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Isotherms of methane-aethane mixtures at 0°, 25° and 50°C up to 60 atmospheres
Physica
Ii I
VI, no 7
Juli
1939
ISOTHERMS OF METHANE-AETHANE MIXTURES AT O”, 25” AND 50°C UP TO 60 ATMOSPHERES by A. MICHELS 64th
publication
and G. W. NEDERBRAGT
of the Van der Wmls-fund,
An&dam
5 1, The method. As in the pressure-temperature-range, where the mixtures were to be investigated, the occurrence of liquid-gas equilibrium could be expected, it was not advisable to use the method generally applied in the v. d. W a a 1 s-laboratory for the measurements of compressibility-isotherms of gases. Under these circumstances the apparatus, as described by M ich els and N e de rb r ag t ‘) for the measurement of dewpoints and modified later by M i c h e 1 s, B 1 a i s s e and M ic h e 1 s “) for the determination of 9-v-T relations in the neighbourhood of the critical pount, offers special advantages. It was essential however that the method should be altered so as to allow a synthesis of the mixtures to be carried out in situ. The principle of the compressibilitymeasurements can be understood from the diagrammatical scetch given in fig. 1. Three steel containers A, B and C are coupled with capillaries as shown in the drawing. A and B are placed in a thermostate. The gas under examination is contained under a glashood in A and the rest of the apparatus is filled with mercury as indicated. Pressure is applied on the mercury surface in C with compressed nitrogen form reservoir F, which passed through filter G and cooler H. This pressure is measured with a pressure-balance and the pressure of the gasmixture in A can then be obtained from the pressure of nitrogen by applying corrections for the position of the mercuryheads. The gasvolume can be known from the amount of mercury in C after correction for small volumechanges of the reservoirs resulting from pressure- and temperatureexpansions. For the determination of the quantity of mercury in C this bomb is placed on the scale of a -
656 -
ISOTHERMS
OF
METHANE-AETHANE
MlXTURES
65.7
zbalance; the steel capillaries coupled to C, being choosen so flexible, that the sensitivity of the balance reading is within 0.2 g. on a total load of 15 kg.
Fig.
1.
Reservoir B has no function during the measurements of the compressibility and is only added to the apparatus to allow the synthesis of the mixture to be made. In principle this is affected in the following way. With valve 3 shut and 1 and 2 opened CH., is allowed in A and the quantity of CH4 is determined from the increase in weight of C and the nitrogenpressure, after equilibrium has been. .reached. In a similar way with valves 1 and 2 shut and 3 opened an amount of C,H, is filled into B and measured. This quantity is then transferred from B to the glashood in A by increasing the N,-pressure in C and opening valve 2. The calibrations were carried out with the same amount of mercury, as used during the measurements. For the filling with mercury C is opened and the mercury poured in. After closing; the three reservoirs are evacuated simultaneously, whereupon nitrogen is allowed in bomb C, thus forcing the mercury to fill the other reservoirs completely. During the measurements only the pressure of N, in C and the weight of this reservoir can directly be measured. From these data the volume and the pressure of the gas in B, during the synthesis of the mixtures, and of the mixture in A, during the measurements of the compressibilities, must be determined. Hence it is advisable to express all corrections in terms of the weight of the reservoir C. 3 2. The calibrations.
Physica
VI
42
658
A. MICHEJ++ND
G. Wt. NEDERBRAGT
The pressure corrections. ~sthepressurebf~h~~~~ is equal to the pressure of the nitrogen in C, corrected for the’difference in mercury heads between C and B resp. C and A, this difference must be measured as a function of the weight of C. This is carried out by connecting C to one side and B resp. A to the other side of the differential mercury levelling gauge H and reading the pressure differences from G at different positions of the mercury levels in A, B and C. The volume of the gas in A (resp. B). Thevolume of the gas in A can be found directly from the difference in weight of C when A and B are completely filled with mercury and the weight under working conditions. The calibration thus involves determination of the weight of C as function of pressure at different temperatures of the reservoirs A and B, while these are completely filled with mercury. It must be noted that from the weight of C must be subtracted the weight of the nitrogen, filling the space above the mercury. For more detailed description of the apparatus and technique of the measurements may be referred to the previous publications. 5 3. T/W gases. Methane was prepared from a natural gas by low temperature rectification. The boyling point of the fraction used was constant within OO.03. Part of this fraction was used for the determination of the compressibility-isotherms of pure CH4, as described ,by Michels and Nederbragtg). Aethane was obtained by electrolysis of potassium acetate and was purified by passing through concentrated potassium hydroxyde and sulfuric acid, followed by low temperature fractionation. 5 4. The measuremen.ts. Measurements were carried out ivith 4 mixtures containing respectively 2040-60-80 molO/; laethane. The first. three mixtures were composed by adding successive quantities of aethane to the aethane-methane mixture, whereas the synthesis of the last mixture was made with fresh quantities of both gases. The results of the measurements are given in tables Ia, b, c, d.
ISOTHERMS
OF
METHANEdET&ANE
TABLE
TABLE Ia M i fraction methane M -LOIfraction aethane
0.8009 0.194 - 11
p in atm.
d in mol/f.
9.980 19.997 29.924 40.058 49 993 59.896
.4620 .9614 1.4978 2.0952 2.7372 3.4394
.9639 .9278 .a913 .a530 .014a .7769
25.35’C
10.012 20.017 29.835 40.059 50.001 59.901
.4199 .a631 1.3218 1.8274 2.3474 2.8944
.9735 .9468 .9214 .a950 .8696 .e449
49.84oc
lo.028 20.004 29.944 40.069 49.911 59.903
.386 1 .7a57 1.1998 1.6385 2.0806 2.5465
.9aoo .9605 .9416 .9226 .905 1 .8a75
0°C
-
TABLE Molfraction Molfraction p in atm. 0°C
!5.35”C
19.84”C
-
r
Molfraction Molfraction
I 0°C
13.774 20.057 29.969 40.087 49.926 59.907
.5930 .a854 1.3796 1.9322 2.5244 3.1869
.94a3 .924a .a869 .a471 .a074 .7674
49.84oc
13.812 20.028 29.980 40.099 49.936 59.926 ---
S424 .8013 1.236 1 1.7067 2.1934 2.7187
.9608 .9431 .9150 .8865 .a590 .a317
.860 1 .7772 .7071 .6779 .6448 .5967 .5764 .53a3 .5382 .4ao I .429 1 .390 1 .3474 .323a .3191 .315a .3116 .3lOO .3076 .3070 .3oa4 .3121
20.100 30.027 4Q.024 49.946 59.936
.9144 1.4544 2.0846 2.8278 3.7430
.a974 .a429 .783a .7211 .6537
PO.066 ‘30.064 40.059 49.979 59,960.
.a2lG 1.2855 1.796 1 2.3563 2.9839
.9222 .a924 .a416 .a003 .7502
Table Yaction :raction p in atm.
Id
methane aethane
. -I
0.1996 0.8004
d in mol/l.
PWT
20.059 23.312 26.829 29.974 32.538 33.125 36.331 39.929 43.052 45.320 46.151 46.179 47.162 48.085 49.079 49.98 I 54.247 56.644 59.892
1,lOOl 1.3407 1.6394 1.9568 2.5208 2.7562 4.0806 5.975 1 a. I 300 10.138 11.012 11.023 12.379 12.505 12.584 12.636 12.855 12.956 13.079
.8135 .7757 .7301 .6a35 s740 .5362 .3972 .2981 .2363 .1994 .la70 .1869 .I700 .1716 .1740 .1765 .1aa3 .I950 .2043
!5.35”C
20.115 30.026 40.021 49.038 59.915
.9483 1.5548 2.3452 3.3713 5.7694
.a660 .7805 .6967 s939 .4204
k9.84oc
20. i4a 30.063 40.051 SO.237 59.948
.a454 1.3423 1.9215 2.6253 3.4450
.a993 .a450 .7065 .7220 .6566
PvlRT
1.037 1 1.7227 2.3541 2.6328 2.9661 3.4838 3.7187 4.1372 4.1382 4.9605 5.9 138 6.8515 a. 1556 9.1363 9.3822 9.6011 9.9296 10.101 10.429 10.714 11.160 1 I.552
PvlRT
25.35’C
M M
19.994 30.011 37.307 40.009 42.868 46.592 48.041 49.920 49.924 53.375 56.ea3 59.904 63.492 66.300 67.101 67.958 49.362 70.193 71.917 73.733 77.158 80.808
I
.9300 .a975 .84?2 .7820 .7193 6515
I
d in mol/l.
.6583 .9933 1Sa66 2.286 I 3.0951 4.1012
0.4006 0.5994
d in mol/l.
0.60 10 G.39! PO
13.734 19.981 29.947 40.068 49.897 59.890
Ic
methane aethane
Ib
methane aethane
PvIRT
1
659
MIXTURES
OY
660
A. MICHELSlAND
G, W. NEDERBRAGT TABLE
IIa
~
T
-~ !5”C
-~ jO°C
loo ii 40 50 60
.4572
1. .9759
.4629
1. .9633
.4695
1.4413 .9372 1.9726 2.5306 3.1159
.9520 .9233 .9047 .a315 .a591
1.5022 .9617 2.0917 2.7373 3.3 136
.9273 .a910 .a532 .a143 .7765
I .9943 .5393 2.2309 3.1043 4.1132
0 :: 30 40 SO 60
1. .a470 .4161 1.2932 1.7546 2.23 14 2.7225
1.
.9a24 1 .965 .9432 .9313 .9159 .9003
0
.a635 .4199 1.3313 1.a276 2.3513 2.906 1
1.
.9735 .9467 .9207 8946 .a690 .a439
1. .9502 .?3974 .a419 .7324 .7136 .6503 1.
.a340 .4246 I .3337 1.9310 2.5353 3.2025
1.
.9626 .9247 .3362 .a467 .306 1 .7653 1.
. I .<
II~~~T~~~~~~~
o-c
-~ ?5-c
-~ 50-C
0 10 32:
1.7219 1.0374
1. .7773 .3601
1.9594 1.0959
.6831 .a142
40 50 60
2.6322 4.1556 6.3833
.6730 .5368 .3339
6.0209 12.646 13.122
.2964 .I766 .2046
0 :: 30 40 50 60
1. .9103 1.4556 2.0333 2.3414 3.7666
0 10 3”: 4rJ 50 60
1.
.a975 .a424 .7329 .7192 .6511 j
1.2314 .a175 1.7913 2.3552,. 2.9323
1. .9435 t.5567 2.35 13 3.5426 5.9010
1. .a829 .9226 .a421 .a006 .7536
.5005 1.1364
.a664 .7377 .6952 .5769 .4156
1. .9393 .4447 1.7227 3.03 17
.9191 .8263 .7113 .5305
4020
1. .9331
1.4203 .366 1 2.1222 3.1079 4.8531
.a703 .7963 .71oa .6C67 .4662
1. I .a379 .3379 1.9163 2.6037 3.4435
:9002 .e456 .7372 .7242 .657 1
1. .a914 .7521
ISOTHERMS
OF
METHANE-AETHANE
I 0.2
Ol CH,
I 0.6
I 0.4
66
MIXTURES
I 0.8
I C&
Fig. 2.
0
I
I 20
I 40 Fig. 3.
I 60
-
Pm
662
ISOTHERMS
OF METHANE-AETHANE
MIXTURES
To facillitate comparison, the values of fw/RT have been calculated for integral temperatures and pressures and are given in table IIa, b, together with the values for methane and aethane. The data for methane have been taken from data published by M i c h e 1 s and N e d e r b r a g t “) and for aethane from unpublished measurements carried out by M i c h e 1 s and G e r v e r. In fig. 2 and 3 the @/RT values at 0°C are plotted against the pressure and the molconcentration CzH6. In fig. 4 the pressureconcentration diagram of the mixtures at 0°C is plotted.
Fig.
3.
In connection with some discussion lately arisen in the litterature, it is of some interest to draw here special attention to the precautions essential for obtaining equilibrium after pressure changes. In presence of only one phase. the time required to reach equilibrium after expansion or compression was only 15 minutes, whereas in the coexistance-region at least a couple of hours is required. This confirmed the results obtained during the measurements with COa. Here, in the close neighbourhood of the critical point even time intervals of 12 to 18 hours were necessary, although the crossection of the glashood was 30 mm and the height of the gascolumn in this case only about 80 mm. Received
May
26th
1939. REFERENCES
1) A. Miche
andG.W. B. Blaisse andG. W.
Nederbragt, andC. Nederbrachf
Michels,
Ind.Fng.Chem.6, 135, 1934. Proc.roy.Soc.A.100,348,1937. Pbysica,‘s-Grav.2, 1301, 1935;3*
Ii I
VI, no 7
Juli
1939
ISOTHERMS OF METHANE-AETHANE MIXTURES AT O”, 25” AND 50°C UP TO 60 ATMOSPHERES by A. MICHELS 64th
publication
and G. W. NEDERBRAGT
of the Van der Wmls-fund,
An&dam
5 1, The method. As in the pressure-temperature-range, where the mixtures were to be investigated, the occurrence of liquid-gas equilibrium could be expected, it was not advisable to use the method generally applied in the v. d. W a a 1 s-laboratory for the measurements of compressibility-isotherms of gases. Under these circumstances the apparatus, as described by M ich els and N e de rb r ag t ‘) for the measurement of dewpoints and modified later by M i c h e 1 s, B 1 a i s s e and M ic h e 1 s “) for the determination of 9-v-T relations in the neighbourhood of the critical pount, offers special advantages. It was essential however that the method should be altered so as to allow a synthesis of the mixtures to be carried out in situ. The principle of the compressibilitymeasurements can be understood from the diagrammatical scetch given in fig. 1. Three steel containers A, B and C are coupled with capillaries as shown in the drawing. A and B are placed in a thermostate. The gas under examination is contained under a glashood in A and the rest of the apparatus is filled with mercury as indicated. Pressure is applied on the mercury surface in C with compressed nitrogen form reservoir F, which passed through filter G and cooler H. This pressure is measured with a pressure-balance and the pressure of the gasmixture in A can then be obtained from the pressure of nitrogen by applying corrections for the position of the mercuryheads. The gasvolume can be known from the amount of mercury in C after correction for small volumechanges of the reservoirs resulting from pressure- and temperatureexpansions. For the determination of the quantity of mercury in C this bomb is placed on the scale of a -
656 -
ISOTHERMS
OF
METHANE-AETHANE
MlXTURES
65.7
zbalance; the steel capillaries coupled to C, being choosen so flexible, that the sensitivity of the balance reading is within 0.2 g. on a total load of 15 kg.
Fig.
1.
Reservoir B has no function during the measurements of the compressibility and is only added to the apparatus to allow the synthesis of the mixture to be made. In principle this is affected in the following way. With valve 3 shut and 1 and 2 opened CH., is allowed in A and the quantity of CH4 is determined from the increase in weight of C and the nitrogenpressure, after equilibrium has been. .reached. In a similar way with valves 1 and 2 shut and 3 opened an amount of C,H, is filled into B and measured. This quantity is then transferred from B to the glashood in A by increasing the N,-pressure in C and opening valve 2. The calibrations were carried out with the same amount of mercury, as used during the measurements. For the filling with mercury C is opened and the mercury poured in. After closing; the three reservoirs are evacuated simultaneously, whereupon nitrogen is allowed in bomb C, thus forcing the mercury to fill the other reservoirs completely. During the measurements only the pressure of N, in C and the weight of this reservoir can directly be measured. From these data the volume and the pressure of the gas in B, during the synthesis of the mixtures, and of the mixture in A, during the measurements of the compressibilities, must be determined. Hence it is advisable to express all corrections in terms of the weight of the reservoir C. 3 2. The calibrations.
Physica
VI
42
658
A. MICHEJ++ND
G. Wt. NEDERBRAGT
The pressure corrections. ~sthepressurebf~h~~~~ is equal to the pressure of the nitrogen in C, corrected for the’difference in mercury heads between C and B resp. C and A, this difference must be measured as a function of the weight of C. This is carried out by connecting C to one side and B resp. A to the other side of the differential mercury levelling gauge H and reading the pressure differences from G at different positions of the mercury levels in A, B and C. The volume of the gas in A (resp. B). Thevolume of the gas in A can be found directly from the difference in weight of C when A and B are completely filled with mercury and the weight under working conditions. The calibration thus involves determination of the weight of C as function of pressure at different temperatures of the reservoirs A and B, while these are completely filled with mercury. It must be noted that from the weight of C must be subtracted the weight of the nitrogen, filling the space above the mercury. For more detailed description of the apparatus and technique of the measurements may be referred to the previous publications. 5 3. T/W gases. Methane was prepared from a natural gas by low temperature rectification. The boyling point of the fraction used was constant within OO.03. Part of this fraction was used for the determination of the compressibility-isotherms of pure CH4, as described ,by Michels and Nederbragtg). Aethane was obtained by electrolysis of potassium acetate and was purified by passing through concentrated potassium hydroxyde and sulfuric acid, followed by low temperature fractionation. 5 4. The measuremen.ts. Measurements were carried out ivith 4 mixtures containing respectively 2040-60-80 molO/; laethane. The first. three mixtures were composed by adding successive quantities of aethane to the aethane-methane mixture, whereas the synthesis of the last mixture was made with fresh quantities of both gases. The results of the measurements are given in tables Ia, b, c, d.
ISOTHERMS
OF
METHANEdET&ANE
TABLE
TABLE Ia M i fraction methane M -LOIfraction aethane
0.8009 0.194 - 11
p in atm.
d in mol/f.
9.980 19.997 29.924 40.058 49 993 59.896
.4620 .9614 1.4978 2.0952 2.7372 3.4394
.9639 .9278 .a913 .a530 .014a .7769
25.35’C
10.012 20.017 29.835 40.059 50.001 59.901
.4199 .a631 1.3218 1.8274 2.3474 2.8944
.9735 .9468 .9214 .a950 .8696 .e449
49.84oc
lo.028 20.004 29.944 40.069 49.911 59.903
.386 1 .7a57 1.1998 1.6385 2.0806 2.5465
.9aoo .9605 .9416 .9226 .905 1 .8a75
0°C
-
TABLE Molfraction Molfraction p in atm. 0°C
!5.35”C
19.84”C
-
r
Molfraction Molfraction
I 0°C
13.774 20.057 29.969 40.087 49.926 59.907
.5930 .a854 1.3796 1.9322 2.5244 3.1869
.94a3 .924a .a869 .a471 .a074 .7674
49.84oc
13.812 20.028 29.980 40.099 49.936 59.926 ---
S424 .8013 1.236 1 1.7067 2.1934 2.7187
.9608 .9431 .9150 .8865 .a590 .a317
.860 1 .7772 .7071 .6779 .6448 .5967 .5764 .53a3 .5382 .4ao I .429 1 .390 1 .3474 .323a .3191 .315a .3116 .3lOO .3076 .3070 .3oa4 .3121
20.100 30.027 4Q.024 49.946 59.936
.9144 1.4544 2.0846 2.8278 3.7430
.a974 .a429 .783a .7211 .6537
PO.066 ‘30.064 40.059 49.979 59,960.
.a2lG 1.2855 1.796 1 2.3563 2.9839
.9222 .a924 .a416 .a003 .7502
Table Yaction :raction p in atm.
Id
methane aethane
. -I
0.1996 0.8004
d in mol/l.
PWT
20.059 23.312 26.829 29.974 32.538 33.125 36.331 39.929 43.052 45.320 46.151 46.179 47.162 48.085 49.079 49.98 I 54.247 56.644 59.892
1,lOOl 1.3407 1.6394 1.9568 2.5208 2.7562 4.0806 5.975 1 a. I 300 10.138 11.012 11.023 12.379 12.505 12.584 12.636 12.855 12.956 13.079
.8135 .7757 .7301 .6a35 s740 .5362 .3972 .2981 .2363 .1994 .la70 .1869 .I700 .1716 .1740 .1765 .1aa3 .I950 .2043
!5.35”C
20.115 30.026 40.021 49.038 59.915
.9483 1.5548 2.3452 3.3713 5.7694
.a660 .7805 .6967 s939 .4204
k9.84oc
20. i4a 30.063 40.051 SO.237 59.948
.a454 1.3423 1.9215 2.6253 3.4450
.a993 .a450 .7065 .7220 .6566
PvlRT
1.037 1 1.7227 2.3541 2.6328 2.9661 3.4838 3.7187 4.1372 4.1382 4.9605 5.9 138 6.8515 a. 1556 9.1363 9.3822 9.6011 9.9296 10.101 10.429 10.714 11.160 1 I.552
PvlRT
25.35’C
M M
19.994 30.011 37.307 40.009 42.868 46.592 48.041 49.920 49.924 53.375 56.ea3 59.904 63.492 66.300 67.101 67.958 49.362 70.193 71.917 73.733 77.158 80.808
I
.9300 .a975 .84?2 .7820 .7193 6515
I
d in mol/l.
.6583 .9933 1Sa66 2.286 I 3.0951 4.1012
0.4006 0.5994
d in mol/l.
0.60 10 G.39! PO
13.734 19.981 29.947 40.068 49.897 59.890
Ic
methane aethane
Ib
methane aethane
PvIRT
1
659
MIXTURES
OY
660
A. MICHELSlAND
G, W. NEDERBRAGT TABLE
IIa
~
T
-~ !5”C
-~ jO°C
loo ii 40 50 60
.4572
1. .9759
.4629
1. .9633
.4695
1.4413 .9372 1.9726 2.5306 3.1159
.9520 .9233 .9047 .a315 .a591
1.5022 .9617 2.0917 2.7373 3.3 136
.9273 .a910 .a532 .a143 .7765
I .9943 .5393 2.2309 3.1043 4.1132
0 :: 30 40 SO 60
1. .a470 .4161 1.2932 1.7546 2.23 14 2.7225
1.
.9a24 1 .965 .9432 .9313 .9159 .9003
0
.a635 .4199 1.3313 1.a276 2.3513 2.906 1
1.
.9735 .9467 .9207 8946 .a690 .a439
1. .9502 .?3974 .a419 .7324 .7136 .6503 1.
.a340 .4246 I .3337 1.9310 2.5353 3.2025
1.
.9626 .9247 .3362 .a467 .306 1 .7653 1.
. I .<
II~~~T~~~~~~~
o-c
-~ ?5-c
-~ 50-C
0 10 32:
1.7219 1.0374
1. .7773 .3601
1.9594 1.0959
.6831 .a142
40 50 60
2.6322 4.1556 6.3833
.6730 .5368 .3339
6.0209 12.646 13.122
.2964 .I766 .2046
0 :: 30 40 50 60
1. .9103 1.4556 2.0333 2.3414 3.7666
0 10 3”: 4rJ 50 60
1.
.a975 .a424 .7329 .7192 .6511 j
1.2314 .a175 1.7913 2.3552,. 2.9323
1. .9435 t.5567 2.35 13 3.5426 5.9010
1. .a829 .9226 .a421 .a006 .7536
.5005 1.1364
.a664 .7377 .6952 .5769 .4156
1. .9393 .4447 1.7227 3.03 17
.9191 .8263 .7113 .5305
4020
1. .9331
1.4203 .366 1 2.1222 3.1079 4.8531
.a703 .7963 .71oa .6C67 .4662
1. I .a379 .3379 1.9163 2.6037 3.4435
:9002 .e456 .7372 .7242 .657 1
1. .a914 .7521
ISOTHERMS
OF
METHANE-AETHANE
I 0.2
Ol CH,
I 0.6
I 0.4
66
MIXTURES
I 0.8
I C&
Fig. 2.
0
I
I 20
I 40 Fig. 3.
I 60
-
Pm
662
ISOTHERMS
OF METHANE-AETHANE
MIXTURES
To facillitate comparison, the values of fw/RT have been calculated for integral temperatures and pressures and are given in table IIa, b, together with the values for methane and aethane. The data for methane have been taken from data published by M i c h e 1 s and N e d e r b r a g t “) and for aethane from unpublished measurements carried out by M i c h e 1 s and G e r v e r. In fig. 2 and 3 the @/RT values at 0°C are plotted against the pressure and the molconcentration CzH6. In fig. 4 the pressureconcentration diagram of the mixtures at 0°C is plotted.
Fig.
3.
In connection with some discussion lately arisen in the litterature, it is of some interest to draw here special attention to the precautions essential for obtaining equilibrium after pressure changes. In presence of only one phase. the time required to reach equilibrium after expansion or compression was only 15 minutes, whereas in the coexistance-region at least a couple of hours is required. This confirmed the results obtained during the measurements with COa. Here, in the close neighbourhood of the critical point even time intervals of 12 to 18 hours were necessary, although the crossection of the glashood was 30 mm and the height of the gascolumn in this case only about 80 mm. Received
May
26th
1939. REFERENCES
1) A. Miche
andG.W. B. Blaisse andG. W.
Nederbragt, andC. Nederbrachf
Michels,
Ind.Fng.Chem.6, 135, 1934. Proc.roy.Soc.A.100,348,1937. Pbysica,‘s-Grav.2, 1301, 1935;3*