466
JAMESC. M. LI AND KENNETH S.PITZER
I n the case of molecular diffusion the analog of the product fK is given by the theory of Brownian motion as kT where T is the absolute temperature and k is Boltzmann's constant. This no longer applies but if the colloid particles behave hydrodynamically like macroscopic bodies then a number of semi-empirical expressions for f can be substituted in eq. 24. I n general fK >> kT and hence the contribution of 2 is much smaller than in the case of molecular diffusion.
V. Discussion A number of relations characterizing the coagulation of particles in turbulent flow have been obtained on the assumption that the turbulence is homogeneous and isotropic and the fluid remains macroscopically a t rest. The possibility of extension of these results to fluids which are displaced
Vol. 60
with a constant velocity, say along the x-axis, can be relatively easily formulated since the basic turbulent diffusion theory exists.1.8.8 A more serious dif6culty is encountered when an analog of these results is required for turbulence fields which are neither homogeneous nor isotropic. It is hoped that a t least a first guide to the type of coagulation behavior to be expected in turbulent media is given by the simpler types of fields. Finally the possibility suggests itself that a global (as opposed to a local property) characteristic of field of turbulence could be established based on the coagulation time of a sufEcien tly well-characterized colloidal suspension. Needless to say, extensive further investigation would be required to implement such a suggestion. (8) E. W. Hewson, "Int. Symp. on Atmospheric Turbulence in the Boundary Layer," Geophys. Ree. Paper fl9, Cambridge, 1952.
ENERGY LEVELS AND THERMODYNAMIC FUNCTIONS FOR MOLECULES WITH INTERNAL ROTATION. IV. EXTENDED TABLES FOR MOLECULES WITH SMALL MOMENTS OF INERTIA' BY JAMESC. M. LI AND KENNETH S. PITZER Contribution from the Departmat of Chemistry and Chmical Engineering, Univemity of California, Berkeley, Califmiu Received October 3, IS66
Additional theoretical developments have made it ossible to present general tables for the thermodynamic properties of a restricted rotator which are ap licable to molecures with very small moments of inertia. Additional eigenvalues for the Mathieu equation were obtainelfor these calculations. These eigenvalues w i l l be of me also in the interpretation of spectral data for restricted rotators.
When the general tables for the thermodynamic functions for a restricted internal rotator were presented in Paper I of this series,2 it did not appear to be practical to extend general tables into the region of low moments of inertia where the partition function depends on the over-all rotation quantum numbers. Recently, however, in treating the case of methyl alcohol in which the partition function for internal rotation does depend significantly upon over-all rotation quantum number, Halford3showed that it was still possible to separate the internal and external rotational partition functions in good approximation. We will show that this separability is generally valid for molecules comprising coaxial tops. From the nature of the energy level pattern for less symmetrical molecules it seems virtually certain that the same approximation is still valid and that the extended tables to be presented herewith are applicable to all molecules which consist of a symmetrical top attached to a rigid frame. We shall find that the partition function for internal rotation is a periodic function of a boundary value parameter p which depends on the over-all (1) This work waa assisted by the American Petroleum Institute through Reaearoh Project 50 and by the Atomio Energy Commission through the Radiation Laboratory. (2) K. S. Pitzer and W. D. Gwinn, J . Cham. Phys., 10,428 (1942); see also K. S. Pitaer, ibid., 6, 469 (1937): 14, 239 (1946); J. E. Kilpatrick and K. 9. Pitaer, ibid., 17, 1064 (1949). (3) J . 0. Halford, ibid., 18, 1051 (1950).
rotational state. The essential approximation is that the actual molecules a t temperatures of interest be widely distributed over external rotational levels and as a result uniformly distributed over the range in p . Therefore we take the average over p for our final partition function for internal rotation. The reduced moment of inertia and similar quantities were d e h e d in Paper I. Extended Tables of Thermodynamic Functions for Internal Rotation.-We first give in Tables I-IV supplements to Tables I, 111, V and VI in Paper I for smaller moments of inertia and hence larger values of the reciprocal of the partition function for free rotation. In terms of current values of physical constants4 we have Qr = 2.7935( 10"BIT)'h/n
(2)
where I is the reduced moment of inertia for internal rotation. In the present tables R is 1.9872 which does not differ significantly from the value 1.9869 used previously. It should be noted that equation 2 is the.classical approximation to the partition function for free rotation, We use this expression as a definition of &S even down to &f = 1 where the classical approxi(4) F. D. Rossini. F. T. Guoker. Jr., H. 1,. .Jolmston, L. Pauling and G . W. Vinal, J . A m . Chem. SOC.,7 4 , 2699 (1952).
467
THERMODYNAMIC FUNCTIONS FOR MOLECULES WITH INTERNAL ROTATION
April, 1956
TABLE I ( -F/T) \
V/RT
‘I
0 0.2 .4 .6
0.55
0.60
1,190 1.182 1.164 1.136 1.099 1.056 .937 .817 ,705 .608 .525 .455 .398 ,347 .273 .218 .179 .149 .124 .092 ,069 .053 .042 .033
.8
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
0.65
1.014 1.009 .997 .974 .947 ,912 .815 .713 .616 .BO
.457 .393 .340 .297 .230 .181 .145 .120 .loo .071 .052 .039 .030 ,023
0.856 .852 .842
.826 .804 .777 .700 .615 .534 .458 .395 .339 .290 .253 .193 .149 .118 .095 .079 ,054 .038 .028 ,022 ,017
0.70
0.75
0.710 .707 .699 ,687 .670 .647 .588 .521 .454 .390 .336 .288 .247 .214 .161 .123 .096 .078 .063 .042 ,030 .021 .016 .012
0.575 .570 .565 .555 .543 .526 .481 .428 .375 .324 .278 .239 .205 .177 .131 .loo .078 .062 .049 .033 ,023 .016 .012 .009
0.80
0.443 .441 .438 .431 .424 .411 .379 .338 .298 .258 .222 .190 .162 .139 .lo3 .078 .060
.047 .037 .025 .016 .012 .008 .006
0.85
0.323 .321 .318 .315 .310 .302 .277 .249 .219 .191 .165 .140 .117 .lo2 ,074 .056 .042 .032 .026 .018 .012 .008 .006 .004
0.90
0.208
.207 .206 .204 .200 .195 .178 .160 .141 .12% .lo5 .088
.074 ,063 .045 .032 .024 .019 ,015 .OlO .007 ,004 .003 .002
0.95
0.102 .lo1 .099 .097 ,096 .094 .084 .074 .063 .053 ,042 .034 .027 .020 .012
1.00
0.001 .Ooo - .002
- ,002 - .002 - ,004
- .009
- .011
- .015 - ,018 - .021 -
.008
-
.005 .004 .002 .001
-
.Ooo
-
.ow .Ooo
.om
.024 ,026 .025 .024 .022 .018 .014 ,012 ,008 ,007 .004 .003 .002
TABLE I1
s V/RT
0
0.2 .4 .6 .8
1.0 1.5 2.0 2.5 3.0
3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
\
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
2.182 2.180 2.170 2.151 2.125 2.094 1.997 1.874 1.739 1.576 1.458 1.328 1.209 1.097 0.915 .774 ,660 .570 .496 .388 .309 .251 ,205 .170
2.009 2.003 1.996 1.980 1.957 1.928 1.833 1.718 1.589 1,450 1.323 1.199 1.086 0.982 ,808 .672 .566 .483 .414 .315 ,247 .196 ,158 .129
1.850 1.848 1.837 1.823 1.800 1.774 1.685 1.578 1.456 1.330 1.206 1.087 0.978 ,881 ,715 .588 ,486 .407 .348 ,255 .196 .155 ,121 ,097
1.703 1.701 1.691 1.677 1.654 1.629 1.552 1.450 1.335 1.217 1.100 0.988
1.567 1.563 1.555 1.541 1.523 1.499 1.428 1.332 1.224 1.114 1.004 0.901 .804 .716 .568 .453 ,366 ,300 .248 .176 .126 ,092 .072 .056
1.438 1.433 1.428 1.415 1.399 1.377 1.310 1.224 1.126 1.021 0.919 .821 .730 .648 .509 ,401 .320 .258 ,211 .146 .loo .075 ,056
1.316 1.312 1 .307 1.295 1.284 1.262 1.201 1.122 1.031 0.936 .841 .748 ,662 .588 .457 .357 .281 .223 .180 ,122 .084 .059 ,042 .032
1.203 1.196 1.193 1.184 1.171 1.153 1.094 1.024 0.942 ,855 ,769 .683 .607
1.097 1.091 1.085 1.076 1.068 1.052 1,000 0.936 ,860 ,779 ,703 ,623 .551 .486 ,372 .285 .220 ,171 ,134 ,084 .056 ,038 ,026 ,018
0.995 ,991 .983 .976 ,966 .952 ,906 ,846 .779 .710 ,638 .568 .501 .442 .339 .257 ,195 ,150 .116 .073 .046 .029 .020 .014
,884
.794 .637 .516 .422 .350 .293 .213 .157 .119 ,093 .073
mation is clearlv invalid. Also we find that the average of & ( p ) ” over p can give values less than unity. We will see t,hat &(O) cannot be less than One, but &(PI can be lower for other Values of P. This is the explanation of the negative values in the last colunin of Table I. For 7‘ ‘I’ and Ivy the functions &’ and &” are needed. These were calculated by
,042
,535
.412 .319 .248 .195 .154 .lo1 ,069 .048 .034 ,024
standard methods and treated in exactly the method to be described for Q. The energy levels for the restricted rotator are eigenvalues for the Mathieu equation. These eigenvalues are available5 for p = 0 and r. The ei(5) “Tsblea Relating to Mathieu Functiona,” from tbe Computation Laboratory of the National Bureau of Standards, Columbia Univemity preas, New York, N. Y.,1951.
468
JAMESC. M. LI AND KENNETH S. PITZER
Vol. 60
TABLE 111 HIT V/RT
\ I/"
0 0.2 .4 .6 .8
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0.994 0.996 1.006 1.014 1,026 1.038 1.059 1 ,057 1.032 0.988 .933 .872 ,810 .750 ,644 ,554 ,480 .421 .370 .296 .240 .19s ,164 ,138
0.994 ,994 ,999 1.004 1,009 1.014 1.019 1.005 0,.972 ,924 .868 .806 .744 .685 ,580 ,491 .420 ,363 .314 .244 .195 ,157 .128 ,105
0.994 .994 .994 .995 ,996 ,996 .987 .962 ,922 ,870 .811 .749 .687 . G28 .523 ,437 .368 .312 .269 .202 .158 .I27 .099 ,080
0.994 ,994 .992 .990 ,984 ,982 ,962 .928 .882 ,828 ,765 .701
0.994 ,992 .990 .987 .980 .972 ,945 ,904 .850 ,791 ,727 . G61 .599 .540 ,437 .354 .290 ,240 ,200 .I43 ,103 ,076 .060 .047
0.994 ,992 .98a .984 .976 .965 ,932 ,886 ,827 ,763 ,697 ,630 .567 .508 .406 ,324 .261 ,211 ,174 ,121 .084 .061 ,047 ,036
0.994 .991 .988 .982 .974 .962 .922 .873 ,811 .744 .676 .609 .545 .485 .383 .302 .239 ,191 .154 ,104 ,072 .051 .036 .028
0.994 .990 .986 .980 .972 ,960 ,916 ,864 ,801 .732 ,663 .595 .531 ,470 ,368 .286 ,223 ,176 .140 ,091 ,062 ,044 ,029 .022
0.994 ,989 ,985 .979 .971 .959 .915 .860 .796 ,728 .659 .590 ,526 .465 ,361 .279 .215 ,168 .132 ,084 .056 .038 ,026 ,018
0.994 ,990 .985 .979 ,970 .960 .917 ,861 ,797 .729 ,662 ,592 ,528 ,467 .363 ,279 .213 ,165 ,129 ,081 ,053 ,034 ,023 ,016
.638 ,580 .476 .392 .326 ,273 .231 .I70 .127 ,098 .077 .061
TABLE IV C V/RT\
0 0.2 .4 .6 .8
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
'/"
0.50
0.55
0.60
0.994 1.000 1,018 1,049 1.092 1.144 1.299 1.465 1.619 1.732 1.803 1.834 1.832 1.808 1.711 1.568 1.468 1.362 1.262 1.107 0.978 ,873 ,780 ,701
0.994 1.000 1.017 1.046 1,084 1.131 1.273 1.424 I . 562 1,663 1.727 1.754 1.749 1.718 1.614 1.487 1.366 1,250 1.151 0,989 ,855 ,749 .G57 ,580
0.994 1.000 1.015 1.041 1,075 1,118 1.247 1.382 1.504 1,597 1.654 1.674 1.664 1.631 1.520 1.390 1.262 1.144 1.045 0.877 .744 ,639 ,549 ,477
0.65
0.994 1,000 1.013 1.036 1.067 1.105 1.218 1.341 1.448 1.532 1.580 1.593 1.578 1.543 1,429 1,296 1.164 1.048 0.943 .774 ,644 ,542 ,456 ,389
0.70
0.994 I ,000 1.012 1,031 I . 058 1.091 1,192 1,300 1.393 1.460 1.506 I . 513 1,496 1.457 1.342 1.207 1.074 0.956 ,850 .682 ,554 ,457 ,378 ,316
genvalues for p = a/2 (Le., the Mathieu functions periodic in 4 a but not in 2a) were kindly computed for us by Robert Pexton of the Theoretical Section, Livermore Laboratory, Radiation Laboratory. University of California. Table V gives these eigenvalues which should be of value in calculating spectral frequencies in internal rotator problems in addition to the present problem and possibly others.
0.75
0.994 1.000 1.010 1.026 1.049 1.078 I . 165 1.258 1.341 1.401 1.432 1.435 1.413 1.373 1.255 1.120 0,988 .869 ,765 .600 ,479 ,387 .312 .256
0.80
0.994 1.000 1.008 1.021 1.040 1.065 1.141 I . 218 1.289 1.337 1.361 1.359 1.333 1.292 1.173 1.040 0.908 .789 ,688 ,528 .411
.324 ,259 .208
0.85
0.90
0.95
1.00
0.994 0.999 1.007 1.017 1.031 1.052 1.115 1.180 1.238 1.276 1.293 1.286 1.259 1.214 1.096 0.962 ,834 .717 ,618 ,463 ,352 .272 .215 ,168
0.994 0,999 1.005 1.014 1.025 1.040 1.090 1.146 1.190 1.217 1.226 1.215 1.185 1.140 1.022 0.890 .765 ,652 .556 .407 .303 .229 .175 .135
0.994 0.999 1.004 1.011 1.020 1,031 1.070 1.113 1.146 1.164 1.165 1.148 1.115 1.068 0.954 .826 .704 .593 .499 .358 .262 .194 .144 .lo9
0.994 0,999 1,004 1.010 1.015 1.024 1,053 1,083 1.105 1.114 1.108 1.085 1.051 1.004 0.888 .767 .647 ,542 .450 .313 .226 .162 .119 .089
In addition to Tables I-IV, the thermodynamic functions are presented in Tables VI-IX in terms of variables related directly to Mathieu function parameters. These tables, which will be discussed below, will be found more convenient for certain calculations because the temperature appears in only one variable, rather than both of the variables of Tables I-IV.
469
THERMODYNAMIC FUNCTIONS FOR MOLECULES WITH INTERNAL ROTATION
April, 1956
TABLE V MATHIEU EQUATION 1ZIGENVALUE :S e 0.25 ,50 .75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00
eo,l
eO,Q
0.208949 .092337 .084899 .307285 .562707 .a42670 1.141382 - 1.454857 - 1,780282 2.115613 - 2.459323 2.810233 3.167422 3.530152 - 3.897815 - 4.269911 - 4.646019 5.025777 - 5.408889 5.795073 - 6.184114 6.575788 - 6.969928 7.366371 - 7.764973 - 8.165598 - 8.568133 - 8.972473 9.378517 9.786184 -10.195386 -10.606050 -11.018110 -11.431503 -11.846159 -12.262037 -12.679082 -13.097248 -13.516487 -13.936760
-
-
-
2.274369 2,340760 2.433751 2.537181 2.638247 2.727976 2.800509 2.852375 2.881905 2,888745 2.873486 2.837294 2.781657 2.708185 2.618455 2.513956 2.396052 2.265952 2.124731 1,973339 1.812598 1.643238 I ,465902 1.281160 1.089517 0.891417 ,687267 ,477423 .262230 ,041912 .la3173 ,412823 .646818 .884933 1.126996 1.372820 1,622247 -1.875117 -2.131302 2.390669
-
-
-
-
h a 6.255967 6.274030 6.304650 6.348449 6,406033 6.477777 6.563612 6.662826 6.774095 6.895458 7,024558 7.158754 7.295355 7.431776 7,565455 7.694393 7.816734 7.930712 8.035237 8.129310 8.212282 8.283713 8.343425 8,391462 8.427866 8.452919 8.467041 8.470638 8.464031 8.447799 8.422363 8.388190 8.345675 8.295253 8.237319 8.172240 8.100359 8.022001 7.937465 7.847028
OW 12.252779 12.261122 12.275058 12.294629 12.319901 12,350990 12.387986 12.481050 12.480344 12.536090 12.598502 12.667837 12.744301 12.828091 12.919413 13.018332 13,124883 13.238932 13.360270 13.488511 13.623158 13.763515 13.908874 14.058307 14.210958 14.365680 14.521570 14.677645 14,832845 14.986332 15.136962 15.284212 15.427032 15.565008 15.697350 15.823622 15,943356 16.056300 16.162150 16.260728
eoS4 20,251624 20.256504 20.264620 20.276003 20.290654 20.308586 20,329813 20.354375 20.382265 20.413524 20.448182 20.486275 20,527825 20.572907 20,621561 20.673846 20.729853 20.789622 20.853250 20.920797 20.992400 21.068172 21.148188 21.232537 21.321411 21.414838 21.513003 21.615939 21.723756 21.836492 21.954224 22.076962 22.204611 22.337188 22.474465 22.616434 22.762836 22.913155 23.067546 23.225194
834
eo,,
h 7
30.251068 30.254274 30.259618 30.267102 30.276728 30.288500 30.302421 30.318497 30.336733 30.357135 30.379711 30.404466 30.431411 30.460556 30.491911 30.525488 30,561298 30.599360 30.639683 30.682286 30.727185 30.774402 30.823948 30.875858 30.930150 30.986859 31.046001 31.107612 31.171714 31.238356 31.307586 31.379426 31.453905 31.531119 31.611060 31.693839 31.779462 31.868010 31.959538 32.054086
42.250758 42.253030 42.256819 42,262124 42.268945 42.277286 42.287145 42,298526 42.311429 42.325857 42.341813 42.359298 42.378317 42.398871 42.420962 42.444598 42.469779 42.496512 42.524800 42.554647 42.586060 42.619043 42.653608 42,689751 42.727484 42.766812 42.807745 42,850280 42.894439 42,940224 42.987645 43.036710 43.087445 43.139837 43.193907 43.249649 43.307102 43.366286 43.427191 43.489820
56.250566 56.252263 56.255091 56.259051 56.264143 56.270367 56.277724 56.286215 56.295841 56.306601 56.318498 56.331531 56.345703 56.361014 56.377466 56.395061 56.413799 56.433683 56.454710 56.476889 56.500219 56.524701 56.550339 56.577134 56.605088 56.634204 56.664485 56.695942 56.728563 56.762359 56.797331 56.833484 56.870821 56.909335 56.949047 56.989954 57.032058 57.075365 57.119894 57.165624
BQ,C 72.250439 72.251755 72.253950 72,257024 72.260977 72.265811 72.271527 72,278125 72.285606 72.293877 72.303095 72.313193 72.324171 72.336029 72.348768 72.362389 72.376891 72.392276 72.408543 72.425694 72.443730 72.462649 72.482536 72.503247 72.524849 72.547343 72.570729 72.59501 1 72.620188 72.646263 72.673238 72.701113 72,729815 72.759478 72.790044 72.821513 72.853889 72.887170 72.921361 72.956533
BO,* 90.250360 90.251401 90,253153 90.255606 90.258761 90.26261 9 90.267179 90.272443 90.278411 90.285083 90.292460 90.300438 90.309198 90.318660 90.328823 90.339690 90,351259 90.363531 90,376506 90.390184 90.404566 90.419652 90.435443 90.451938 90.469138 90.487043 90.505746 90,525082 90.545128 90,565883 90.587350 90.609528 90.632420 90.656025 90.680344 90.705380 90.731131 90.757601 90.784687 90.812574
TABLE VI -F/RT 0 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 25 30 35 40
4.5
4.0
3.5
3.0
2.5
2.0
1.75
1.5
1.25
1. o
0.6313 .6270 ,6148 ,5960 ,5723 ,5455 .5170 .4877 .4588 ,4307 .4037 .3541 .3107 ,2732 ,2413 ,2140 .1621 .1265 . I012 .0827
0.5724 .5679 .5551 ,5355 ,5110 .4833 ,4542 .4248 ,3959 .3682 .3419 .2944 .2539 .2197 .1918 ,1673 .1231 .0938 .0735
0.5056 .5010 .4877 .4674 .4422 .4141 .3849 ,3559 .3277 .3011 .2763 .2325 .1962 .1665 .I423 .1225 .0870 .0644 .0491 .0383
0.4285 ,4238 ,4102 .3896 .3643 ,3366 ,3083 ,2806 ,2543 .2300 .2077 ,1696 .1392 .1153 ,0963 .0813 .0552 .0393 .0288 .0217
0.3374 ,3326 .3193 ,2993 .2752 ,2494 ,2236 ,1990 ,1764 ,1560 .1380 ,1082 ,0857 ,0687 ,0558 ,0458 ,0294 .0198 .0138 ,0099
0,2258 ,2216 ,2097 ,1924 .1721 ,1511 .1310 ,1126 .0965 .0826 ,0708 ,0525 ,0396 ,0304 ,0238 ,0189 .0112 .0070 ,0045 .0030
0.1590 ,1554 ,1453 ,1308 .1143 ,0977 ,0823
0.0820 ,0794 ,0723 ,0624 .0517 ,0416 ,0325 ,0256 ,0200 ,0157 .0124 ,0081 ,0056 ,0041 .0031 .0024 ,0014
-0.0092 - .0100 - ,0119 - ,0142 - ,0160 - .0169 ,0172
.oO08
,0004 .o002
-0.1208 - ,1183 - ,1117 - .lo18 - ,0900 ,0776 - .0657 - .0548 .0454 ,0373 - ,0306 - ,0206 - ,0139 - .oo95 - .0065 - ,0045 - ,0019 .0009 - .0004 ,0002
,0589
Mathematical Treatment.-The essential derivations will be aresented brieflv for the case of two coaxial symmetrical tops. In"addition to the work Of Halforda and the papers Of this series,2 reference should be made to the important work of
.0688 .0573 .0477 ,0400 ,0285 .0208 ,0156 ,0119 ,0093 ,0052 ,0031 .0019 ,0012
.0005
.0003
-
-
-
,0160
.0144 ,0128 ,0111 ,0081 ,0058 .0040
,0028
- ,0020 - .OW8
-
.oooo
-
-
-
Dennison and collaborators6 on the energy levels of molecules of this type and to a recent revision of (6) J. 8. Koehler and D. M. Denniaon. Phhud. RBU..I T . 1006 (1940): D. G . Burkhard and D. M . Dennkon, ;bid.: 84, 408 (1951); E. V. Ivmh and D. M. Dennison, J . Chem. Phya.. a i , 1804 (1953).
470
JAMESC. M. LI AND KENNETH S. PITZER
Vol. 60
TABLE VI1 SIR 4e\+
0 1 2 3 4 5 6
7 8
9 10 12 14 16 18 20 25 30 35 "4
4.5
1.1313 1.1290 1.1222 1.1111 1.0959 1.0770 1.0548 1.0297 1.0024 0.9731 .9425 .8789 .8147 .7523 .6931 .6383 .5215 .4316 .3628 .3094
4.0
1 ,0724
1.0696 1.0614 1.0479 1.0297 1.0071 0.9808 .9515 .9199 .8866 .8521 .7821 .7135 .6487 .5897 .5352 .4249 .3437 .2834 .2375
3.6
3.0
1.0056 1.0022 0.9920 .9754 .9531 .9258 ,8945 .8601 ,8235 .7857 ,7472 .6714 .5998 .5347 .4768 .4260 .3266 .2568 .2066 .1692
0.9285 .9241 .9112 .8904 .8627 .8295 .7920 .7516 ,7096 .6672 .6253 .5456 ,4739 .4117 .3589 .3138 .2300 .1743 .1355 .lo75
2.5
1.75
1.5
1.25
1.0
0.7258 .7180 ,6956 .6608 ,6169 ,5672 ,5152 ,4636 .4143 ,3685 .3270 .2570 .2030 .I619 ,1306 .lo66 ,3671 ,0443 ,0303 ,0212
0.6590 ,6499 .6239 ,5841 ,5350 .4812
0.5819 .5712 ,5409 ,4958 ,4420 .3854 ,3306 .2804 ,2362 ,1983 ,1665 .1181 ,0851 ,0625 ,0468 ,0356 ,0190
0.4908 ,4783 ,4436 .3937 .3370 ,2807 .2289 .1851 ,1488 ,1193 .0958 ,0627 .0420
0.3792 .3653 .3274 .2759 ,2215 ,1719 ,1305 .0979 ,0731 ,0547 .0412 ,0238 ,0143 .0089 .0057 .0037 .0014
.0108
.0034 ,0018 .0011
2.0
1.75
2.0
0.8374 .8316 ,8148 ,7882 ,7534 .7125 .6677 .6210 .5740 ,5282 ,4845 ,4054 .3388 .2842 .2399 ,2041 ,1408 .loll .0748 .0566
.4268
,3745 ,3264 ,2833 ,2454 .1846 ,1402 ,1079 ,0843 ,0668 ,0393 ,0243 ,0157 ,0104
,0064 .0040
.0289
,0203 .0146 ,0068
,0006
,0003 .0001
TABLE VI11 H/RT 48\7
0 1 2 3 4
5 6
7 8
9 10 12 14 16 18 20 25 30 35 40
4.6
0.5000 .5020 .5074 .5150 .5235 .5315 ,5379 .5420 .5435 .5424 .5388 ,5248 .5040 .4790 .4519 .4243 .3594 ,3051 .2615 ,2267
4.0
0.5000 .5017 .5062 .5124 .5187 .5238 ,5266 .5268 .5240 .5184 .5102 ,4876 ,4596 .4289 .3979 .3679 .3018 ,2499 .2099 ,1786
3.5
0.5000
.5012 .5043 .5080 .5109 .5117 ,5096 .5043 .4958 .4845 ,4709 ,4389 .4036 ,3682 ,3345 .3035 ,2396 ,1924 .1575 .I309
3.0
0.5OOO .5004 .5010 ,5008 .4984 .4928 .a36 .4710 .4553 .4373 .4176 .3759 ,3347 ,2964 ,2623 ,2325 ,1748 .1350 ,1067 .0858
2.5
0.5000
,4990 .4955 .4888 .4781 .4631 ,4441 ,4220 ,3976 .3722 ,3465 ,2971 ,2531 ,2155 ,1841 ,1582 .1114 ,0813 ,0610 ,0467
the calculations for methanol.' The asymmetry of the OH group is so small that the model of two coaxial tops is still satisfactory for methanol. However, it is important t o distinguish between examples such as ethane or methylchloroform where both tops have strict n-fold symmetry and exmnples like methanol where one top is asymmetric but of a nature to leave two moments of inertia near?y equal. We s h d also mention the probable effects in cases of large asymmetry. For the case of coaxial tops, the external rotation perpendicular t o the symmetry axis is readily separated and the partition function integrated. The complete rotational partition function in good approximation is then (7) E. V. Ivauh, J. C. M. Li snd E. 8. Pitzer, J . Cham. Phys., 33, 1814 (1955).
0.5000 ,4965 .4859 ,4685 ,4448 ,4161 .3843 .3510 .3178 ,2859 ,2562 ,2045 ,1634 .I315 ,1069 ,0878 ,0559 ,0373 ,0258 .0182
0.5000
,4945 .4786 .4533 .4207 ,3836 ,3445 .3058 ,2691 ,2355 ,2054 ,1561 .1194 ,0924 ,0724 .0576 .0340 ,0212 .0137 .0092
1.5
1.25
1.0
0.5000 ,4919
0.5000 ,4883 ,4555 ,4080 ,3531 ,2976 .2461 ,2011 .I632 .I321 ,1070 ,0708 ,0478 .0329 .0231 ,0165 ,0076
0.5000 .4836 .4391 .3777 ,3115 ,2495 ,1962 .1527 ,1185 ,0921 .0718 ,0444 ,0282 .0h3 ,0122 .0083 .0033 .0014 ,0007 ,0003
.4686
.4334 .3903 .3439 ,2978 ,2548 ,2162 .1826 .1541 ,1100 .0795 .0585 ,0437 ,0332 ,0177 ,0100 .0059 ,0036
,0038
,0020 ,0011
where A is the total moment of inertia perpendicular to the symmetry axis, C, the total moment about the symmetry axis, U , the symmetry number for the internal rotation over-all rotation, and E~,,(R) energy for the indicated level. It is to be noted that E depends upon the quantum number K , which gives the total angular momentum about the symmetry axis, as well as the quantum number for internal rotation, 1. The sum in K runs from to+-o). Koehler and Dennisons showed that the variation of E with K was periodic. Let us define a --o)
I
47 1
THERMODYNAMIC FUNCTIONS FOR MOLECULES WITH INTERNAL ROTATION
April, 1956
TABLE IX 48\7
0 1
2 3 5 G 7 8 9 10 12 14 16 18 20
40
4.5
4.0
3.5
3.0
2.5
0.5000 ,5038 ,5152 ,5335 ,5577 ,5868 ,6194 ,6540 ,6894 ,7241 ,7572 ,8150 ,8580 ,8850 .a967 ,8955 .a543 ,7880 ,7188 ,6556
0.5000 ,5046 ,5180 ,5394 ,5675 .6005 ,6366 ,6740 ,7108 ,7457 ,7773 ,8277 ,8587 ,8710 ,8682 .8517 .7831 ,7027 ,6282 ,5640
0.5000 .5055 .5215 .5468 .5792 .6163 ,6556 .6946 .7311 ,7635 ,7906 .8269 ,8396 ,8234 ,8108 .7798 ,6871 ,5978 ,5220 ,4593
0.5000 ,5067 ,5260 ,5557 .5928 ,6335 .6743 ,7120 .7444 ,7698 ,7877 ,8006 ,7881 .7580 .7178 ,6733 .5642 ,4730 ,4006 ,3426
0.5000 ,5082 ,5313 ,5658 ,6066 ,6483 ,6862 ,7168 .7380 ,7492 .7508 ,7301 ,6880 ,6356 ,5807 .5278 .4155 ,3314 ,2682 ,2195
variable P in which E will have the Deriod 2n. The used in I is related by p = i p / ~ . - For examples where both tops have the same n-fold symmetry (as CHZ-CFI) or where only one top has a strict symmetry (as CH3-OH), the expression for p in terms of K is p
p =
(2d1/nC)K
+ 2?rp/n
(4)
where C1is the moment of inertia of one top (one which is strictly symmetrical), n is the symmetry number for internal rotation, and p is an integer such that 0 2 IpI 5 4 2 . The value of p depends upon the nuclear spin orientation of the symmetrically placed nuclei. I n a CHI group = 0 for the spin species of symmetry A, and p = 1 for species E. If both tops have strict symmetry, either may be taken for C1, but for optimum accuracg in the calculation to follow the top with smaller moment of inertia should be selected. Let us now sum the partition function over the internal rotational energy levels for various values of .p. The results may be expanded in Fourier series.
*
&(p) = 1 am
e-Ft,p =
&(PI
=
2 a, m
cos mp
(5)
cos mpdp, m # 0
Only the cosine terms are needed since Q is an even function of p. We may now insert this result in the total partition function QT
=
8iT2AkT
-
e-6K' am cos [mp(K)]dK (7) X
m
Since 6 is small in all cases of interest, we may replace the sum over K by an integral provided the cosine factor also varies slowly with K . The latter condition will hold for m = 0 and 1 at least.
-
cos [mp(K)] = cos ( 2 & n / n ) cos (2i~C,mK/nC) sin (27~Bm/n)sin (27~CtmKlnC)
2.0
1.75
1.5
1.25
1.0
0.5000 .5097 ,5366 .5742 .6143 ,6495 ,6743 ,6863 .6853 ,6728 .6514 .5918 .5239 ,4580 .3988 .3479 .2502 ,1844 ,1384 ,1053
0.5000 ,5104 ,5381 .5748 .6102 ,6361 .6481 ,6450 ,6276 .6007 .567 1 .4913 ,4170 .3514 ,2963 ,2508 ,1691 .1173 .0831 .0598
0.5000 .5105 ,5372 .5689 .5938 .6038 ,5965 ,5738 .5392 .4990 ,4556 ,3713 .2986 .2400 .1939 .1578 ,0971 .0619 .ON4 .0270
0.5000 ,5090 .5300 .5492 .5444 .5403 .5088 .4655 .4166 ,3673 ,3202 .2400 ,1793 .1349 ,1023 ,0788 .0424 ,0239 ,0139 ,0083
0.5000 .5042 .5096 .5030 .4760 ,4310 .3755 ,3179 ,2637 ,2163 ,1763 ,1166 ,0780 ,0530 ,0366 ,0257 .0112 .0052 ,0025 ,0013
-
QT =
8aaAkT -h2u [a0
();
'14
+ ...I
(8)
+ al cos (2i~p/n)e-*'Cl*/n*C~6+ . . .
(9)
+ a,
CON
&E
QI = a0
(2i~fl/n)e-*'C1*/n'C'6
-(;)
8a2AkT = h%
'1%
The result in equation 8 is immediately factorable into the standard partition function for external rotation QE and the desired result for internal rotation QI, equation 9. The subscript I will be omitted hereafter. I n Paper I, Q as defined in equation 5 was assumed to be independent of p whereupon uo = Q and all other a's are zero. The tables were discontinued whenever Q(0) and Q ( r )were observed to be significantly different. The next approximation is the first term in the series in equation 9, ie., Q = ao. This corresponds to the average of Q ( p ) over p. Halford's result is essentially this approximation, although he further concluded that a. = '/zQ(O) 4l/@(n). He verified that the additional terms were negligible for methanol at 200°K. ; however, it is instructive to note their magnitude. With the molecular data of Ivash and Dennison, we obtain st 200"K., a. = 1.190, al = 0.007, and the exponent in equation 9 is -23.5 for the ul term. For the A nuclear spin species, p = 0, and the second term contributes a correction of +lO-loo/o. For the E species, p = f1, and cos(2np/n) = - 1/2. Thus the correction is half as large and of the opposite sign. We have also studied molecules where the two tops have different orders of symmetry. The expression for p becomes more complex in these cases but a8 K increases the values of p always oscillate rapidly over their full range in a manner to make Q r = a0 a good approximation. For example if nl = 3 and 12% = 2, one may express the energy levels by taking (2rCt/6C)K (~TC;/SC)K
p =
p =
-
T
for even K for odd K
Vol. 60
JAMES C. M. LI AND KENNETH S. PITZER
472
TABLE X a0 1.00
1.25
1.50
1.75
2.00
~~
0 I 2 3 4 5 6 7 8 9 10 12 14
0.8862 ,8883 ,8942 ,9032 ,9139 ,9253 ,9364 ,9466 ,9556 ,9633 .9698 .9796 ,9862 .9906 .9935 ,9955 .9981 .9991 .9996 .9998
16 18 20 25 30 35 40
2.5
~~
0,9908 9901 98b 1 ,9858 ,9841 ,9832 ,9829 .9841 .9857 .9873 ,9889 .9919 .9943 ,9957 ,9972 ,9980 ,9992 .9996 ,9998 .9999
1 ,0854 1 0826 1,0750 1.0644 1.0531 1.0424 I , 0333 1.0259 1.0202 1.0158 1.0125 I . 0081 1.0056 1.0041 1.0031 1.0024 1.0014 1,0008 1.0005 1.0003
I ,i n 4 1,1681
1,1564 1.1398 I . 1211 1.1026 1,0858 1.0712 1.0590 1.0490 1.0408 I . 0289 1.0210 1.0157 1.0120 1.0093 1.0053 1.0031 1,0019 1.0012
~
1.2533 1 2480 1.2333 1.2124 1.1878 1.1631 1.1399 1.1192 1.1013 1.0861 1 0733 1.0539 1.0404 1.0309 1 0240 1,0190 1.0112 1.0070 1.0045 1.0030
3.0
3.5
4.0
4.5
1 ,5350 1.5277 1.5071 1.4764 1.4396 1.4002 1.3611 1.3239 1.2896 1 ,2586 1.2309 1,1849 1,1494 I . 1222 1.1011 1.0847 1.0568 1.0400 1.0293 1.0220
1.6580 1.6503 1.6285 1.5958 1.5560 1.5131 1.4695 1.4274 1.3878 1.3514 I . 3183 1,2617 1,2168 1.1811 1,1529 1.1303 I . 0910 1,0665 1.0503 1.0390
1.7725 1.7646 I . 7422 1.7033 1 ,G669 1.6214 1.5749 1.5292 1.4857 1.4451 1.4076 1.3424 1.2890 1,2458 1.2107 1,1821 1.1310 1.0984 1.0763 1.0607
1.8800 1.8720 1.8494 1.8149 1.7724 1.7254 1.6769 1.6286 1.5822 1.5383 1.4974 1.4249 1.3644 1.3142 1.2728 1.2386 1.1759 1.1348 1.1065 1.0862
~~
1,4012 1.3947 1.3761 1,3489 1.3168 1.2832 1.2505 I . 2202 1.1929 1.1689 1,1479 1,1143 1,0895 1.0713 1.0574 1.0469 1.0298 1.0200 1.0139 1.0100
TABLE XI al x 104 48 \ r
0 1 2 3 4 5 6 7
8 9 10 12 14 16 18 20 25 30 35
40
1.0
1.25
1.5
1.75
2.0
2.5
3.0
3.5
4.0
4.5
1503 1468 1369 1228 1067
907 888 836 760 672 582 500 417 348 290 240 164 113 78 54 38 16 7 3 2
536 526 499 459 412 363 314 270 229 194 164 116 82 58 41 29 13 6 3
312 308 293 272 248 22 1 194 169 146 126 108 79 57 41 30 22 10 5 2 1
180 178 170 160 146 132 118 104 91 79 69 52 38 28 21 16 8 4 2 1
59 58 56 53 49 4!J 41 37 33 30 26 21 16 13 10 7 4 2 1 1
19 18 18 17 16 15 14 13 12 10 10 8 6 5 4 3 2 1 1 0
6 6 6 5 5 5 4 4 4 4 3 3 2 2 2 1 1 0
2 2 2 2 2 2
1 1 1 0
90G 756 624 512 418 341 227 152 103 71 49 21 9 4 2
1
These expressions are for symmetry A for both of the tops. The calculation leading from equation 7 to equation 8 now yields a zero coefficient for a ] , and the coefficient of a2 will be extremely small in any practical example. We are unable to think of any real molecules (at meaningful temperatures) in which the terms beyond no in equation 9 will be significant.* Nevertheless, we give in the Appendix values of al and ( 8 ) Halford (ref. 3) indicates concern about molecules auoh as 1butyl alcohol in which C becomes very large and therefore the exponent in equation 9 might become small. However, he presumably thought CI could be the OH group moment, which is not correct since the OH group does not have the threefold symmetry. Actually CI S C in this c u e so that the argument in the cosine in equation 5 changes by 21/3 for successive K values with a given nuclear spin speciw. Halford’e detailed arguments lead to this aame conclusion.
1
1 1 1 1 1 1 1 1 0
a2, in addition to ao, which should certainly suffice
to establish the value of Q in all cases. I n addition to the partition function itself, we need the related functions written as Q‘ and Q” and defined as the sums of Fle-Fl and Fl2e-FI, respectively, where Fl is the quantity defined in equation 3. These functions are similarly related to the variable p and are treated correspondingly. The following expressions, which should be of fully adequate accuracy, were used in the numerical calculations. [&(O)
+
+ +
2Q(n/2) Q(r)l/4 [Q(O) - &(r)1/2 a2 = I&(O) - 2Q(r/2) &(.r)1/4 ao =
a1 =
(10)
The equations for the a‘ and an are of identical
THERMODYNAMIC FUNCTIONS FOR MOLECULES WITH INTERNAL ROTATION
April, 1956
473
TABLE XI1 --al' x 104 r
48
1.0
1.25
1.75
1.5
2.0
2.5
3.0
3.5 ~~~~~
2957 2867 2621 2278 1903 1544 1230 968 757 592 463 287 181 117 78 53 21 9 4 4
0 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 25 30 35 40
2344 2282 2114 1873 1602 1334 1088 876 699 555 440 276 175 112 73 49 19 8 4 2
1193 1170 1105 1010 898 782 669 566 474 394 326 222 150 101 69 47 18 8 3 2
1716 1678 1571 1416 1238 1055 884 730 598 486 394 256 167 109 72 48 18 8 3 2
799 786 748 691 623 551 480 414 356 300 254 179 126 88 61 43 18 8 3 2
333 328 315 296 273 247 22 1 196 173 151 132 99 74 55 41 30 14 7 3 2
129 128 124 117 110 101 92 83 75 67 59 47 37 28 22 17 9 5 2 1
4.0
4.5
17 17 17 16 15 14 13 12 11 10 10 8 7 6 5 4 2 1 1 0
6 6 6 6 5 5 5
~
48 47 46 44 42 39 36 33 30 27 25 20 16 13 10 8 5 3 2 1
4 4 4 4 3 3 2 2 2 1 1 0
TABLE XI11 -al" x 104 48
\I
0
I
1 2 3 4
1.0
-848 -876 -937 -987
- 994
- 944 -845 -730 - 603 -487
5 6 7 8 9 10 12 14 16 18 20 25 30 35 40
1.25
'
-384 -227
- 129 - 72 - 40 - 22 - 5 - 1 - 0
916 842 652 402 157 - 41 - 180 259 -293 -295 277 -216 152 - 102 - 63 - 42 - 13 - 4 - 1
-
-
-
0
1.75
1.5
1793 1719 1516 1237 936 656 422 240 103 17 -41 -91 95 81 62 - 46 - 19 - '7 - 3 - 1
2.0
2013 1952 1783 1544 1274 1009 768 561 405 277 181 59 0 24 -31 -29 18 - 8 - 4 - 2
-
1856 1812 1686 1504 1294 1079 878 700 547 420 318 173 86 37 11 - 6
-
- 9 - 7 - 4 - 2
2.5
1191 1170 1112 1025 920 810 701 598 504 423 351 238 158 103 66 41 11 2 0 0
3.0
3 ..5
4.0
4.6
624 616 59 1 555 510 462 413 365 320 279 241 179 131 95 68 49 20 8 3 1
29 1 288 278 264 247 227 207 187 168 150 134 105 81 62 48 36 18 9 4 2
126 124 121 116 109 102 94 86 79 72 65 53 43 34 27 22 12 7 4 2
51 51 50 48 46 43 40 37 34 32 29 24 20 17 14 11 7 4 2 1
TABLE XIV 40
0
1 2 3 4
5 6 7 8 9 10
12 14
\r
a¶ X 104
-as'
X 10'
at'
1
1.25
1
1.25
1.5
0.9 .9 .8 .7 .6 .5 .4 .3 .2 .2 .1 .I .1
0.1 .1 .1
8.6 8.3 7.6 6.5 5.4 4.4 3.4 2.6 2.0 1.5 1.1 0.6
1 .o 1.0 0.9 .8 .7 .6 .5 .4 .3 .2
0.1 .1
.1
.1 .1
.1
.o .o .o .o .o .o
0.3
.2
.1 .1
.1 .1
.1 .1 .1 .1
.o .o .o .o .o
1
62.9 60.7 54.8 47.7 37.9 29.5 22.3 16.5 11.1 8.6 6.1 3.0
1.5
X 104 1.25
1.5
10.1 9.5 9.1 8.0 6.9 5.5 4.7 3.5 2.7 2.0 1.6 1.1 0.5
1.4 1.4 1.3
1.2 1.0 0.9 0.7 0.6 0.5 0.4 0.3 0.1 0.0
474
BERNHARD GROSSAND RAYMOND M. Fuoss
form. For these quantities it was more convenient to use variables which relate more closely to the Mathieu equation parameters. 40 = 5.000 X 103YV0/na = rVo/RT r = 4 Q Z / r = 32raIkT/n2ha
(11)
Here VOis the potential barrier height in cal./mole, and I is in g. cm.2. In these terms the Mathieu equation is M” + (e, + 28 COB 22)M = 0 (12) M(z
+
T) =
eGM(z)
and the partition function for a given
p
is
m
Qb)=
z=o
exp [eo,o(O)
-
~ O . Z ( P ) ~ / T
(13)
Since we do not believe it will be necessary to use the a1 and a2 quantities in practical cases, we shall not present the detailed formulas for thermodynamic properties involving these terms. The derivations follow in a straightforward fashion from equation 9. The 0 and 7 variables are more convenient for some thermodynamic calculations than Qfand ( V / RT) since T appears in both the latter but in only 7 and not 0. Thus in calculating properties for a given molecule at a series of temperatures one has only to vary r and may hold 13 constant. Both for this reason and because they represent the original calculated values, we present thermodynamic functions in Tables VI to IX as functions of 7 and 4 0.
Vol. 60
The values in Tables I to IV were interpolated from these directly calculated values. The derivation in this section was presented for the example of two coaxial tops. While we are not prepared to present a rigorous derivation for the general case of a symmetric top attached to a rigid frame it seems practically certain that the tables will be applicable in the extended region just as they were in the region discussed in Paper I. The only requirement is that the coupling with overall rotation be of such a nature as to weight & ( p ) equally over the full range of p. Since varying rotational states will require different p values and there will be a broad distribution over various rotational states, it seems safe to assume that the range of p will be covered evenly. Thus we believe it is safe to use the present tables in the same manner as was recommended in Paper I. Appendix In Tables X through XIV are given values of the functions ao,al, a’, a!, a2,a’, a[ as defined in equation 6 and 10 and the accompanying text. The corresponding values of al and a;‘ can be obtained from Tables VI11 and I X by the equations relating the partition function and the thermodynamic properties.
[s + (“)‘I
ao‘ = ao(H/RT) ao“ = ao k
(14) (15)
ELECTRICAL PROPERTIES OF SOLIDS. XIX. CARBON BLACK I N POLAR AND NON-POLAR POLYMERS BY BERNHARD GROSS’AND RAYMOND M. Fuoss Sterling Chemistry Laboratory, Contribution No. 1319, Yale University, New Haven, Connecticut Received October 7 , 1066
The electrical roperties (d.c. conductance K , dielectric constant E and loss tan ent, tan 6 ) of non-polar (butadiene-styrene) and polar (butaJene-acrylonitrile)polymers containing 0-96 parts by weight of various types of carbon black per 100 parts elastomer have been measured; frequency range, 30 c. G f 0.5 Mc. With carbon blacks whose particles average greater than 100 m# in diameter, the dielectric constant can be computed by Bruggemann’s mixture rule up to about 60 parts of carbon by weight; for smaller particles or higher concentrations, the observed dielectric constant is greater than calculated. Chain formation can account for this result. With coarse blacks, K decreases init.ially with concentration of carbon, due to adsorption of electrolyte on the surface of the particles, while fine blacks cause a rapid increase in K. Marked transients were found in K, which satisfy a power law in time; the transients are ascribed to an ionic atmosphere surrounding the carbon particles and t o the semiconducting layer of adsorbed ions. The magnitude of the transient component increases with increasing concentration of coarse blacks. For the polar acrylonitrile polymer, the maximum ax. loss factor appears at about 200 kc., independent of carbon content. I n the non-polar polymer, losses are decreased by initial addition of carbon due to adsorption; the adsorbed layer of ions then gives rise to a pure a.c. loss mechanism.
Introduction Earlier work on the electrical properties of rubber-carbon mixtures has been reviewed by McPherson i2 recent investigations3.* have shown that with regard to conductance these systems (1) On leave of absence from the Instituto Nacional de Tecnologia, Rio de Janeiro, Brazil. Grateful acknowledgment is made to the California Research Corporation and to the National Research Council of Brazil for research grants for the academic year 1954-55. (2) A. T. McPherson, “Electrical Properties of Rubber,“ Chapter in C. C. Davies and J. T. Blake, “Chemistry and Technology of Rubber,” Reinhold Publ. Corp., New York, N. y.,1937. (3) B. B. 8. T. Boonstra and E. M. Dannenberg, I n d . Eng. Chem.
46, 218 (1954). (4) P. E. Wack, R. L. Anthony and E. Guth, J . Appl. Phys., 18, 456 (1847).
possess two clearly distinct regions, one in which their behavior approaches that of an insulator and another where it approaches that of a conductor. These regions are separated by a rather narrow interval, across which a relatively small increase in concentration of the black causes the conductance to increase by about 4-5 orders of magnitude. The present paper deals only with the first zone. Conductance, like other electrical properties, depends mainly on particle size and concentration of the (carbon) black. For sufficiently high concentrations and/or sufficiently small particle size, it increases strongly due to chain formation by the particles, but before this happens it may show a
