The thermodynamics and hindered rotation of occluded co in the β-quinol clathrate

J. Phys.

Chem. Solids

THE

Pergamon

Press 1963. Vol. 24, pz. 1435-1450.

THERMODYNAMICS

OF OCCLUDED

AND

CO IN THE G. L. STEF’AKOFFt Department

Printed

in Great Britain.

HINDERED

P-QUINOL

ROTATION

CLATHRATE*

and L. V. COULTER

of Chemistry,

Boston University

(Receiwed 24 Muy 1963)

Abstract-The

heat capacities of four p-quinol clathrates of CO, yC0 * 3CsHa(OH)z, having compositions in the y = 0.4-08 range, have been measured from 15” to 300°K in an adiabatically operated calorimeter. At 298,15”K, the molar heat capacity and entropy of clathrated CO are 9.76 and 20.94 cal deg-1 mole-r, respectively. The pressure isotherm constant for the reaction CO(g) % COl~-su~nol, 8), is p(l -y)/y = &sac = 8.9 atrn. The heat capacity in the 15-40”K range is interpreted in terms of a three-dimensional harmonic oscillator (Y = 104.3 cm-l) for the vibrational motion of CO in the cavity and a two-dimensional hindered rotator with Vs = 745 cal mole-r. At 298.15”K, the thermodynamic properties are correlated by the use of the classical cell model for the rattling motion of CO corrected for the effect of hindered rotation. Agreement between calculated and observed properties is obtained for Zi(/k) II2 = 390 and 2 = 3.22 A for the interaction parameters of the cavity wall, and YO = 2867 cal mole-l for the potential hindering the rotation of occluded CO. The aonarent increase of ootential with temoerature perhaps results from increased alignment of the CO molecules with the cavity walls. INTRODUCTION IN A preliminary

communication(l) the molar heat capacities of CO and Ns in their respective &quinol clathrates have been given graphically for the 15-100°K range, and an interpretation of the data has been offered in each case based on a model consisting of a three-dimensional isotropic harmonic oscillator for the rattling or vibrational motion of CO in the host cavity and a two dimensional harmonic oscillator for the librational motion. For clathrated CO the librational frequency was found to be 43.1 cm-1 and the barrier restricting the rotation was calculated to be 691 cal mole-l. This paper now presents the complete results of the measurements of the heat capacities of four /Lquinol clathrates of CO of various compositions for the 15-300°K range from which the molar heat capacity and entropy of clathrated CO have

been obtained. In addition, an interpretation of these data and the vapor pressures of p-quinol clathrates of CO at 25°C is given in terms of the VANDER WAALS(~) theory for the vibrational motion of CO enclosed in the host cavity, and the properties of a two-dimensional hindered rotator for which the appropriate partition function and statistical thermodynamic equations have been derived.

EXPERIMENTAL The clathrates of CO employed in this work were prepared under isothermal conditions by subjecting a slurry of doubly vacuum sublimed Eastman hydroquinone (99.99 per cent by CeIV ion titration)@) in n-propyl alcohol to a predetermined constant pressure of CO gas (Matheson, 99.5 per cent) in a thermostated high pressure autoclave. The slurry was agitated by rocking the bomb for several days at 25*O+O*l”C to insure synthesis of * Supported in part by a grant from the National _ homogeneous samples. Following the conversion Science Foundation. t Present address: Manlabs, Inc., Cambridge, Mass. of the u-hydroquinone to the /3-quinol clathrate 1435

1436

G.

L.

STEPAKOFF

of CO by the net reaction: 3 ar-quinot(,, + yCO(,, = 3 fi-quinol *yCC+,,, where y is the fraction of cavities filled or number of moles of gas occluded, the mother liquor was filtered from the clathrate crystals in situ under the prevailing equilibrium pressure and the clathrate partially dried by circulating CO at the same pressure through the bed of crystals for about 1 hr in order to avoid reversal of the above reaction. The product was then quickly removed from the reaction vessel and placed on a high vacuum line to remove last traces of solvent. Infra-red spectra and X-ray powder diagrams confirmed the presence of the Bstructure. Prior to calorimetric use, the samples were stored in a vacuum desiccator maintained below 25°C. The gas content of the cIathrates was determined by fusion of a weighed amount of sample in a calibrated vacuum manifold provided with a constant volume manometer for pressure observations. Pressures of the order of 0.50.8 atm were measured to ) 0.005 cm with a Gaertner cathetometer. During the course of the preparation of the clathrates, the pressures for the equilibrium: + CU,_,,j,,~, 8) were also observed CC+,, and were used to-calculate the 25°C pressure isotherm constant &?j”c = P@O)(l -y)/y.(s) Strictly speaking, the observed pressures must be regarded as tentative equilibrium pressures since in all cases the “equilibrium” preparations were approached from only one direction and, accordingly, cannot yet be regarded as true equilibrium preparations until a more rigorous equilibrium study is completed. The heat capacities of four CO clathrates, having 46.0, 62.6, 75 ~7 and 81 .O per cent of the cavities filled, were measured from 15 to 300°K in an adiabatically operated calorimeter and cryostat which have been described previously.f‘n The ice point was taken as 298,E”K and one defined calorie aas taken as 4-1840 abs J. RESULTS

AND

DLSCUSSION

The pressure isotherm constant at 25.O”C, calculated as the slope of the plot of P@bs) vs. y/l -y at low pressures (Fig. l), was found to be 8.9 f 0.3 atm. The three phase pressure at y = O-34 was calculated to be 4.6 atm, assuming the validity of the van der Waals theory for the composition of the cfathrate for the three-phase ~quilibrium.~~~

and

L.

V.

CUULTER

Although the dependence of P, or fugacity, on y/l-y deviates from linearity above N 10 atm, these deviations cannot yet be properly attributed to the breakdown of the isotherm function or the effect of pressure on the stability of the p-quinol lattice, since the systems were not rigorousIy tested for equilibrium. The experimental heat capacities of the cfathrates of CO are given in Tables 1-4. No discontinuities were found in the 1%300°K range. From 15 to 120”K, the heat capacity of each clathrate is curved substantially towards the X axis while above lZO”K, C, is almost linearly dependent on temperature. Since the heat capacity of

Fro. 1. EguiIibrium pressure isotherm for COroti -S CU~g-4nino~, _+ Lawerieftsegment of dotted fine-r&on of instability of &&rate relative to a-qtinoi and CO gas. Camplete dotted line-apparent tow pressure limiting slope.

enclosed gas is obtained from the difference of large numbers, the measurements of heat capacities in temperature regions of interest were often repeated to insure reproducible results. The average deviations of the data from a smooth curve amounted to about + W-O*2 per cent above 90°K and increased to about & 0.5 per cent below 50°K. Two of the samples lost a small amount of CO presumably during the soldering operations on the loading port of the calorimeter. This was evidenced as a small anomaly near the triple point of carbon monoxide (N 66°K). SampIe 3 with a final y value of 0.757 had an initial value of 0*771 before loading the calorimeter while the _vvalue of

HINDERED

ROTATION

OF

OCCLUDED

CO

IN

THE

/3-QUINOL

CLATHRATE

1437

Table 1. Experimental

heat capacity ofcurbonmonoxide cluthrute No. 1: 3C&4(OH)z * O-460 CO in cul deg-1 3 moles /LquinoZ-1 (O-1113 moles of cluthrute in the calorimeter) Series I T”K G 57.54 27.62 61.03 28.92 64.41 30.10 67.71 31.22 70.94 32.26 75.19 33.66 80.12 35.18 84.63 36.49 89.08 37.76 93.63 39.00 98.14 40.16 102.32 41.36 108.29 42.99 114.15 44.71 118.56 45.97 123.29 47.34 128.34 48.80 133.36 50.23 138.39 51.66 143.62 53.13 148.90 54.69 154.27 56.38 159.71 57.97 165.04 59.62 170.45 61.19 175.65 62.86

Series I-co&d. T”K 217.94 725 223.65 78.05 229.45 80.00 235.15 81.88 240.93 83.78 246.78 85.70 252.50 87.61 Series II T”K 242.09 247.62 253.39 259.20 264.87 270.64 276.50 282.44 288.50

CP 84.18 86.05 87.96 89.81 91.68 93.67 95.78 97.64 99.74

Series III T”K 13.28 14.96 16.37 18.79

G 1.91 3.14 4.26 6.17

181.06

64.54

21.46

8.29

183.64 188.97 194.72 200.68 206.53 212.28

65.42 66.98 68.70 70.65 72.48 74.37

23.46 25.29 26.90 28.74 31.02 33.48

9.79 11.01 12.04 13.10 14.49 16.00

Series III-contd. T”K CP 35.89 17.44 38.21 18.77 40.44 19.98 43.20 21.43 46.72 23.07 SO.65 24.80 54.74 26.53 59.23 28.26 64.27 30.06 Se&s IV T”K G 15.26 3.41 16.76 4.58 18.66 6.11 21 .oo 7.96 23.29 9.67 25.69 11.31 28.48 12.95 31.57 14.88 34.54 16.66 37.83 18.55 41.31 20.38 44.67 22.10 48.55 23.89 52.85 25.70 Series V T”K 273.93 279.84 285.62 291,49 297.75

C, 94.82 96.80 98.76 100.76 103.04

_____

sample 4 went from 0.816 to O-810. When more care was taken to keep the calorimeter cool during soldering operations, subsequent samples did not undergo any decomposition. The heat capacity data have been corrected for the presence of solid and gaseous CO as well as for the heat effect associated with the sublimation of the CO. These corrections did not exceed 0.2 per cent below 50°K and O-1 per cent above 70°K. The molar heat capacity of the enclosed CO in the clathrate was obtained from the slope of the heat capacitycomposition isotherms, i.e., plots of the heat capacity of 3-p quinol. yC0 against y, for selected temperatures. With few exceptions the points fell on a straight line for each temperature in the

15-300°K range with the deviations falling within the range of the experimental uncertainties of the observed values for C,. Values of the molar heat capacity of the clathrated CO obtained in this manner for selected temperatures are given in Table 5 along with smoothed values for the same temperatures. The dependence of the heat capacity on temperature is illustrated in Fig. 2. The heat capacity of 3 moles of /3-quinol at selected temperatures is also listed in Table 5 and has been determined from the y = 0 intercepts of the heat capacity-composition isotherms for selected temperatures. The entropy of each of the CO clathrates at 298+15”K was evaluated in the usual fashion by

1438

G.

L.

STEPAKOFF

and

L.

V.

COULTER

Table 2. Experimentalheat capacity of carbon monoxide clathrate No. 2.3CsH4(0H)~ *0*626 CO in cal deg-1 3 moles ,t?-puinol-1 (O-1230 moles of clathrate in the calorimeter)

Series I T”K 59.16 62.03 65.11 68.39 71.94 75.55 79.22 83.04 87.32 91.95 96.40 100.93 105.62 1 IO.28 115.07 120.13 125.32 130.54 135.78 141.03 146.31 151.61

c, 29.41 30.50 31.62 32.75 33.85 34.98 36.13 37.40 38.67 39.99 41.21 42.42 43.84 45.11 46.59 48.07 49.58 50.97 52.56 53.88 55.56 57.16 Series II

T”K 149.47 154.66 160.07 16552 170.86 176.26 181.71 187.22 192.77

CP

56.46 58.20 59.75 61.47 62.86 64.62 66.27 67.83 69.68

Series III T”K 54.97 57.91 60.94

C, 27.80 29.06 30.10 Series IV

T”K 15.02 16.03 17.20 19.18 21.46 23.57

CD

3.37 4.29 5.18 6.84 8.77 IO.37 Series

T”K 13.93 15.64 17.06

V CP

2.67 3.90 5.10

Series T”K 19.11 21.60 23.74 25.83 28.32 30.96 33.42 35.88 38.34 40.97 43.80 46.73 49.74 52.79 55.96 59.32 62.85 66.60 Series T”K 19.07 2158 23.92 26.62 29.52 32.07 34.53 40.11 43 .Ol 45.99 49.17 52.58 Series T”K 190.16 195.79 201.93 207.88 213.86 219.71 225.60 231.92 238.47 244.99 251.35 257.54 263.62 269.71 275.99 282.35 288.60 294.57 299.98

V-contd. C, 6.81 8.80 IO.52 II.94 13.50 15.16 16.77 IS.33 19.67 21.18 22.73 24.14 25.55 26.91 28.17 29.45 30.81 32.14 VI C, 6.71 8.75 10.57 12.44 14.26 15.84 17.46 20.74 22.31 23.78 25.29 26.80 VII C, 68.94 70.63 72.70 74.51 76.40 78.26 80.18 82.28 84.46 86.58 88.74 90.87 92.82 94.81 97.06 99.27 101.07 103.30 105.77

Series T”K 210.70 221*31 227.74 234.03 240.20

VIII CD 75.36 78.74 80.90 82.97 85.16

Series IX T”K 58.36 61.17 64.08 67.17

CP

29.13 30.17 31.31 32.34 Series X

T”K 187.93 193.69 199.87 206.24 213.70 219.72 225.80

C, 68.03 70.02 71.91 73.89 76.52 78.51 80.40

Series XI T”K 178.16 183.42 189.02 194.88 200.84 206.84 212.87 219.11 225.37 231.56 237.69 243.88 250.15 256.32

C, 65.22 66.92 68.63 70.35 72.24 74.30 76.10 78.08 80.13 82.35 84.31 86.36 88.31 90.36

Series XII T”K C-9 188.14 68.25 70.03 193.78 71.86 199.65 205.72 73.65 75.77 211.99 77.75 218.14 79.78 224.42 230.97 82.14 237.39 84.13 243.69 86.20 249.87 88.39 90.14 256.13

HINDERED

ROTATION

OF

OCCLUDED

CO IN

THE

p-QUINOL

CLATHRATE

Table 3. Exprketztal heat capacity of carbon monoxide clathrate No. 3,3CaH4(OH)z * O-757 CO in cal &g-l 3 moles /Quinol-1 (0.1275 moles of clathrate in the cakrimeter) ----._

--l_l_ Series IV

Series I T”K 53.91 5667 59.24 61.82 64.57 67.81 71.55 75.35 79.21 83.32 5768 92.29 97.13 102.00 106.74 111.63 116.92 122.31 127.53 133.34 139.40 144.94

G 28*31 29.38 3044 31.43 32.43 33.71 34.91 35.99 37.28 38.60 40.06 41-34 42.65 44.07 45.42 46.87 48.47 49.97 51.55 53.30 55-08 56.61

Sm*es II T”K 143.22 148.75 154.67 160.33 165.80 171.55 177.30 182.90 188.71 194.73 202.70 208.49 211.92 217.79

CP 56.17 57.91 59.53 61.09 62.82 64.59 66.38 68.15 69.93 71.67 74.22 76.13 76.99 78.86

Series 111 T”K 215.60 221.33 227.28 236.43

G

78.15 79.93 81.90 84.93

T”K 226.17 232.07 237.97

CP

81.81 83.71 85.53 Series V

T”K 235.74 242.03 248.22 254.29 260.63 266.92 273-02 279.23 285.52 291.91

CP 85.09 87.18 89.22 91.08 93.32 95.35 97.48 99.46 101.60 103.92

Sevies VI T”K 54.72 2260 58.27 29.99 62.06 31.66 65.80 32.98 69.50 34.19 73.23 35-34 76.82 36.37 80.23 37.89 85.02 39.18 89.05 40.51 93.48 41.59 98.33 43.05 103.27 4444 105.89 45.12 110.85 46.67 115.95 48.13 121.18 49.70 128.37 51.77 134.09 53.23 139.95 55.22 145.62 56.75 151.17 58.46 156.92 60.28 162.69 61.94 168.34 63.62 174.07 65.43 179.85 67.25 188.18 70.02 194.12 71.75 199.95 73.57 205.97 75.49

Series VII T”K CP 3.51 14.91 4.70 16.31 6.39 18.33 8.36 20.69 9.91 22.50 11.13 24.16 12.38 25.88 13.90 28.13 15.78 30.95 17.97 34.28 19.95 37.47 21.85 40.83 23.86 44.51 25.61 48.18 27.53 51.94 29.05 55.77 30.52 5944 Series VIII T”K G 4.71 16.32 6.30 18.22 20.41 8.15 9.74 22.33 11.23 24.26 26.89 13.04 30.04 15.20 17.32 33.31 19.56 36.83 21.67 40.45 23-84 44.63 26.14 49.13 Series IX T”K CP 54.94 28.65 58.45 30.13 31.47 62.06 33.02 65.99 34.47 70*00 35.70 74.05 Se&s X T”K 280.60 286.55 292,26 298.47

G 99.97 102.03 103.99 106.30

1439

1440

G.

L.

and L.

STEPAKOFF

V. COULTER

Table 4. Experimental heat capacity ofcarbon monoxide clathrate No. 4.3Cs&(OH)a * 0.810 CO in cal deg-1 3 moles /3-quittoE- (0.1355 moles of clathrate in the calorimeter) -

_...._z~~ T”K

Series I CP

%*I6 59.07 61~9.5 64.87 67.69 70.58 74.16 78.74 83~80 88.99 94.09 99.13 104-14 109.82 115-82 121.20 126.43 132.13 138.28 143.98 149.21

T”K

29.53 30.80 31.95 33.06 34.05 34.92 36.05 379.51 39.27 40.92

157.96 163.40 168.69 174-16 179.85 185.52 191.07 196.51 202.18 208.09

60.76 62.48 63.97 65.67 67.52 69.24 70.93 72.70 74.70 76.61

42.25 43.71 45.11 46.78 48 *53 SO-13 51.64 53.27 54.77 56.57 58.22

213~88 219.88 226.38 232.81 238.82

78-14 80.26 82.37 84.22 86.25

Series II CP

147.41 152.64

Series II (continued) T”K G

57.72 59.26

Series II1 (continued) T”K 282.27 288,54 294.37 299.77

c, 101.20 103.17 105.44 107.29

Series IV T’K 52.99 56.05 59.06 62.06 64.90 67.82 70.78 73.76

Series III T”K 239.21 245.07 251.14 257.07 263.22 269.57 275.91

c, 28.20 29.51 30.77 31% 33.08 34.09 35.10 35.94

Series V c, 86.74 88.55 90.52 92.76 94.83 96.77 98.64

T”K:

14.35 15.62 16-75 18.75 21.25 23.34 25.37

G

3-25 4.06 5.05 6-77 8.96 10.79 12.24

_______ ._-

the graphical integration of &-fog T plots from 14.79°K to 298.15”K and from 0 to 14*79”K by extrapolating the heat capacity data with the function :

caI deg-1 mote-l, assuming the validity of the third law of thermod~ami~s for these clathrate systems. From they = 0 intercept of the same isotherm, the entropy of three moles of /I-quinol at 298 - 15 “K was

cp = 3[3(240/T)~+3(81~7/T)~]-ty[3(140/T)~ +

~(Q/T)EI

which represents C, for 3CsH4(OH)s *yCO over the 15-60°K range to within 2 1 per cent. The terms in the first bracket consist of three Debye and three Einstein functions per mole of &quinol whereas the second bracketed expression applies to the y moles of guest CO with which are associated one three-dimensional Einstein function for the vibrational or “rattling” heat capacity and one twodimensional function for the librational motion. The molar entropy of clathrated CO at 298*15”K obtained from an entropy-composition isotherm at this temperature was found to be 20,942 0.20

FIG. 2. Molar

heat capacity of occluded &quinol clathrate.

CO

in the

HINDERED

ROTATION

OF

OCCLUDED

CO

IN

THE

,!?-QUINOL

CLATHRATE

1441

Table 5. Smooth values of the heat capacity of the CO clathrates in cal deg-1 3 moles /?-quinol-1 at selected temperatures T”K

c y = 0.460

15 20 25 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 298.15 300

0.626

0.757

3.20 7.17 10.82 13.91 16.94 19.75 22.27 24.53 26.63 28.55 30.31 31.96 33.48 35.15 37.98 40.72 43.49 46.52 49.39 52.14 55.02 58.05 61.06 64.26 67.30 70.42 73.62 76.90 80.19 83.51 86.82 90.08 93.44 96.88 100.24 103.20 103.86

3.40 7.54 11.37 14.56 17.76 20.68 23.30 25.66 27.83 29.74 31.59 33.25 34.82 36.38 39.43 42.20 45.12 48.02 50.91 53.73 56.66 59.74 62.74 65.73 68.80 72.04 75.24 78.44 81.71 84.94 88.30 91.63 95.03 98.40 101.88 104.84 105.48

3.56 7.79 11.56 15.17 18.40 21.41 24.41 26.55 28.70 30.78 32.70 34.35 35.94 37.52 40.63 43.52 46.42 49.33 52.27 55.14 58.18 61.08 64.12 67.37 70.40 73.45 76.58 79.76 83.04 86.59 89.78 93 .lO 96.42 99.86 103.22 106.21 106.84

*Interpolated

values.

found to be 100.28 cal deg-1 to be compared with 100.4 and 99.8 cal deg-1, reported by PARSONAGE and STAVELEY(~) for the entropy of 3 moles of /3-quinol based on the heat capacities of the clathrates of methane, and argon respectively. The heat capacity of clathrated CO at low temperatures It is assumed that the heat capacity of clathrated molecules at low temperatures is composed of two

0.810 3.65 7.90 11.96 15.32* 18.65* 21.70* 24+42* 26.87 29.05 31.15 33.10 34.80 36.30 37.89 41.06 44.08 47.00 49.88 52.68 55.52 58.44 61.38 64.38 67.53 70.63 74.00 77.14 80.26 83.50 86.73 90.23 93.75 96.91 100.28 103.86 106.72 107.37

contributions which, to a rough approximation, are taken to be separable.@) These contributions consist of the three-dimensional oscillations of the center of mass or so called rattling motion, and the two-dimensional librations of the molecular axis in a hindering potential of the form, I’ = (L’o/2)(1- cos 20). It has been found previously(l) that these contributions, to the harmonic oscillator approximation, accounted for the heat capacity from 15 to 80°K if the rattling frequency was taken

1442

G.

L.

STEPAKOFF

as 97.3 cm-1 and the librational frequency as 43.1 cm-t, corresponding to a barrier Vs = 691 cal mole-l restricting the rotation. The proper analysis of the heat capacity data should take into consideration the anharmonicity of the rattling motion as well as the hindered rotational behavior of the guest CO molecules which previously has been approximated as a librational motion with the associated heat capacity of a twodimensional harmonic oscillator,(l) For the latter motion we now introduce the more realistic twodimensional hindered rotator model and associated

I21

ooooooooooo

IO t

-0

OO

”

-I

s-

&

f

3

s-

z

and

L.

V.

COULTER

used by PITZER and GWINN(~) for the onedimensional case, the internally restricted rotation of co-axial tops, and later by HILL(~) for similar type problems of localized adsorption. The approximation consists of writing the partition function for the hindered rotator in the form Qhr = Qf qamo/qc, where Qf is the classical phase integral for the rigid rotator in the potential field, and qgmoand qc are partition functions for a quantum mechanical and a classical harmonic oscillator, respectively, in two dimensions. The derivations of the partition function and thermodynamic functions are given in the appendix and result in functions for the two-dimensional hindered rotator which are sums of contributions of a two-dimensional harmonic oscillator with a frequency v = (l/rrc)( VI$I)~/~ plus functions expressed in terms of probability integrals in the form extensively studied by MILLER and GORDON:(s) .> F(X) = exp( -X)a

At,.

L

u- 4-

where X = (Vo/kT)l/2

{exp(rs) 0

dr

in our case.

2-

1

0

I

50

1

I

100

I50

1

200 T. OK

1

250

1

Thus for the heat capacity of the two dimensional hindered rotator:

300 C2Oh.r

FIG. 3. Comparison of smoothed molar heat capacity of occluded CO in the p-quinol clathrate with the calculated heat capacity. (a) Upper

left

curve: Ctota1 = C3%.0.(~ = 104.3 cm-l) + CZ~U.(VO = 745 cal mole-r). (b) Upper right curve: CtotiLl = CCC.tf+C2°h.r.(~0 = 2867 cal mole-l). (c) Lower left curve: C20n.~. with Yo = 745 calmole-‘. (d) Lower right curve: Cs%r. with Vo = 2867 cal mole-r. O-Smoothed experimental heat capacity; a-lower left: c exp-C3~~.~.(v=~04.3cm-l) = C&t; A-lowerright: Crot = Cexp - CCCM (heat capacity of CO according to classical cell model with A/k = 3906” and (V*/V)2 = 0.95).

equations for use in the calculation of its contributions to the thermodynamic properties of clathrated guest diatomic molecules. The partition function of the two-dimensional hindered rotator has been obtained by an approximation method analogous to the procedure first

=

;[s- 1-j -

K[l 4F(X)

+X/F(X)]

+

c,,,

(1)

is the heat capacity for a twowhere C,,, dimensional harmonic oscillator. Although this equation presumably takes into account the effect of hindered rotational motion of guest diatomic molecules over the entire temperature range of concern, an accurate calculation of the contribution of the rattling motion to the heat capacity is somewhat less certain. At low temperatures we assume again that the vibrational motion of the CO molecule in the host cavity may be described as a three-dimensional isotropic harmonic oscillator. On this basis, with v = 104.3 cm-l for the rattling vibration and a barrier, Vs = 745 cal mole-l, hindering the rotational motion, the heat capacity is well accounted for in the 15-40”K range as indicated in Fig. 3.

HINDERED

ROTATION

OF

OCCLUDED

comparison with the previous analysis,@) the frequency of the rattling motion has been increased by 7 cm-l and the barrier increased by 54 cal mole-l. It is to be noted, however, that this analysis accounts for the heat capacity only up to 40°K prhereas the previous analysis,@) based in part on the harmonic oscillator approbation for the librational motion, accounted for the heat capacity from 15” to -90°K. It may be that the latter correlation of the heat capacity to as high a temperature as 90°K by the In

CO

IN

THE

fi-QUINOL

CLATHRATE

use of the harmonic oscillator approximation was fortuitous since presumably the hindered rotator model, which approaches the harmonic oscillator model at low temperatures, should be more appropriate at higher temperatures where libration becomes free rotation to an increasing extent with increasing temperature. Accordingly, assuming the validity of equation fl), it would appear that this semiempiric~ analysis is an overly simplified approximation which fails to take into account complexities that become significant above the vicinity of -40°K.

Table 6. The partial molal heat capacity of clathrated CO and the heat capacity of three moles of &quinoE at selected temperatures T”K

15 20 25 30 35 40 45 50 55 60 65 70 7s 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 298.15 300

Ccofobs.) Cco (smoothed) (cal deg-l mole-r) (Caldeg-1 mole-l) (cal clegzs moles-r) 1.20 2.15 3.11 4.04 4.90 5.68 6.23 6.72 7.10 7.48 8.01 8.16 8.22 8.52 8.90 9.23 9.43 9.65 9.87 9.90 10.06 10.15 10.23 10.05 10.26 10.17 loa 10.01 9.85 9.89 9.89 10.09 9.93 9.98 9.74 9.78 9.75

1443

1.20 2.15 3.11 4.04 4-90 5.68 6.23 6.72 7.13 7.48 7.79 8.06 8.30 8.52 8.90 9-20 9-44 9-67 9.86 10.01 10.12 10.18 10.22 10.23 10.22 10.17 10-12 10.03 9.97 9.92 9.86 9.84 9.80 9.79 9.78 9.76 9.76

266 6.18 9.39 12.06 14.69 17.16 19-40 21.44 23.34 25.07 26.61 28.18 29.66 31.06 33.90 36.47 39.21 42.03 44.76 47.58 50.40 53.39 56.36 59,58 62.52 65.76 69-00 72-24 75.58 78.85 82.21 85.41 88.86 92.22 95,79 98.72 99-42

1444

G,

L.

STEPAKOFF

and

The thermodynamic properties of clatkrated CO at 25°C Consideration of the high molar heat capacity (- 10 cal deg-1 mole-l) of occluded CO in the range 100-300°K compels one to abandon the assumption of free rotation of CO for which the combined rotational and vibrational heat capacity of CO in the host cavity would not exceed -4R. It is in fact necessary to assume that the barrier restricting rotation in the high temperature region is substantially larger than the barrier of 745 cal mole-1 found in the low temperature region since the heat capacity associated with a restricted rotator of the latter nature has already fallen to -2.3 cal mole-l at 300”K, and in this sense is essentially a free rotator at 300°K. Accordingly, since we are dealing with hindered rotation, the assumption of separability of the rotational and rattling motions is, strictly speaking, not valid, a possibility anticipated by VAN DER WAALS.@) For the present, we propose an interpretation of the heat capacity and entropy data for clathrated CO, and the pressure isotherm constant, based on the classical equations of VAN DER WAALS~) for the thermodynamic properties of clathrates of monatomic, or freely rotating polyatomic, molecules corrected for restricted rotation by the use of the equations presented in the appendix for the twodimensional hindered rotator. Thus, for the total molar heat capacity of clathrated CO: C =

CCCizIf

C2"h.r

(2)

where the last term is given by equation (1) and Cccaf is the molar heat capacity of the occluded molecule according to the classical cell model:(a)

c CCM = Rr~.5+(A/kT)2(V*/v)4(1/g2)[(V*/V)4ggll + 4ggmm - (Y*/Y)4g;

- 4&

- 4f V*/V)2ggl~~ + 4( ;2 = Zc~a+;4;

V* =[(e+

v*/~‘>2glgwl; cco)j213

(3)

; j7 = 2-1isrt3,

(with a, the cavity radius = 3.95 A@)); and the ‘g functions” are complicated integrals dependent on the dimensionless parameters AjkT and V*/V. ~co and ace are the characteristic energy parameters of the familiar Lennard-Jones expression for pure CO, and E and or are the average cor-

L.

V.

COULTER

responding parameters for the Z atoms comprising the cavity wall. Application of equation (2) to determine C&,. r. and, accordingly, the barrier restricting the rotation of guest CO molecules, depends on a knowledge of the parameters AjkT and V*/V and hence on Z&a and 3 for the lattice as well as ~co and oco for the guest CO molecule. Although values for the latter based on second virial coeflicient data are available{“@ it is necessary to assign values to Z@a and o which will not only account for the heat capacity but the other thermodynamic nronerties as well. ’ In order to facilitate the determination of the appropriate set of the dimensionless parameters, A/kT and V*/V, the ‘g functions” which are dependent on these parameters have been calculated with an IBM 7090 computer at closely spaced intervals of AjkT and V*/V over a ranLre of values that encompasses those for the usual gue”st molecules of clathrate systems. In turn, the parts of each of the thermodynamic functions dependent on the same dimensionless parameters have been computed for the same selected values of the parameters. The resulting values of the “g functions” which we have obtained are in good agreement with the previously published table+) limited to a narrower range of AjkT and V*V. Application of equation (2) for the heat capacitvi and -the corresponding equations for the entropy and the equilibrium pressure of formation of the CO clathrate then involves an iterative procedure in which the assignments are made for AjkT and V*/V and the hindering potential Vo which will account for the experimental values for the heat entropy and the pressure isotherm capacity, constant. For the best correlation of the observed data, we find (V*/V)2 = 0.95 and A/k = 3906” from which we obtain Z(
HINDERED

ROTATION

Table

7.

OF

OCCLUDED

CO IN

THE

j3-QUINOL

CLATHRATE

Comparison of the obsemed asd ca~~l~ted~ t~~ody~a~~ properties of CO in the /I-quinoE clathrate at 25°C ~. _-...

Property K = p(l -y)/y

C cal deg-1 mole-1

* Based on Z(c/k)l/”

8.9

8.89

9.76

9.64

20.94 -____~-

S cal deg-1 mole-1

-

Calculated

Observed atm

1445

19.83 ~.

-

= 390; ij = 3.22 ii; VII = 2867 cal mole-l.

For the proposed “best” assignment the calculated properties are given in the final column in Table 7 for comparison with the observed properties listed in the second column. Although very good agreement exists for the pressure isotherm constant and the heat capacity at 25”C, the calculated entropy is -5 per cent too low. To some extent the direct contrasting of the “low” and “high” temperature barriers restricting the rotation of guest CO is probably not justified since the basis for the calculation of the contribution to the vibrational heat capacity is different in the two cases and CrOt is accordingly obtained by two different procedures. Nevertheless, it appears quite likely for the following reasons that the barrier indicated for the “high temperature region” is significantly larger than the barrier restricting rotation in the “low temperature region”. The interpretation of the data at 25°C leading to a barrier of -2.9 kcal is in qualitative agreement with the spectroscopic observations of BALL and McKEAN@~) who have observed strong absorption for the /3-quinol clathrate of CO at 2133 cm-r in the region of the forbidden Q branch of CO with quite uncertain indications of P and R branches. It is difficult to reconcile the “low temperature” barrier of 745 cal mole-1 with the spectroscopic data at room temperature since for this low barrier more prono~lnced P and R branches would be expected. Moreover, it is only possible to account for the rather large heat capacity of clathrated CO in the vicinity of 300°K by the use of a large barrier unless we assume that drastic modification of the host lattice is occurring for which there is no evidence. The low value obtained for the barrier restricting the rotation of clathrated CO in the region of low

temperatures is consistent with the values found by a variety of methods for the barriers for other diatomic molecules in their respective clathrates. Magnetic data(lsJ*) indicate barriers of 127 cal mole-l for Os(ra) and -420 cal mole-r for N0;(14) pure quadrupole resonance spectra(l5) and heat capacity data(r) for clathrated Ns indicate 960 and 511 cal mole-l, respectively, for Vs at low temperatures. With the exception of the higher value for the barrier for Ns, based on the temperature dependence of the pure quadrupole resonance spectrum of Ns in the ,8-quinol clathrate of Ns,(f@ the barriers restricting rotation of these molecules, and CO, in the corresponding clathrates increase more or less regularly as indicated in Fig. 4 with increasing 8, the collision diameter, which is a measure of “snugness of fit” of the guest molecule in the cavity. Alternatively, for this group of diatomic molecules, including CO, the restricting barrier increases with increasing quadrupole moment in a regular manner as shown graphically on the left side of Fig. 4. Accordingly, a barrier of - 1 kcal, or less, restricting the rotation of CO in the clathrate at low temperatures would appear to be quite reasonable in comparison with the barriers for other diatomic molecules in their /3quinol clathrates at low temperature. The apparent increase of the restricting barrier with increasing temperature is also plausible from a theoretical point of view and has been anticipated by VAN DER WAALS.@) For the excited vibrational states of the occluded CO in the host cavity, the probability density will be larger near the cavity wall in comparison with the ground state. Hence with increasing temperature an increasing fraction of CO molecules will be aligned along the host cavity wall and accordingly be subjected to higher

1446

G.

L.

STEPAKOFF

barriers restricting rotation than is characteristic of the ground state. In view of the apparent dependence of the restricting barrier on temperature as well as the uncertainty associated with the extent of anharmonicity of the vibrational motion of the guest molecules in the cavities, an accurate interpretation

and L.

V. COULTER

the vibrational frequency and potential restricting the rotation of CO, accounts accurately for the heat capacity and entropy of the occluded CO at 25°C and holds reasonably well for the heat capacity down to N 120°K. For a vibrational frequency, v = 74 cm-l for the three dimensional isotropic harmonic oscillatory motion of CO in the cavity- and a restricting barrier of 2027 cal mole-l, the calculated heat capacity and entropy are 9.78 cal deg-1 mole-l and 20.97 cal deg-1 mole-l, respectively, at 25”C, essentially in exact agreement with the observed values. Hence by this approximation as well, the barrier restricting rotation in the high temperature region appears to be in the vicinity of 2 kcal mole-l. Acknowledgements-The authors wish to thank the Computation Center, Massachusetts Institute of Technology, for an allocation of time on the IBM 7090 computer for the calculation of the “g functions” for the classical cell model. The authors also thank Mr. G. ROPER for his assistance with the measurements.

REFERENCES 1. COULTER L. V., STEPAKOFFG. L. and ROPER G. C.,

J. Phys. Ckem. Solids 24, 171 (1963). 2. VA~\I DER WAALS J. H., Trans. Faraday Sot. 52, 184 (19%); VAN DER WAALS j. H. and PLATTEEUW -----4x lo%“--. -or L--J. C., Advanc. them. Phys. 2, 1 (1959). Also see PARSONAGEN. G. and STAVELEYL. A. K.. Nol. FIG. 4. Dependence of barrier restricting rotation of Pkys. 2,212 (1959) for the equations for C and S. diatomic molecules in &quinol clathrates on (a) quad3. FURMAN N. H. and WALLACE J. H., JR., J. Amer. rupole moment, Q-left; (b) collision diameter, o-&ht. them. Sot. 52, 1443 (1930). Quadrupole moments: A-GORDY W., SMITH W. V. 4. COULTERL. V., SINCLAIR J. R., COLE A. and ROPER and T&MBARULO R., Microzuave Spectroscopy, p. 345. G. C., J. Amer. them. $0~. 81, 2986 (1959). Wiley, New York (1953); 0-JANSEN L. and DEWETTE F. W., Physica 21, 83 (19Sj); l HILL R. M. and Sh%xTH 5. PARSONAGEN. G. and STAVELEE’L. A. I(., Mol. Rhys. 3, 59 (1960). W. V., Phys. Rev. 82,451 (1951) ; Collision diameters6. VAN DER WAALS J. H., J. Phys. Ckem. Solids 18, 82 Ref. 10. Vo data: CO-this research; Ns-Ref. 1; (1961). NO (approx.)-Ref. 14; Oz-Ref. 13. 7. P~VZERK. S. and GWINN W. D., J. them. Phys. 10, 428 (1942). 8. HILL T. L., J. them. P&s. 16, 181 (1948); Introof the dependence of the heat capacity on temperad&don do Statistical Thermodynamics, p. 172. ture in the intermediate range is precluded at this Addison-Wesley, Reading, Mass. (1960). time. Nevertheless, we find that the apparent bar9. MILLJZRW. L. and GORDONA. R.,J. phys. Ckem. 35, rier of 2867 cal mole-l accounts fairly well for the 2788 (1931). heat capacity of clathrated CO, down to 250°K as 10. HIRSCHFELDERJ. O., CURTIS C. F. and BIRD R. B., Molecular Theory of Gases and Liquids, p. 1111. illustrated in Fig. 3, below which the calculated WiIey, New York (1954). rotational heat capacity is too small by -6 and 11. LENNARD-JONESJ. E. and DEVONSHIREA. F., Puoc. 10 per cent at 200 and 180”K, respectively. Roy. Sot. Al& 53 (1937); 165, 1 (1938); PRIGOAs an alternative to the calculation of the vibraGINE I. and GARIKIAN G., T. Chim. Phvs. 45. 273 (1948); WENTORF R. H.,~R., BUEHLE~ R. J.; tional heat capacity, CCCM, according to the classiHIRSCHFELDER J. 0. and CURTXSSC. F., J. them. cal cell model, it is of interest to note that the model Pkys. 18, 1494 (1950). employed to account for the heat capacity at low 12. BALL D. F. and MCKEAN D. C., Spectrochim. Acta temperatures, but with a different assignment for 18, 933 (1962).

HINDERED

ROTATION

OF

OCCLUDED

13. MEYER H., O’BRIEN M. C. M. and VANVLECK J. H., Proc. Roy. Sot. A243, 414 (1957). 14. VAN VLECK J. H., J. Phys. Chem. Solids 20, 241 (1961). 15. MEYER H. and SCOTT T. A,, J. Phys. Chem. Solids 11, 215 (1959). 16. STERNE T.. E., .Proc. Roy. Sot., Lond. A130, 551 (1931): FLAMMERC.. SPheroidal Wave Functions. Stanfbid University Press (1957); STRATTONJ. A.; MORSE P. M., CHU L. J., LITTLE D. C. and CORBATO F. J., Spheroidal Wave Functions, Technology Press, Cambridge, Mass. (1956).

APPENDIX

CO

IN

THE

Qiu. = Qf ?!Y!f

CLATHRATE

1447

VLECK.(~~) Thus,

the positions of B = 0 and ~9= ir are positions of equal minimum potential energy, which is in accord with the relative orientation of the diatomic molecule with respect to the oxygen hexagons of the cavity.(rs) Thus the classical Hamiltonian for rotation becomes : H

_

P$ I 2I e

Pg

+

(1/2)~0(1-cm24

(M)

The classical integral can be written as:

Qf = -&

The two-dimensional hindered rotator The rigorous formulation of the equations for the hindered rotational properties of a clathrated diatomic molecule (or a diatomic molecule in its condensed state) necessitates the summation over the eigen values of the spheroidal wave function.(ls) Since this would be a sizable task even with high speed computers, it was considered adequate to utilize the approximation technique for the construction of the quantum mechanical partition function for the hindered rotator as described by PITZER and GwINN.(~) This technique has also been used by HILL@) for the problem of localized adsorption and hindered rotation on an adsorbing surface. The approximation consists in writing the partition function for the hindered rotator in the form:

fi-QUINOL

7 I/

rexp(-H/kT)dp,dp+d*ddQ

(A3)

0 o-00--g) Substitution of (2) into (3) and integration over the momenta and + yields:

2+IkT

Qf =F

=

I 0

exp{ - Vo/kT(l - cos 28)) sin 0 d0 WI

The factor 2 in the denominator of (3) arises from the fact that there are two positions of equal minimum potential energy in the cavity. By use of the following transformations : (l/2)(1 - cos 2e> = 1 - co&9 .z = cos e

Vo/kT = X2

(Al) (4) becomes: 4c 2&Ik T where QJ is the classical phase integral for the exp( - Xs) / exp(Xszs) dz (AS) Qf = h2 rigid rotator in the potential field, and qgmoand -1 qc are partition functions for a quantum mechanical and a classical harmonic oscillator, respectively which can be further simplified by defining the (in two dimensions, in this case). Thus, as summarized by HILL,@) in the limit as Vo/kT -+ CD, new variable y = Xz and redefining the limit of integration. Thus: Qf must approach a classical oscillator, qc, so that Qh.r. approaches qqmo. On the other hand, as Vo/kT -+ 0, qqmo approaches qC SO that Qh.,.. 4GIkT exp( -Xs) x becomes Qf. In order to formulate Qj in this case, exp(r2)dy W) Qf=y x s it is assumed that one can employ the axially 0 symmetrical function, In order to investigate the limiting properties V = (1/2)Vs(l -cos28) of (6) as Vo/kT approaches infinity, use is made of the relation between the barrier Vs and the as discussed by MEYER, O’BRIEN and VAN librational frequency, (l/rr)( Vo/21)l/2. Thus, if one

G. L.

1448

STEPAKOFF

defines the term [(27?1kT)/hsJ as B, then: Vo/kT = B(hv/kT)z Substitution

or 3 = X2qc

(A7)

of (7) into (6) gives: Qf = 2XqJJ(Xj

F(X)

(A8j

= exp( - Xs) jexp( y sj dy

(A91

0

The F(X) function has been studied extensively by MILLER and GORDON(g)and its limiting properties and numerical integration have been given by these workers. The relations of immediate use here are the following : lim XF(X)

X-t=3

-dF(X) =

= l/2

W*)

l -2XF(X)

(All)

dX

Thus, when the limit of (AS) is taken as X -+ to, Qf = qc which justifies the correct asymptotic behavior of Qf. If (A8) is substituted into (Al>, one has finally: Qh.r. = 2XF(X)qpno

and L.

COULTER

the function C(X) approaches -R cal deg-1 mole-1 so that equation (16) approaches R, the value for free rotation. On the other hand, for large X, the function C(X) vanishes so that the major contribution to the heat capacity arises from the quantum mechanicat oscillator. The function C(X) has a maximum in the vicinity of X = 2.4. Since the harmonic oscillator term for intermediate values of X has almost attained its limiting value (liberational frequencies for clathrated molecules are known to lie in the 20-60 cm-r range) the total hindered rotational heat capacity can rise above the 2R value for the classical oscillator for appropriate values of Va and I. It is noted with interest that below the maximum of the heat capacity curve, equation (A16) is in quantitative agreement with the heat capacity calculated by multiplying PITZER’~ tables by 2. However, above the maximum (which is of the order of 4.3 cal deg-1 mole-r) the rotational heat capacity curve defined by equation (16) falls to R faster than the Pitzer function, for the given value of Yo and I. (See Fig. 5.) I 6

(Al2j

and the usual statistical From equation (All) mechanical relations, the following relations for the thermodynamic functions are obtained : F=

V.

-~~(lnX+lnF(~+ln2j~F~~~

t

(A13)

U = -RT[l/2+(1/2)X/F(X)-~]+

U,,,

(A14j

s = -R[1/2+(1/2)X~F(X)-Xs] + R[ln X+ In F(X) + In 21 -I-Sllmo

(A15)

C = (1/2)R[X3/F(X)--I] - [(1/4)~X/F(X)J[l

0

f X/F(X)]

+ Gmc (AIS)

Values of the function F(X) for selected values of X, taken from the extensive listing of MILLER and GORDON,cg)are given in the second column of Table A-l. In columns 3 and 4 the heat capacity and entropy functions in equations (A16) and (A15), exclusive of the harmonic oscillator contributions, are tabulated for selected values of X, where X = (~0~RT). 112It is seen that for small X,

40

60

I 120

160

200

240

I 280

-ii “K

FIG. 5. Comparison of the two-dimensional hindered rotator heat capacity function with 2x Pitzer’s function. To = 745 cal mole-l I = 14.55 x lo-a0 g cm3 Upper curve: Pitzer’s function. 1,ower curve: Two-dimensional hindered rotator.

It is found that the entropy decrease from free rotation calculated from equation (15) is of the order of one e.u. greater than that calculated from twice the value of Pitzer’s tables, for barriers and

HINDERED

ROTATION

OF

OCCLUDED

CO

IN

THE

&QUINOL

CLATHRATE

1449

Table A-l. The heat capacity and entropy junction of the laindere~ rotator C(X) = Ch.*. - C2-qmo and S(X) = Sh.,. - SsoqntOwhere X = (V~,lkT)llz.

X

F(X)*

C(XYt

S(X) $

OGO 0.20 0.40 0.60 0.80 1 .oo 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2-60 2.80 3 .oo 3 *20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5*50 6 .OO 6.50 7.00 8 .oo 10.00

0-000000 0‘194751 0.359943 0.474763 0.532101 0~538080 0.507273 0.456507 0.399940 0.346773 0.301340 0.264511 O-235313 0.212165 o-193551 0.178271 0.165462 0.154524 0.145042 0.136721 0.129348 O-122761 O-116835 O-111472 0.106593 0.102134 0.092493 0.084543 0.077868 0.072181 0.063000 0.050254

- 1.987 - 1.987 - 1,982 -1.963 - 1.907 - 1.786 - 1.564 -1.217 -0.7562 - 0.2480 +0.2033 o+io53 0.6320 0.6221 0.5410 0.4417 0.3534 0.2850 0.2340 0.1970 0.1690 0.1473 0.1302 0.1161 0.1044 0.0946 0.0754 0.0620 0.0519 0.0449 0.0332 0.0217

-a, - 7-006 - 4.254 -2.652 -1.536 - 06799 - 0.0916 + 0.3420 0.6086 0.7280 0.7317 0.6615 0.5594 0.4586 0.3718 0.3042 0.2532 0.2146 0.1849 0.1616 0.1429 0.1276 O-1146 0.1041 0.0941 0.0865 0.0703 0~0581 0.0488 0.0424 0.0317 0.0199 -

* Ref. 9. t C(x) = 0.9935[X3/F(X) - l] - 0.49675[X/F(X)][l +(X/F(X)] $ s(X) = -0.9935[1 +X/P(X)-2Xz]+4~576[Iog1~X~(X)+0.3010].

moments of inertia relevant to clathrated diatomic molecules. For the theoretical equilibrium pressure for the reaction DER x2~P-~~i~~l, 8) VAN X2(g) = SAABS assumed free rotation of guest molecules in the lattice cavities and obtained:

p

_ T-2z

kT expWoW~ g

Y

__ 1-Y

= K~y/l-y

the potential energy of the guest molecule at Y = 0 the center of the cavity; and y is the fraction of host cavities filled by the guest molecules. On taking into account the restricted rotational behavior of diatomic molecules in the host cavities equation (17) becomes

where a = 3.95 A, the average radius of the cavity; g is a complicated integral dependent on the dimensionless parameters A/kT and ~*~~,W~O~ is

x

&=2-

(A17)

F(x)

Qco Y -J&----Qqmo 1 -Y

6418)

Note added in proof STAVELEY [Advanc.

Chem. Series: Nonstoichiometric Compounds 39, 218 (1963)] has estimated V’s - 1200

cal mole-1

based

on unpublished

data of Grey

and

1450

G.

L.

STEPAKOFF

Staveley. To some extent the difference between the barrier estimated by Staveley and that obtained in this research may result from a lack of agreement with respect to the experimental heat capacity data or the assignment of parameters employed in the calculation of Cock according to equation (2). However, it is to be noted that

and L.

V.

COULTER

Staveley’s estimate which has been made with the use of the tables of PITZER and GWINN(~)is expected to be less than the “high temperature” barrier reported in this work since [email protected]., calculated by equation (l), is less than 2 x CI0h.r. (where CI”h.r. is given by PITZER and GwINNc71) above 80°K as indicated in Fig. 5 for CO.