Hydrocarbon Gases

On the basis of kinetic theory monatomic gases have a heat capacity of 1/2 R per mole per degree of freedom; having three degrees of transla- tional f...
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HEAT CAPACITIES OF

Hydrocarbon Gases DANIEL R. STULL AND F. DREW MAYFIELD

......

TheDow Chemical Company, Midland, Mich.

freedom is included. A table of bonding frequency contributions to the molar heat capacity for temperatures from 250" to 1500" IC;. is presented; the nature of these contributions as a function of temperature is easily observed in Figure 1. The calculated heat capacities of twenty-nine hydrocarbons over the temperature range 250" to 1500' K. are compared with previously published data and show an average difference of * 4 per cent.

The method of calculating gaseous heat capacities from spectroscopic data as developed by Bennewitz and Rossner, and modified by Dobratz, has been extended to a wider temperanne range. By reassignment of certain frequencies, the method has been brought into closer agreement with literature data on heat capacities; it has also been extended to acetylene derivatives. A table of solutions to the Einstein function for one degree of

T

cules have been treated satisfactorily thus far. Some ten years ago Mecke (20)proposed that the approach be simplified by combining the multitude of vibrational frequencies associated with a given valence bond into two simple frequencies. One of these frequencies represents the vibrational contributions which act in the line of the vibrating bodies (so-called valence vibration, v) ; the other frequency represents the vibrational Contributions which act perpendicular to the valence vibrations (so-called deformation vibration, 8). Mecke also found from spectroscopic data that a given bonding exhibited remarkably similar frequencies regardless of the compound in which it occurred. More recently Bennewitz and Rossner (2) applied Einstein functions to these vibrations in order to evaluate the vibrational contributions to gaseous heat capacities. They also experimentally measured gaseous heat capacities of a variety of organic compounds, and evaluated the valence and deformation frequencies for the bondings involved. Bennewits and Rossner used the following expression to calculate the heat capacity of a gas a t zero pressure:

H E heat capacity of any gas may be evaluated as the sum of three types of energy absorption-translational, vibrational, and rotational. On the basis of kinetic theory monatomic gases have a heat capacity of 1/2 R per mole per degree of freedom; having three degrees of translational freedom, C, at zero pressure becomes 3/2 R and Cp a t zero pressure is then 5 / 2 R. Polyatomic molecules have the ability to absorb energy in two additional manners-rotation and vibration. For the first of these, the contribution to heat capacity has a maximum value of 1/2 R per degree of rotational freedom, while the second has a maximum contribution of R per degree of Yibrational freedom. The problem of calculating heat capacities of polyatomic gases then resolves itself into an evaluation of the temperature function of each of these rotational and vibrational degrees of freedom. Thirty-five years ago Einstein (9) proposed from theoretical considerations of a three-dimensional crystal that the heat capacity of a crystal be related t o its degrees of freedom in the following manner:

These authors presented a table of valence and deformation vibrational bonding contributions to the heat capacity for temperatures ranging from 290" to 690" K. Dobratz (7) noted that the calculated values of Bennewitz and Rossner were consistently low a t temperatures below 400" K. He suggested that this might be corrected by assuming the rotational degrees of freedom t o be already excited at temperatures of 300" K. To obtain a better fit of the experimental data, Dobratz modified Equation 2 to the following form: aR (3) c; 4R f 3 Z(acv) 4- @(&a)

Extensive experimental studies of the heat capacities of solids since that time have borne out the validity of this relation. This Einstein function appears to be equally applicable to the vibrational degrees of freedom encountered in a gas. Whereas the three vibrational degrees of freedom of a crystal may be treated in one expression (as in Equation l), each of the vibrational degrees of freedom of a gas must be treated separately. Tables of solutions of the Einstein Equation 1 are usually calculated for three degrees of freedom (18); but for this work it is desirable to have a table of solutions to the Einstein function based on a single degree of freedom (Table I) to facilitate the treatment of each of the vibrational degrees of freedom separately. The rigorous evaluation of individual degrees of vibrational freedom is time consuming and requires extensive spectroscopic data. For this reason only simple polyatomic mole-

+

Dobrats presented quadratic equations to be employed in evaluating the various valence and deformation vibrational bonding contributions to the heat capacity for the same temperature range as Bennewitz and Rossner. However, 639

INDUSTRIAL AND ENGINEERING CHEMISTRY

640

Table I.

.

Solutions t o Einstein Function for One Degree of Freedom

X

Cvib

X

cai b

X

OIO0O

1.98700 1.98699 1,98698 1.98696 1.98693 1.98690 1.98685 1.98680 1.98674 1.98666 1.98659 1.98650 1.98640 1.98630 1.98619 1,98607 1.98594 1.98580 1.98566 1.98551 1.98534 1.98518 1.98500 1,98481 1.98462 1.98441 1,98420 1.98399 1.98376 1.98352 1.98328 1,98303 1 ,98277 1.98250 1.98222 1,98194 1.98164 1.98134 1.98103 1.98071

0.500 0,505 0.510 0.515 0.520 0.525 0.530 0.535 0,540 0,545 0.550 0.555 0.560 0,565 0,570 0.575 0.580 0.585 0,590 0.595 0.600 0.605 0.610 0,615 0.620 0.625 0.630 0.635 0.640 0.645 0.650 0.655 0.660 0.665 0.670 0.676 0.680 0.685 0.690 0.695 0,700 0,705 0.710 0.715 0.720 0,725 0.730 0.735 0.740 0.745 0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0,790 0.795 0.800 0,805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 0.845 0.850 0.855 0.860 0.865 0.870 0.875 0 880 0.885 0.890 0.895 0.900 0.905 0.910 0.915 0.920 0,925 0.930 0.935 0.940 0.946 0.950 0.955 0,960 0,965 0,970 0.975 0,980 0.985 0.990 0.995

1.94612 1.94530 1,94449 1,94366 1,94283 1,94198 1.94113 1.94028 1,93941 1,93854 1.93766 1.93677 1.93588 1.93497 1.93406 1.93315 1,93222 1.93129 1.93036 1,92940 1,92845 1.92748 1,92652 1.92554 1.92455 1.92356 1.92256 1.92156 1.92054 1.91952 1.91849 1,91746 1.91642 1,91537 1.91431 1.91324 1.91217 1,91109 1.91001 1.90891 1.90781 1.90671 1.90559 1.90447 1,90334 1.90221 1.90106 1,89991 1.89875 1,89759 1.89642 1.89524 1.89406 1.89287 1,89167 1,89046 1.88925 1.88803 1.88680 1,88557 1.88433 1.88309 1.88183 1.88057 1.87931 1.87803 1,87675 1,87547 1.87417 1.87287 1.87157 1 ,87025 1.86893 1.86761 1.86627 1.86493 1.86359 1.86224 1.86088 1.85951 1 ,85814 1.85676 1.85538 1,85399 1.85259 1.85118 1 ,84977 1.84836 1.84693 1.84561 1,84407 1.84263 1.84118 1.83973 1.83827 1.83680 1.83533 1.83385 1.83233 1.83087

1.000 1.005 1.010 1.015 1.020 1.026 1.030 1.035 1.040 1.045 1,050 1.055 1.060 1.065 1.070 1.075 1.080 1.085 1.090 1.095 1.100 1.105 1.110 1.115 1.120 1.125 1.130 1,135 1.140 1.145 1.150 1.155 1.160 1.165 1.170 1.175 1.180 1,186 1.190 1.195 1.200 1,205 1.210 1.215 1.220 1.225 1.230 1.235 1.240 1.245 1,250 1.255 1.260 1.265 1.270 1.275 1.280 1.285 1,290 1.295 1.300 1,305 1.310 1,315 1.320 1.325 1,330 1.335 1.340 1,345 1.350 1.353 1.360 1.365 1.370 1.375 1.380 1.385 1.390 1.395 1.400 1.405 1.410 1,415 1,420 1.425 1.430 1.435 1.440 1.445 1,450 1.455 1.460 1.465 1.470 1.475 1.480 1,485 1.490 1.495

0,005 0,010 0,015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0,070 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140 0.145 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.185 0.190 0.195 0.200 0.205 0.210 0.215 0,220 0.225 0.230 0.235 0.240 0,245 0.250 0.255 0.260 0.265 0.270 0.275 0.280 0.285 0.290 0.295 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.366 0.360 0.365 0,370 0.375 0.380 0.386 0.390 0.395 0.400 0.405 0.410 0.415 0.420 0.425 0.430 0.435 0.440 0.445 0.450 0.456 0.460 0.465 0.470 0.475 0.480 0.485 0.490 0 495

1.98039 1,98006 1.97971 1.97936 1,97900 1,97864 1.97826 1.97788 1.97749 1.97709 1.97668 1.97627 1.97584 1.97541 1,97497 1.97453 1.97407 1.97361 1,97313 1.97265 1.97216 1,97167 1.97116 1.97065 1.97013 1,96960 1.96006 1.96882 1.96797 1.96741 1.96684 1,96626 1.96568 1.96509 1.96449 1.96388 1,96326 1,96264 1.96201 1.96137 1.96072 1.96006 1,95940 1.95873 1,95805 1.95736 1.95666 1,95696 1.95525 1,95453 1.96381 1,95307 1,95233 1.95158 1 ,95082 1.96006 1.94928 1,94850 1.94772 1.94692

Vol. 35, No. 6

Cvib

1.82938 1.82787 1.82640 1.82485 1.82333 1.82180 1.82027 1.81873 1,81719 1.81864 1.81408 1,81252 1.81095 1.80938 1.80780 1.80622 1.80463 1.80303 1.80143 1.79982 1,79817 1.79659 1,79496 1.79333 1.79170 1.79006 1.78841 1.78676 1.78510 1.78344 1.78177 1.78010 1.77842 1.77673 1.77504 1.77335 1.77165 1,76994 1.76823 1.76651 1.76479 1.76307 1.76134 1.75960 1.75786 1.75611 1.75436 1.76260 1 ,75084 1.74907 1.74730 1.74552 1.74374 1 .i4195 1.74016 1.73837 1.73657 1,73476 1,73295 1.73113 1,72929 1.72749 1.72566 1.72382 1.72199 1.72014 1.71829 1.71644 1.71458 1.71272 1.71085 1.70898 1.70711 1 ,70523 1.70334 1.70146 1.69956 1.69763 1.69576 1.69386 1,69195 1.69003 1.68811 1.68619 1.68426 1.68233 1.68040 1.67846 1.67651 1.67456 1.67262 1.67066 1.66870 1.66674 1.66477 1.66280 1.66082 1.65884 1.65686 1.65487

X

1.500 1.505 1.510 1.515 1.520 1.525 1.530 1.535 1.840 1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585 1,590 1.595 1.600 1.605 1.610 1.615 1.620 1.625 1.630 1.635 1.640 1.645 1.650 1.655 1.660 1.665 1.670 1.675 1.680 1.685 1.690 1.695 1.700 1.705 1.710 1.715 1.720 1.725 1.730 1.735 1.740 1.745 1.750 1.755 1.760 1.765 1.770 1.775 1.780 1,785 1.790 1.795 1.800 1.805 1.810 1.815 1.820 1.825 1.830 1.835 1.840 1.845 1.850 1.855 1.860 1.865 1.870 1.875 1.880 1.885 1.800 1.895 1.900 1.905 1.910 1.915 1.920 1.925 1.930 1.935 1.940 1.945 1.950 1.955 1.960 1.965 1.970 1.975 1,980 1.985 1.990 1.995

Ceib 1.I35288 1.65089 1.64889 1.64689 1,64488 1.64287 1.64086 1.63884 1.63682 1.63480 1,63277 1.63074 1.62871 1.62667 1.62463 1,62258 1.62053 1.61848 1.61643 1.61437 1,61231 1.61024 1.60818 1,60610 1,60403 1.60196 1.59987 1.59779 1,59570 1.59361 1.59151 1.58942 1,58732 1.58522 1.58311 1,58100 1.57889 1,57678 1.57466 1,57254 1.57042 1.56829 1.56616 1.56403 1.56189 1.55976 1.55762 1 ,55548 1.55333 1.55118 1.54903 1 ,54688 1,54472 1.54257 1,54041 1.53824 1.53608 1,53391 1.53174 1,52957 1,52739 1.52521 1,52303 1.52085 1.51867 1.51648 1.51429 1.51210 1 ,50991 1.50772 1.50552 1,50332 1.50112 1.49891 1.49671 1.49450 1.49229 1.49008 1.48786 1,48565 1.48343 1.48121 1.47899 1.47677 1.47454 1.47231 1.47009 1.46785 1.46562 1.46339 1.46116 1.45892 1.45668 1 ,45444 1.45219 1.44995 1.44771 1.44546 1.44321 1.44096

X

2.000 2.005 2.010 2.015 2.020 2,025 2,030 2.035 2.040 2.045 2.050 2.055 2.060 2.065 2.070 2.075 2,080 2.085 2.090 2.095 2.100 2.105 2.110 2.115 2.120 2.125 2.130 2.135 2.140 2.145 2.150 2.155 2.160 2.165 2.170 2.175 2.180 2.185 2,190 2.195 2.200 2.205 2.210 2.215 2.220 2.225 2.230 2.235 2.240 2.245 2,250 2.255 2.260 2.265 2.270 2.275 2.280 2.285 2,290 2.295 2.300 2.305 2.310 2.315 2.320 2,325 2.300 2.335 2.340 2.345 2.350 2.355 2.360 2.365 2.370 2.375 2.389 2.380 2.390 2.395 2.400 2.405 2.410 2.415 2.420 2.425 2.430 2.435 2.440 2,446 2.450 2.455 2.460 2,465 2.470 2.475 2.480 2,486 2.490 2.495

Cwb 1,43871 1,43646 1,43420 1.43195 1.42969 1.42743 1,42517 1,42291 1.42065 1.41839 1.41612 1.41386 1.41159 1.40932 1.40705 1.40478 1.40261 1,40024 1.39797 1.39569 1.39341 1.39114 1.38886 1.38658 1.38430 1.38202 1.37974 1.37746 1,37517 1.37289 1.37061 1.36832 1.36603 1.36376 1.36146 1,35917 1,35688 1.35459 1.35230 1.35001 1.34772 1.34543 1,34313 1 ,34084 1.33855 1.33625 1.33396 1.33166 1.32937 1,32707 1.32477 1.32248 1.32018 1.31788 1.31558 1.31329 1.31099 1.30869 1.30639 1.30409 1.30179 1.29949 1.29719 1.29489 1.29259 1.29029 1.28799 1.28569 1 ,28339 1.28109 1.27879 1.27649 1.27419 1.27189 1.26959 1.26729 1.26499 1.26269 1.26039 1.25809 1,25579 1.25349 1.25119 1.24889 1.24659 1.24429 1.24200 1.23970 1.23740 1,23510 1,23281 1,23051 1,22821 1,22592 1.22362 1.22133 1.21904 1.21674 1.21445 1.21216

X

2.500 2,505 2.510 2.515 2,520 2,525 2,530 2.535 2,540 2.545 2.550 2.555 2.560 2.565 2.570 2.575 2.580 2.585 2.590 2.595 2.600 2.605 2.610 2.615 2.620 2.625 2.630 2.635 2.640 2 645 2.650 2.655 2.660 2.655 2.670 2.675 2.680 2,685 2.690 2.695 2.700 2.705 2.710 2.715 2,720 2.725 2.730 2.735 2.740 2.745 2.750 2.755 2.760 2.765 2,770 2,775 2.780 2.785 2,790 2.795 2.800 2.805 2.810 2.815 2.820 2.825 2.830 2.835 2.840 2.845 2.850 2.855 2,860 2.865 2.870 2.875 2.880 2.885 2,890 2.895 2 I900 2.905 2.910 2.915 2.920 2.925 2.930 2.935 2.940 2.945 2.950 2.955 2.960 2.965 2.970 2.975 2.980 2.985 2.990 2.995

C*ib 1.20986 1 ,20757 1.20528 1.20299 1,20070 1,19841 1,19612 1.19384 1.19155 1.18926 1.18698 1.18469 1,18241 1.18012 1,17784 1,17556 1.17328 1,17100 1.16872 1.16644 1.16416 1.16189 1,15961 1.15733 1.15506 1.15279 1.15051 1.14824 1.14597 1,14370 1,14144 1,13917 1.13690 1.13464 1.13237 1.13011 1.12785 1,12559 1,12333 1.12107 1.11882 1,11656 1,11431 1.11205 1,10980 1.10755 1.10530 1,10305 1.10080 1.09856 1,09631 1.09407 1.09183 1.08959 1.08735 1.08511 1.08287 1.08064 1.07835 1.07618 1.07395 1.07172 1.06949 1.06726 1.06504 1.06282 1.06059 1,05837 1.05616 1,05394 1.05173 1.04951 1.04730 1.04609 1.04288 1,04068 1.03847 1.03627 1.03407 1.03187 1.02967 1.02747 1.02528 1.02308 1.02089 1.01870 1.01652 1.01433 1.01215 1.00096 1.00778 1.00661 1.00343 1.00125 0.90908 0,99691 0.99474 0.99258 0.98041 0.98825

-

INDUSTRIAL AND ENGINEERING CHEMISTRY

June, 1943

Solutions to Einstein Function for One Degree of Freedom (Concluded)

Table I. 1:

3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.66 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94

641

Cvib

a:

l i

0.98609 0.98177 0.97746 0.97316 0.96887 0.96468 0.96031 0,95604 0.95178 0.94753 0.94329 0.93906 0.93484 0.93062 0.92641 0.92222 0.91803 0.91385 0.90968 0.90552 0.90137 0.89723 0.89310 0.88897 0.88486 0.88076 0.87666 0.87258 0.86851 0.86444 0.86039 0.85635 0.85231 0.84829 0.84428 0.84028 0.83628 0.83230 0.82833 0.82437 0.82042 0.81648 0.81255 0.80863 0.80473 0.80083 0.79695 0.79307 0.78921 0.78536 0.78151 0.77768 0.77386 0.77006 0.76626 0.76247 0.75870 0.75493 0.75118 0.74744 0.74371 0.74000 0.73629 0.73260 0.72891 0.72524 0,72158 0.71793 0.71430 0.71067 0.70706 0.70346 0.69987 0.69629 0.69273 0.68917 0.68563 0.68210 0.67858 0.67507 0.67158 0.66810 0.66463 0.66117 0.65772 0.65429 0.65086 0.64745 0.64406 0.64067 0.63730 0.63393 0.63058 0.62725 0.62392

3.95 3.96 3.97 3.98 3.Q9 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70 4.71 4.72 4.73 4.74 4.75 4.76 4.77 4.78 4.79 4.80 4.81 4.82 4.83 4.84 4.85 4.86 4.87 4.88 4.89

4.90 4.91 4.92 4.93 4.94 4.95 4.96 4.97 4.98 4.99 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 6.34 5.35 5.36 5.37 5.38 5.39 5.40 5.41 5.42 5.43 5.44 5.45 5.46 5.47 5.48 5.49 5.50 5.51 5.52 5.53 5.54 5.55 5.56 5.57 5.58 5.59 5.60 5.61 5.62 5.63 5.64 5.65 5.66 5.67 5.68 5.69 5.70 5.71 5.72 5.73 5.74 5.75 5.76 5.77 5.78 5.79 5.80 5.81 5.82 5.83 5.84

0.55690 0.55384 0.55079 0.54776 0.54474 0.54173 0.53873 0,53575 0,53276 0.52982 0.52687 0,52393 0.52101 0.51810 0.51520 0.51231 0.50944 0.50658 0.50372 0.50088 0.49806 0.49524 0.49244 0.48965 0.48687

0.45709 0.45445 0.45183 0.44921 0.44661

0.41877 0.41631 0.41386 0.41142 0.40900 0,40658 0.40418 0.40178 0.39940 0.39703 0.39467 0.39232 0.38998 0.38766 0.38534

Cwib

0.33926 0.33719 0.33512 0.33307 0.33102 0.32899 0.32696 0.32495 0.32294 0.32095 0.31897 0.31700 0.31503 0,31308 0.31114 0.30920 0.30728 0.30537 0.30347 0.30158 0.29970 0.29782 0.29596 0.29411 0.29226 0.29043 0.28861 0.28679 0.28499 0.28319

0.25576 0.25412 0.25249 0.25087 0.24926 0.24766 0.24607 0.24449 0.24291 0.24135 0.23979 0.23824 0.23670 0.23517 0.23365

0.21746 0.21604 0.21462 0.21322 0.21182

0.20360 0.20226 0.20092 0.10960 0.19828

2

5.85 5.86 5.87 5.88 5.89 5.90 5.91 5.92 5.93 5.94 5.95 5.96 5.97 5.98 5.99 6.00 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 6.35 6.36 6.37 6.38 6.39 6.40 6.41 6.42 6.43 6.44 6.45 6.46 6.47 6.48 6.49 6.50 6.51 6.52 6.53 6.54 6.55 6.56 6.57 6.58 6.59 6.60 6.61 6.62 6.63 6.64 6.65 6.66 6.67 6.68 6.69 6.70 6.71 6.72 6.73 6.74 6.75 6.76 6.77 6.78 6.79

Cuib

0.19697 0.19566 0,19437 0.19308 0.19180 0.19052 0.18926 0.18800 0.18675 0.18550 0.18427 0.18304 0.18181 0.18060 0.17939 0.17819 0.17700 0.17581 0.17463 0.17346 0.17230 0.17114 0,16999 0.16884 0.16771 0.16658 0.16545 0.16433 0.16322 0.16212 0.16102 0,15993 0.15885 0.15779 0.15670 0.15564 0.15458 0.15353 0.15249 0.15145 0.15042 0.14939 0.14837 0.14736 0.14635 0.14535 0.14436 0.14337 0.14239 0.14141 0,14044 0.13948 0.13852 0.13757 0.13662 0.13568 0.13475 0,13382 0.13289 0,13198 0.13107 0.13016 0,12926 0.12837 0.12748 0.12660 0.21572 0.21485 0.12398 0.12312 0.12226 0.12141 0.12057 0.11973 0.11889 0.11807 0.11724 0.11642 0.11561 0.11480 0.11400 0.11320 0.11241 0.11162 0.11084 0.11006 0.10929 0.10852 0.10774 0.10700 0.10625 0.10550 0.10476 0.10402 0.10329

2

cuis

X

Cuib

X

6.80 6.81 6.82 6.83 6.84 6.85 6.86 6.87 6.88 6.89 6.90 6.91 6.92 6.93 6.94 6.95 6.96 6.97 6.98 6.99 7.00 7.01 7.02 7.03 7.04 7.05 7.06 7.07 7.08 7.09 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29 7.30 7.31 7.32 7.33 7.34 7.35 7.36 7.37 7.38 7.39 7.40 7.41 7.42 7.43 7.44 7.45 7.46 7.47 7.48 7.49 7.50 7.51 7.52 7.53 7.54 7.55 7.56 7.57 7.58 7.59 7.60 7.61 7.62 7.63 7.64 7.65 7.66 7.67 7.68 7.69 7.70 7.71 7.72 7.73 7.74

0.10256 0.10184 0.10112 0.10040 0.09969 0.09829 0.09899

7.75 7.76 7.77 7.78 7.79 7.80 7.81 7.82 7.83 7.84 7.85 7.86 7.87 7.88 7.89 7.90 7.91 7.92 7.93 7.94 7.95 7.96 7.97 7.98 7.99 8.00 8.05 8.10 8.15 8.20 8.25 8.30 8.35 8.40 8.45 8.50 8.55 8.60 8.65 3.70 8.75 8.80 8.85 8.90 8.95 9.00 9.05 9.10 9.15 9.20 9.25 9.30 9.35 9.40 9.45 9.50 9.55 9.60 9.65 9.70 9.75 9.80 9.85 9.90 9.95 10.00 10.05 10.10 10.15 10.20 10.25 10.30 10.35 10.40 10.45 10.50 10.55 10.60 10.65 10.70 10.75 10.80 10.85 10.90 10.95 11.00 11.05 11.10 11.15 11.20 11.25 11.30 11.35 11.40 11.45

0.05145 0.05107 0.05032 0.05069

11.50 11.55 11.60 11.65 11.70 11.75 11.80 11.85 11.90 11.95 12.00 12.05 12.10 12.15 12.20 12.25 12.30 12.35 12.40 12.45 12.50 12.55 12.60 12.65 12.70 12.75 12.80 12.85 12.90 12.95 13.00 13.05 13.10 13.15 13.20 13.25 13.30 13.35 13.40 13.45 13.50 13 55 13.60 13.65 13.70 13.75 13.80 13.85 13.90 13.95 14.00 14.05 14.10 14.15 14.20 14.25 14.30 14.35 14.40 14.45 14.50 14.55 14.60 14.65 14.70 14.75 14.80 14.85 14.90 14.95 15.00

0.09759 0.09690 0.09621 0.09533 0.09485 0.09418 0.09351 0.09284 0.09218 0.09153 0.09088 0.09023 0.08958 0.08831 0.08895 0.08768 0.08705 0.08643 0.08581 0.08520 0.08459 0.OS398 0.08338 0 .OS278 0.08219 0.08160 0.08101 0.08043 0.07985 0.07928 0.07871 0,07814 0.07758 0.07702 0.07646 0.07591 0.07536 0.07482 0.07428 0.07374 0.07321 0.07268 0.07215 0.07164 0.07111 0.07059 0.07008 0.06957 0.06907 0.06857 0.06807 0.06757 0.06708 0.06659 0.06611 0.06562 0.06467 0.06515 0.06420 0.06373 0.06326 0.06280 0.06234 0.06189 0.06143 0.06098 0.06054 0.06009 0.05965 0.05921 0.05878 0.05835 0.05792 0.05749 0.05707 0.05665 0.05623 0.05582 0.05541 0.05500 0.05459 0.05419 0.05379 0.05339 0,05300 0.05261 0.05222 0.05183

0,04994 0.04967 0.04921 0,04884 0.04848 0.04812 0.04776 0.04741 0.04670 0.04705 0.04635 0.04601 0,04567 0.04533 0.04499 0.04465 0.04432 0.04399 0.04366 0.04333 0.04301 0.04269 0.04111 0.03959 0.03813 0.03672 0.03535 0.03404 0.03277 0.03154 0.03036 0.02922 0,02812 0.02706 0.02605 0,02506 0.02411 0.02320 0.02232 0.02147 0.02065 0.01987 0.01911 0.01838 0.01767 0,01700 0.01634 0.01572 0.01511 0.01453 0.01396 0.01342 0.01291 0.01240 0.01192 0.01146 0.01101 0.01058 0.01017 0.00977 0.00939 0,00902 0.00867 0.00833 0.00800 0.00768 0.00738 0.00709 0.00681 0.00654 0.00628 0.00603 0.00579 0.00556 0.00534 0.00513 0.00492 0.00473 0.00454 0.00436 0.00418 0.00402 0.00385 0,00370 0.00355 0.00341 0.00301 0.00289 0.00277

0.00143 0.00137 0.00131 0.00126 0.00121 0.00116 0.00111 0.00106 0.00102 0* 00098 0.00094 0.00090 0.00086 0.00083 0.00079 0.00076 0.00073 0.00070 0.00067 0.00064 0.00061 0.00059 0.00056 0.00052 0.00054 0.00050 0.00047 0,00045 0.00044 0.00042 0.00040 0.00038 0.00037 0.00035 0,00034 0.00032 0.00031 0.00030 0.00028 0.00027 0.00026 0.00025 0.00024 0.00023 0.00022 0.00021 0.00020 0.00019 0.00018 0.00018 0.00017 0.00016 0.00015 0.00015 0.00014 0.00014

INDUSTRIAL AND ENGINEERING CHEMISTRY

642 Table 11. T o K. 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1300 1400 1500

C-H(A1) Y = 2914 0 = 4182 0.0000 0,0004 0.0018 0.0063 0.0159 0,0324 0.0907 0.1812 0,2940 0.4187 0,5474 0.6713 0.7868 0.8932 0.9907 1.0786 6 = 1247

0 = 1790

The value of R was taken equal to 1.987 calories.

Contribution of Each Bonding Frequency to Heat Capacityn C-C(A1) C=C(AI) (Sy) C=C(Al)(Un) Y = 989 Y = 1618 Y = 1064 a = 1419 e = 2321 e = 2387 0,2214 0.0161 0.0129 0.3991 0.0520 0,0442 0,5866 0.1156 0,1009 0,2040 0.7634 0.1827 0,9206 0.3076 0.2810 1.0569 0.4210 0.3899 0.6126 1.2721 0.6489 0.8169 1.4268 0,8548 0,9942 1.5383 1.0290 1,1415 1.6218 1,1732 1,6848 1.2620 1.2923 1.7327 1.3623 1.3894 1.7710 1,4447 1.4680 1.8007 1.5120 1,5335 1.8249 1,5683 1.5881 1,8441 1.6339 1.6162 6 = 390

a

= 560

6 = 421 B = 604

6 = 599

0 = 860

1.2491 250 1.3296 0 . 8044 0.0793 1.4980 1.4336 300 0.1826 1 0443 1,5593 0.3160 1 2298 1.6120 350 1,6920 1.6490 0.4638 I 3710 400 1.7496 1,7140 0.6109 1,4786 450 0,7514 1 5614 1.7916 1.7624 500 0.9937 1 6793 1.8272 1.8481 600 1.1833 1,7553 1.8686 1.8844 700 1.3305 1 8060 1.8957 1,9080 800 1.8415 1.8162 1,9238 900 1.4438 1,8691 1.9272 1000 1.5313 1.9354 1.8914 1.9375 1,9446 1100 1.6001 1.9045 1.9456 1,9510 1200 1.6566 1.9161 1.9514 1300 1.7011 1.9565 1,9283 1400 1.7371 1.9564 1.9610 1.9606 1.9331 1500 1.7674 1.9650 Naphthenic ring hondings were treated as aromatic.

C=C(Al) Y =

e

1115

= 1600

0.1260 0.2765 0.4388 0.6040 n ,760ti 0.9010 1.1338 1,3084 1.4390 1.8365 1.6120 1.6707 1.7170 1,7542 1.7841 1.8084 8 = 470 B = 675

1.1190 1.3280 1 ,4702

1.5750 1.6530 1.7107 1.7900 1.8388 1.8738 1.8.967 1.9134 1.9257 1.9350 1.9428 1.9487 1,9536

shortly above 700" K. the Dobratz quadratic equations begin to fall off rapidly and become impossible above 800' K. EXTENSIOIV O F TEMPERATURE RANGE

It is desirable to make heat capacity calculations over wider temperature ranges than those of the above authors. For such a program the writers coupled the rotational modification suggested by Dobratz with the Bennewitz and Rossner method and retained the use of Einstein functions. This improves the low-temperature precision and extends the upper temperature range. For reasons stated earlier, Table I, giving solutions to the Einstein function corresponding to one degree of freedom, was calculated from the table of Sherman and Emell (25).

C-H Y

(Ar)

= 3045

0 = 4370

C-C(.4r) Y = 989 B = 1419 0.2214 0.3991 0.6866 0.7634 0,9206 1.0869 1.2721 1.4268 1.5383 1.6218 1.6848 1,7327 1.7710 1.8007 1.8249 1,8441

C=C(Ar) Y = 1618 a = 2321 0.0161 0,0520 0.1156 0.2040 0.3076 0,4210 0.6489 0.8548 1.0290 1.1732 1.2923 1.3894 1.4680 1.5335 1.5881 1.6339

FREQUENCY ASSIGNMENT

The frequency assignments of Bennewitz and Rossner were made on the basis of experimental data at one temperature (410"K.). Since their work, more extensive experimental data have appeared, and it was considered desirable to adjust the frequency assignment in order to repro6 = 1318 6 = 390 6 = 844 e = 1891 B 560 e = 1211 duce the experimental data 0 0591 1.3296 0.3735 with better precision. In 0,5922 0.1460 1,4980 0.2640 1.6120 0.7966 this re-evaluation study it 0.9740 0.3996 1.6920 was felt that the 6 fre0.5420 1.7496 1.1235 0.6790 1.7916 1.2468 quency for the aliphatic 0,9220 1.8481 1.4305 1,8844 C-H bond was too high 1.1190 1.5571 1,2735 1.9080 1,6474 and that the aliphatic C=C 1.9238 1,3940 1.7126 1.4870 1.7613 1.9354 bond was inade1,5623 1.9446 1.7977 quately treated. The 1,6225 1.9610 1.8263 1.6711 1.9565 1.8483 rest of the f r e q u e n c i e s 1.7107 1.9610 1.8678 1.9650 1.8825 1.7438 adolsted are substantiallv the-same as those of Bennewitz and Rossner. The heat capacities of methane and ethane have received careful individual treatment (12, I S , 27) and are known to have eccentricities not common to the other members of the series. For this reason the assignment of the aliphatic C-H 6 frequency was based on the experimental data for propane ( 1 6 ) . Bennewitz and Rossner assumed the C=C bond to be the same wherever encountered, but in this study it n-as found necessary to consider three types of C=C: a symmetrical aliphatic C=C, an unsymmetrical aliphatic C=C, and an aromatic C=C. The symmetrical aliphatic C=C v frequency adopted is the same as that of Bennewitz and Rossner; the 6 frequency was reevaluated on the basis of experimental data on ethylene ( I d , 27), and 2-butene (17'). The unsymmetrical aliphatic C=C Y and 6 frequencies were based on propylene ( 1 6 , 2 6 ) . 0.0000 0.0002 0,0012 0.0043 0.0111 0 0244 0.0728 0,1510 0.2544 0.3716 0.4927 0,6136 0,7292 0,8360 0,9346 1.0238

~

T o K. hIethane Acetylene Ethylene Ethane hlethylacetylene Allene Propylene Cyclopropane Propane Dimethylacetylene Butadiene 1-Butene 2-Butene Butane Isoprene Pentylene Pentane Hexane Heptane Octane Cyclohexene Cyclohexane iYIethylcsdohexane Benzene Toluene Phenylacetylene Styrene Ethylbenzene Propylbenasne

250 8.34 9.35 9.53 11.74 12.97 12.20 13,76 14.40 15.32 16.65 15.88 17 41 16.72 18.96 19.57 21.08 22 60 28.27 29,95 33.63 21.64 23.75 27,22 16.76 20.56 21.90 22.72 24.25 27.94

300 8.86 9.91 10.48 13.05 14.35 13.43 15.43 16.30 17.42 18.84 17.92 19.88 19.29 21.86 22.42 24.3.5 26.30 30.78 35.26 39.74 25.71 28.22 32.34 19.91 24.56 25,84 27.06 29.06 33.55

~~

Table 111.

JIolar Heat Capacities (Cg) of Hydrocarbon Gases

350 9.53 10.49 11.56 14.58 15.76 14.73 17.24 18.27 19.75 21.08 20.05 22.52 22.02 25.02 25.37 27.82 30.28 35.58 40 89 46.19 29.93 32.52 37.60 23.18 28.67 30.00 31.52 34.00 39.32

450 11.07 11.61 13.81 17.70 18.47 17.38 20.93 22.11 24,48 25.37 24.27 27.79 27.45 31.34 31.17 34.67 38.18 45.06 51.94 58.82 38.08 42.00 47.72 29.41 36 31 37.39 39.88 43.41 50.31

400 10.29 11.07 12.69 16.14 17.15 16.07 19.11 20.23 22.15 23.29 22.20 25.20 24.79 28,24 28,33 31.31 34.32 40.43 46.55 52.67 34. OH 37 49 42 78 26 37 32.70 33.83 35.82 38.84 44.97

500 11.83 12.13 14.89 19.20 19.69 18.65 22.68 23.86 26.70 27.30 26.25 30.24 29.96 34.27 33.86 37.83 41.82 49.41 56.99 64.59 41.78 46.08 52.30

600 13.28 13.04 16.87 21.92 21.85 20.98 25.83 26.97 30.67 30.69 29.81 34.63 34.44 39.49 38.65 43.46 48.27 57.10 65.93 74.76 48.29 53.38 60.32 3 2 . 2 1 37.10 40.03 46.16 40.63 46.18 43.64 50.17 47.64 55.01 55.25 63.85

Vol. 35, No. 6

700 14.59 13.82 18.51 24.30 23.66 23.01 28.55 29.62 34.09 33.54 32,87 38.40 38.26 43.95 42.75 48.24 53.76 63.62 73.4s 83,34 53.75 59.64 67.07 41.16 51.17 50.16 55.55 62.11 70.99

800 15.78 14.51 19.96 26.38 25.22 24,78 30,92 31.91 37.05 35.97 35.51 41.63 41.52 47,79 46,25 52.35 58,46 69.20 79.92 90.65 58.37 64.78 72.77 44,54 55.50 54.41 60.09 66.24 76.97

900 1000 1100 16.84 17.79 18.63 15.10 15.63 16.09 21.45 22.62 23.63 28.21 29.82 31.21 27.71 26.55 28.71 27.63 26.32 28,77 32.97 34.75 36.29 35.60 37.10 33.89 4 1 . 8 7 43.80 39.63 38.04 39.83 41.37 37.78 3 9 . 7 3 41.41 48.91 44.27 46.83 46.77 48.87 44.34 51.10 53.98 56.47 49.26 51.84 64.06 55.72 58.92 61.55 6 2 , 5 2 66.02 69.05 7 3 , 9 9 78.13 81.70 85.45 9 0 . 2 3 94.34 107.0 96.92 102.3 65.63 68.54 62.30 76.40 73.06 69.26 8 1 . 7 7 85.38 77.63 51,80 49.75 47.37 64.65 69.06 62.08 60.04 62.21 57.45 69.77 67.06 63.86 7 0 , 5 4 7 4 , 1 8 77.30 82.01 86.28 89,94

~~~~

1200 19.38 16.49 24.51 32.43 29.58 29.76 37.63 38.42 45.50 42.70 42.87 50.73 50 69 58.64 55.98 63.83 71.69 84.81 97.91 111.0 71.04 79.28 88.49 53.54 66.86 64.06 72,08 79.99 93.09

1300 1400 1500 20.03 20.60 21.10 16.83 17.14 17.40 25.92 26.50 25.26 34.42 35,22 33.50 30.32 30.97 31,53 31.34 31.98 30.61 39.79 40.66 38.79 40.56 41.43 39.56 46.96 48,24 49.35 45.72 43.85 44.85 44.11 45.19 46,14 52.28 53,03 54.81 54.79 52 25 53.60 62.15 63.54 60.50 59.06 60.30 57.63 67.48 68.97 65.78 73.95 75.92 77.64 89.79 91.82 87.47 103.6 101 .o 106.0 117.5 114.5 120.1 73.19 75.07 76,70 83.94 85.83 81.77 93.51 91.17 95.54 55.03 56.32 57.43 70 37 71.78 68.74 65.63 66.98 68.15 75.78 77.22 74.06 82.26 84.24 85 95 95.76 98.09 1 0 0 , l

643

INDUSTRIAL AND ENGINEERING CHEMISTRY

June, 1943

d 2

For the aromatic C=C v and 6 frequencies the values of Bennewitz and Rossner were retained. The calculation of acetylenic derivatives was made possible by employing experimental data on methylacetylene (16) and dimethylacetylene (17) to evaluate the v and 6 frequencies for the C=C bond. Acetylene data (16) were ignored in this assignment in an effort to predict the higher acetylenes more reliably. The selected frequencies are listed in Table 11, and by the use of Table I the contribution of each bonding frequency to the heat capacity for every 100" from 250' to 1500' K. was computed and is presented in Table 11. The nature of the temperature functions of these bonding contributions of the heat capacity is easily seen in Figure 1.

c

'13

c

250

500

CALCULATIONS

750

1000

1250

1500

Temperature, OK.

5. C=C (Ar) 6 C=C (Al) (sum.)Y C C (All 6 The use of Table I in comlo' { C=C (Ar) C-C (AI) Y { C-C (Ar) 6 11. C=C (Al) (unsym.) Y piling Table I1 is illustrated by 2. C=C (AI) (unsym.)6 C C (Ar) Y 3 CEC (A1 6 7. C=C (AI) Y 12. C - H (AI) Y considering a vibrational frequency 4: c=c ($ (sum.) 6 8. C-H (Al) 6 13. C-H (Ar) Y of the C-H aliphatic bond. The 9. C H (Ar) 6 Bonding Frequency Contributions to Molar Heat Capacity as valence vibration, v , on the basis Figure 1. Function of Temperature of experimental work, has been assigned a frequency of 2914 cm.-l. Then 0 = 1.435 X 2914 = 4182. With the help of Equation 3 and Table I1 the gaseous heat 0 is known as the characteristic temperature of this vicapacities of a number of hydrocarbons were computed and brational frequency. For the desired values of temperature in ' K.. 8/T = 2: values are comouted. From 2: data are presented in Table 111. in Table 1,'values for Caib (contribution to the molar heat LITERATURE COMPARISON capacity of the given valence vibration) are interpolated and To evaluate the accuracy of this method, Table IV lists listed in Table 11. The procedure is the same for all other the values already published. Graphical interpolation of frequencies. Table I11 was used at the temperatures reported. Where The method of calculating gaseous heat capacities by values were reported for one atmosphere pressure, the correcEquation 3 is illustrated in the following calculation for tion t o zero pressure was determined from the well known propane: From the structural formula for propane, a = 2 , Berthelot equation of state (99). The difference between ~q = 10, n = 11, and 6 = 1.5. From these factors and our values and those of the literature averages *4 per cent. Table 11,we obtain: Exclusive of cvcloorooane, the maximum deviation is found to be - 13 percent inthe case of Pitzer's calculated value for Bond Frequency q 4 400' K. SOOO K. 1200° K. ,, , o,050 2,352 6,294 n-heptane and +8 per cent in the case of Beeck's experimental C H (AI) ; 0 - H (AI) 8 1.5 5.566 16.966 19.879 value for heptane. Y

C-C (AI) C-C (-41)

;

+4R

+R

2 2

... 1.5

1.527 5.076 9 935

3.077 5.724 9.930

3 542 5.853 9 935

22.154

37.054

45,503

- - C;

I n a similar manner the calculation of the gaseous heat capacity of phenyl acetylene, where a = 1, x g = 14, n = 14, qb = 1.5, is: Bond

C-C C-C C--H C-H C-C C-C CZEC CfC C-H C-H

{ti; (Ar)

(AI)

400e K ,

l"requeney 6 6

1.5

i:b ...

i::"e"g$ i:;:: 0.763 1.538

Y

1

(Al)

6

1

6

1 1

Y

1

Y

8

+4R 4-0.5R

4,404 8.218 5.313 8.780

4.383 2.290 7.614

Y

(AI) (AI) (Al) (AI)

12000 K.

3,087 7.413 4.615

i:b ...

Ar) (Ar) All

6

8 o o ~K.

3 3 3 3

1.5

2.538

,,

0.604

.

1.5

i:b c;

2.363

8.586

2.862 1.439 2.811

~:~~~ 8.942 7:::;f

8.942 33.829

54.406

1.771 2.926 1.717 2.902

:g

8.942 64,060

LIMITATIONS OF THE METHOD

The evaluation of the rotational and vibrational degrees of freedom is in question at temperatures below 300" K.; consequently this method is not ~ecommended for temperatures below 250" K. For molecules which have highly strained structures, such as cyclopropane, the method is questionable at lower temperatures. Above the range 1000' to 1100' K., electron interaction develops within some molecules and causes a gradual increase in the true heat capacity. For this reason, values calculated as above may tend to be increasingly low a t temperatures above 1100" K. This method is not recommended for methane, ethane, or acetylene for reasons mentioned earlier. This procedure fails to distinguish between many isomeric structures, such as normal and isobutane and cis- and trans-2-butene. To compute the heat capacities of these isomers with precision, detailed treatment of each isomer is necessary and becomes ~ quite involved.

644

INDUSTRIAL AND ENGINEERING CHEMISTRY

mmm

Vol. 35, No. 6

mmo

mmm

c

These computed heat capacity values a t zero pressure must be converted to values a t some finite pressure by well known thermodynamic equations (19) modified for gas imperfection by means of some equation of state, such as the Berthelot equation (29),residual volume equation (8), or compressibility factors (3, Q, 6). The magnitude of these

corrections to one atmosphere pressure can be observed in Table IV. NOMENCLATURE

a

= number of bonds (AI) = aliphatic bonding (Ar) = aromatic bonding

free rotation

June, 1943

INDUSTRIAL AND ENGINEERING CHEMISTRY = =

= =

=

= =

= = =

= = = =

= =

645

constant-volume molar heat capacity of a gas at zero pressure contribution, according to Einstein function, of vibrator vo to heat capacity, cal./(mole) (" C. or 'K.) deformation frequency, om.-' base of natural logarithms Planck constant gas constant per molecule number of atoms in molecule frequency of simple harmonic vibrator valence frequency, cm.-' (3n - 6 - - 2 9 ) / 2 9 valence bond in a molecule having n atoms total number of valence bonds of all kinds in a molecule having n atoms molar as constant 1.987 cal./ (mole? ( 0 C. or 0 d.1 symmetrical boonding tem erature, K. hv$ = 1.435~~ unsymmetrical bonding

= = -0 = -hvo

T

kT

ACKNOWLEDGMENT

The authors wish to acknowledge with appreciation the permission of The Dow Chemical Company to publish this work. LITERATURE CITED

wmt-dmmm

odmm

d-imm

000-0-0

0000

NNeVNmOO

I I I I I I I

I I I I

dl??????

""C!

1OoYN.?%? +++++I+

m

dh0

t-dwagqmi

idwmdmm

91N.N. .41

?9???"?"???

00000000000

0 0 0 0 0 0 ~

+111++1111+

+IIII++

(1) Beeck, J . Chem. Phys., 4, 680 (1936). (2) Bennewitz and Rossner, 2. physik. Chem., 39B, 126 (1938). (3) Brown, Lewis, and Weber, IND.ENG. CHEM.,26, 325 (1934). (4) Brown, Souders, and Smith, Ibid., 24, 513 (1932). (5) Burick, Eyster, and Yost, J . Chem. Phus., 9, 118 (1941).

(6) Cope, Lewis, and Weber, IND. ENQ.CHEM., 23, 887 (1931). (7) Dobratz, Ibid., 33, 759 (1941). (8) Edmister, Ibid., 30, 352 (1938). (9) Einstein. Ann. Phusik. 141 22. 180 (1907). (IO) Eucken and Lude, 2. physik: Chem... 5B., I

...

413 (1929).

(11) Euoken and Parts, Ibid., 20B, 184 (1933). Frost, Trans. Exptl. Research Lab. "Khenzgas", Materials on Cracking and Chemical Treatment of Cracking Products

(12)

.......... ......................

dmw-...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ....*mmm... .. .. .. .. .. .. .. . . . .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. :;