of OD( T ) of argon and krypton at the higher temperatures is quite the reverse of what has been found for other simple crystals such as some alkali halides and diamond structure elements (Berg and Morrison 1957, Flubacher, Leadbetter and Morrison 1959).

Because of the apparent early onset of anharmonic effects (right from O O K ) , it is not worth while at this stage to attempt an analysis of the heat capacities to obtain moments and other properties of the lattice frequency distributions. It is even difficult to make an unambiguous choice of 0,. The best that can be said from inspection of the @,(T) curves is that 0, for argon probably lies between 8 2 O and 8 5 " ~ , and for krypton between 64" and 6 6 " ~ .

R. H. Beaumont, H . Chihara and J: A. Morrison

2.5. The Enthalpy to Form Vacancies in Solid Argon and Solid Krypton The heat capacities of both solid argon and solid krypton show marked upward

trends in the temperature region below their melting points. At a few tenths of a degree below the melting points, the heat capacities are about 8.5 cal mole-l deg-1 (Tables 1 of this paper and of Flubacher, Leadbetter and Morrison 1961 a). This behaviour might be ascribed to one or more of several causes. For example, it might be due to effects of anharmonicity, to premelting induced by impurities or to the thermal formation of imperfections in the lattices. I n the previous section we have seen that anharmonicity is having the opposite effect on the heat capacities of argon and krypton up to 4 0 " ~ at least, i.e. it is producing a decrease of the heat capacity from that to be expected for harmonic vibrations. The amount of impurity in both the argon and the krypton is known from their behaviour during melting, and its effect on the heat capacity in the region below the melting point can be estimated easily (Sturtevant 1949). For example, for argon at 8 0 " ~ the contribution to the heat capacity is only 0.003 cal mole-l deg-l. We conclude therefore, that the major cause of the rise in heat capacity is the thermal creation of imperfections, in particular the creation of Schottky vacancies. The energy (or enthalpy) to form the vacancies has been calculated from theoy (Kanzaki 1957, Nardelli and Repanai 1958, 1959), and so it is worth while to obtain estimates from experiment for comparison.

I t is very small.

The excess heat capacity due to vacancy formation is (Lidiard 1957)

where ns, the number of vacancies, is given by

ns = Nexp 6) exp( - k) ..

. * . . . (10)

. . . . (11)

and N is Avogadro's number, h, and ss the enthalpy and entropy of formation. I t is obvious that, if In (AC, x T2) is plotted against 1/T, one should get a straight line whose slope is - h,/k.

This may be done by simply extrapolating a graph of Cp against T smoothly from low temperatures so as to obtain an estimate of the ' normal ' heat capacity, We have preferred to use graphs of Cp against 1/T2 for this purpose, and the example of argon is shown in Fig. 9. Such a graph foreshortens the high temperature region, and hence makes extrapolation somewhat easier. show what seemed to be a reasonable spread of possible extrapolations from the

In order to apply the above, we need only to estimate ACp.

The broken curves 1, 2, and 3 in Fig.

Thermodynamic Properties of Krypton

- U' a

e c v -

3 -

1477

7- ?.'<A A

\A\" A \ \ o A \ * \ o A e \ \

5 - \* \\O A \ \* ' \o A \ \ \ \

\

\ O \

Fig. 9. A graph of

1 1 I I I 1.2 15 1.8 2 1

10:: I/T

2 and 3 represent

PiS. 10. A graph ln(ACp x T2) against l / T for argon, based on different estimates of the ' normal ' heat capacity.

1478 R. H . Beaumont, H. Chihara and J. A. Morrison

low temperature region. graphs of In (ACp x T2) against 1/T are illustrated in Fig. 10.

distribution of points corresponding to 3 around a straight line is better. slopes for 2 and 3 give h, = 1150 and 1280 cal mol-l respectively.

AC, was determined using each and the resulting

There is not too much to choose between extrapolations 2 and 3, although the The

We therefore

Table 9. The Enthalpy to Form Vacancies in Solid Argon and Solid Krypton

T ( O K )

Experiment 45 to 83 60 to 83

Theory 80

0

Experiment 60 to 115

Theory 109

hs (cal mole-l) Argon

Reference

1280 f 130 Present paper 1210 Martin (1957)

1880 Nardelli and Repanai (1959) 2033 Kanzaki (1957) Krypton

1770 rt 200 Present paper

2740 Nardelli and Repanai (1959)

take the experimental estimate of hs for argon as 1280 f 130 calmole-l. A similar treatment of the data for krypton gives hs = 1770 & 200 cal mole-l. These results are compared with the values from theory in Table 9, and we see that the experimental ones are smaller by about one third. It is doubtful that the difference can be ascribed to an erroneous estimate of AC, because, in fact, h, is not very sensitive to AC,. Also, it should be noted that the value of h, for argon found by Martin (1957) was obtained using the heat capacity data of Clusius (1936) and a completely different method of extrapolation.

The calculations can be carried further. By using the additional information contained in the intercepts of the lines in Fig. 10, we can obtain explicit expressions for the number of vacancies. These are

(- 1280( & 130)) RT 3 = 30( & 20) exp N . . . . . . (12)

for argon, and ( - 1770( & 200)) R T

. . . . . . (13) ns = 30( k 20) exp R for krypton. Having the number of vacancies at any temperature, it is of interest to correlate it with the expansivity, and we shall do this for argon for which expansivities up to the melting point are available (Dobbs et al. 1956). At these higher temperatures the densities (and hence the expansivities) were determined by a bulk method, and therefore should be particularly sensitive to the change in volume due to vacancy formation. Indeed, the expansivities of Dobbs et d. (1956) show a curl-up in the region T > 5 0 " ~ (solid curve in Fig. 11). If we assume that the volume of a vacancy is equal to the atomic volume (in other words, assume no relaxation of surrounding atoms into the vacancy) and work out cc for a

c '0.

Z ' - - 0

1479

Corrected for vacancy formation assuming V(vacancy):atomic volume

-_-_ Corrected for vacancy formation assuminq V (vacancy)=0.23 x atomic volume

2 - -----

c. \ \ \

'\\

I I I O ' 20 40 60 80

T(OK)

Fig. 11. The expansivity of solid argon.

hypothetical crystal containing no vacancies, we get the result indicated by the dashed curve in Fig. 11 which is completely unreasonable. At worst, cc should flatten off at the higher temperatures; it should not decrease. The middle chain curve was calculated assuming that the volume of a vacancy was only 23% of the atomic volume, and this is much more like what should be expected.

SUMMARY The work which is described in this and in the preceding paper (Flubacher,

Leadbetter and Morrison 1961 a) has had the initial object of providing more com- plete and accurate values of the thermal properties of condensed argon and krypton than have been available heretofore. The experimental results satisfy different tests for internal consistency. For instance, heats of vaporization deter- mined calorimetrically and indirectly from the vapour pressures agree well. Also, zero point energies computed in two ways agree within the uncertainties assigned to the various quantities used.

The latter calculations are of particular interest. In the first place, one of the methods of computation, which uses estimates of certain moments of the lattice frequency distribution, implies harmonic vibrations. The fact that it seems to work suggests that it is not a bad approximation to describe the thermodynamic properties in the low temperature region in terms of an effective harmonic fre- quency distribution. However, the reduced curves of OD( T ) against temperature (Fig. 8) indicate that anharmonic effects definitely come in at higher temperatures (T>@,,/10) and perhaps at all temperatures. A further point about the zero Point energy calculations is that they demonstrate the effect of vacancy formation upon the vapour pressures of the solids.

The enthalpies required to form vacancies, determined from the experimental heat capacities for the solids, are quite appreciably smaller than values estimated form theory. The latter are close to the heats of sublimation (compare Tables 6 and 9), and this comes about because energy effects due to relaxation and distortion around a vacancy are taken to be small (Kankazi 1957). However, a comparison

1480

of heat capacity and expansivity results indicates that relaxation of surrounding atoms into a vacancy is probably large. If this were taken into account it would have the effect of reducing the calculated enthalpies of formation.

Perhaps the least satisfactory aspects of the experimental description of the properties of solid argon and solid krypton are the incompleteness of and inconsistencies in some of the expansivity and compressibility data. Further experiments along these lines are clearly desirable. For the time being we have tried to resolve the difficulties here by using certain results of theory and the principle of corresponding states. This must be viewed as a temporary expedient,

R. H . Beaumont, H . Chihara and J. A. Morrison

ACKNOWLEDGMENTS Throughout the course of this work we have had as theoretical advisers

Dr. T. H. K. Barron, Dr. G. K. Horton, Dr. J. W. Leech and Dr. L. S. Salter. Although they were not always in agreement, their comments and criticisms were valuable, and we should like to thank them for their assistance.

REFERENCES ANDERSON, A., 1960, Thesis, University of Oxford. BARKER, J. R., and DOBBS, E. R., 1955, Phil. Mag., 46, 1069. BARRON, T. H. K., 1955, Phil. Mag., 46, 720. BARRON, T. H. K., BERG, W. T., and MORRISON, J. A., 1957, Proc. Roy. Soc. A, 242,478. BARRON, T. H. K., and MORRISON, J. A., 1957, Can. J. Phys., 35, 799. BEATTIE, J. A., BARRIAULT, R. J., and BRIERLY, J. S., 1952,J. Chem. Phys., 20, 1613, 1615. BERG, W. T., and MORRISON, J. A., 1957, Proc. Roy. Soc. A, 242, 467. BERNARDES, N., 1960, Phys. Rev., 120, 807. DE BOER, J., and MICHELS, A., 1938, Physica, 5, 945. CLUSIUS, K., 1936, 2. Phys. Chem. B, 31,459. CLUSIUS, K. KRUIS, A., and KONNERTZ, F., 1938, Ann. Phys., Lpz., 33, 642. DOBBS, E. R., FIGGINS, B. F., JONES, G. O., PIERCEY, D. C., and RILEY, D. P., 1956,

DOBBS, E. R., and JONES, G. O., 1957, Rep. Progr. Phys., 20,516 (London: Physical Society). FIGGINS, B. F., and SMITH, B. L., 1960, Phil. Mag., 5, 186. ~ U B A C H E R , P., LEADBETTER, A. J., and MORRISON, J. A., 1959, Phil. Mag., 4, 273. - - 1961 b, Proc. VIIth Int. Conf. Low Temperature Physics (Toronto: University

FREEMAN, M. P., and HALSEY, G. D., 1956, J . Phys. Chem., 60, 1119. HENSHAW, D. G., 1958, Phys. Rev., 111, 1470. HIRSCHFELDER, J. O., CURTISS, C. F., and BIRD, R. B., 1954, Molecular Theory of Gases and

KANZAXI, H., 1957, J. Phys. Chem. Solads, 2, 107. KEESOM, W. H., MAZUR, J., and MEIHUIZEN, J. J., 1935, Physica, 2, 669. LIDIARD, A. B., 1957, Handb. d. Phys., 20, part 2, 246. LUDWIG, W., 1958, J. Phys. Chem. Solids, 4, 283. MARTIN, D. L., 1957, Report of 2nd Symposium on Melting, Daffussion and Related Topics,

NARDELLI, G., and REPANAI, A., 1958, Physica Supplement, 24, S182. - SALTER, L. S., 1962, Trans. Faraday Soc., in the press. STEWART, L., 1955, Phys. Rev., 97, 578. STEWART, J. W., 1956, J. Phys. Chem. Solids, 1, 146. STTJRTEVANT, J. M., 1949, Physical Methods of Organic Chemistry, 2nd edition, part 1, edited

WHALLEY, E., and SCHNEIDER, W. G., 1955, J. Chem. Phys., 23, 1644.

Nature, Lond., 178,483.

1961 a, Proc. Phys. Soc., 78, 1449.

Press), p. 695.

Liquids (New York: John Wiley), pp. 162-6.

Ottawa, p. 31.

1959, Report of Solid State Physics Group, University of Milan.

by A. Weissberger (New York: Interscience Publishers), pp. 757-9.

Thermodynamic Properties of Krypton 1481

A P P E N D I X Measured Heat Capacities of Solid and Liquid Krypton

T('K) Cp (cal moled1 deg-l) Series I (0-1586 moles)

2.317 2.697 3.114 3.525 3.969 4.466 4.957 5.710 6.683 7.643 8.605 9.555

10.635 10.905 11.526 11.723 12.307 12.794 13.216 13.852 14.453 14.901 15.870 17.289 18.761 20.765 21.981 23,239 24.856 25.714 27.529 28.293 30.254 30.941 33.146 33.493 35.960 36.041 38.499 38.881 40.983 41.702 44.497 47.286 50.064 51.976 55.055 57.430 57.969 59,611 60,715 61,712 63,347 66,088 68,912 71.642 79,059

Solid 0.01638 0.0257, 0.0420, 0.0616, 0-0927* 0.1389 0.1959 0.3141 0.5159 0.7590 1.016 1.286 1.601 1.648 1.845 1.916 2.086 2.219 2.328 2.507 2.660 2.776 3-001 3.307 3.584 3.940 4.127 4.301 4.502 4.594 4.770 4.835 5.012 5.064 5.228 5-246 5.406 5.404 5.538 5.562 5.651 5.682 5.784 5.880 5.999 6.014 6.1 37 6.179 6.236 6.280 6.325 6.340 6.414 6.489 6.531 6.619 6.801

T('K) Cp (cal mole-1 deg-l) Series I (0.1586 moles)

Solid 81.712 6.844 84.801 6.961 87-792 7.081 90.938 7.179 94.217 7.316 97.395 7.454

100.472 7.611 103445 7.783 106.516 7.939 109.481 8,114 11 2.142 8.292 114.280 8.488

Liquid 116.643 10.325 117.491 10.495 118.430 10.485

Series I1 (0.1143 moles) Solid

2.309 04166, 2.500 0.02093 2.754 0.02875 2.977 0.03738 3.240 0.0484* 3-461 0*0593,, 3.948 0.0918, 4.440 0.1378 4.938 0.1980 5.750 0,3266 6.800 0.5502 7.743 0.7870 8.610 1.031 9.497 1.275 10.444 1.549 11.437 1.844 11.443 1.832 12.914 2.243 14.391 2.646 15.890 3,002 17.264 3.315 18.526 3.550 19.799 3.789 21.061 3.994 64.500 6.420 67.237 6.488 70.058 6,574 72.965 6.670 75.775 6,762 78.498 6.795 81.221 6.874

106.171 7.849 109.073 8.067 111.861 8.250 114444 8.513

Liquid 116.538 10.365 117.273 10,381