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the table, the numbers in the ratio columns of Tables VI. and VII. were

plotted, the mean pressures being taken as abscissae. The points were joined

so as to form curves, and then finding points on the curve whose ordinates

corresponded to a particular number in the first column, the abscissae gave

the numbers required for the third column in Tables X. and XL In this

way the numbers in the third column are rather more uniform than they

would have been had they been the results of actual observation.

Tables X. and XL show that within the limits of accuracy of the

experiments the pressures in the stucco correspond with pressures in the

meerschaum six times as great. This is exactly according to Law V., Art. 7,

from which it appears that the numerical relation between the corresponding

282

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

TABLE X. Showing the pressures of air for which the ratio of the difference

of pressure to the mean pressure is the same for stucco No. 1 and

meerschaum No. 3.

Ratio of mean

pressure to

difference

of pressure

Meerschaum

No. 3.

Pressure

Stucco No. 1.

Pressure

Ratio of

corresponding

pressures

inches

inches

141

31-0

5-1

6-08

138

29-5

5-0

5-90

132

28-5

4-75

6-00

129

27-5

4-6

6-00

116

24-5

4-0

6-1

108

23-0

3-7

6-2

104

21-50

3-5

6-1

94

20-00

3-0

6-6

99

19-50

3-25

6-0

85

18-00

2-6

6-9

90

17-00

2-8

6-05

72

12-5

1-9

6-5

70

11-5

1-8

6-3

57

8-25

1-15

7-0

59

7-8

1-25

6-2

44

5-25

0-5

10-5

44

4-70

0-5

9-0

37

3-40

0-35

9-9

38

3-10

0-35

9-0

29

2-10

0-24

8-5

32

2-00

0-301

6-6

27

1-40

0-20

7-0

,27

1-32

0-20

6-0

pressures is the relation between the diameters of the interstices of the

meerschaum and stucco plates. This fact also is confirmed, for not only does

it appear that the ratio is independent of the mean density of the gas, but it

is the same for hydrogen as it is for air, showing that the relation depends

only on the nature of the plates.

Logarithmic homologues of the curves in figs. 5 and 6.

28. It appeared, however, that as a method of obtaining the corresponding

pressures the comparison of the ratios was not entirely satisfactory, for it

involved the assumption that the ratio of corresponding differences of pressure

should be exactly the same as the ratio of corresponding mean pressures ;

whereas this would only be the case if the differences of temperature were

exactly the same for both plates. It seemed desirable therefore to find a

means of comparing the curves for the two plates on the assumption that the

corresponding abscissae might bear one ratio and the corresponding ordinates

another, or if 1 and 2 are corresponding points, # 2 = ax^ while y 2 = by^.

33]

IN THE GASEOUS STATE.

283

TABLE XI. Showing the pressures of hydrogen at which the ratio of the

difference of pressure to the mean pressure is the same for meerschaum

No. 3 and stucco No. 1.

Katio of mean

pressure to

difference

of pressure

Corresponding pressures

Ratio of

corresponding

pressures

Meerschaum

No. 3

Stucco

No. 1

inches

inches

44

35

5-8

6-0

43

34

5-5

6-2

40

32

5-0

6-4

39

30

4-8

6-2

38

29-5

4-6

6-4

37

29-5

4-4

6-7

36

27

4-2

6-4

32

22

3-4

6-4

29-5

18-5

2-9

6-3

31

18

3-2

5-6

30

18

3-0

6-0

25

12

2-0

6-0

24-6

11-40

1-9

6-0

25

11-50

2-0

5-25

21

7-70

8-0

9-7

24

7-60

1-7

4-5

A graphic method of doing this simply and perfectly was found by com-

paring not the curves themselves, but what may be called their logarithmic

homologues.

Instead of plotting, as in figs. 4 and 5, the mean pressures and differences

of pressure as the abscissae and ordinates of the points on the curve, the

logarithms of these quantities are plotted. Thus, aci=loga; 1 , y 1 ' = \ogy l ,

where ac^ may be taken to be a point on any one of the curves already

plotted, and aj/y/ the corresponding point on the logarithmic homologue. It

is thus seen that if for two curves (1) and (2), # 2 = cw; 1 and y z = byi, then

a; 2 ' = a?/ -f- log a and y? = y^ + log b ; or the logarithmic homologues will all

be similar curves but differently placed with regard to the axes, such that

the one curve may be brought into coincidence with the other by a shift of

which the co-ordinates are log a, and log b.

Fig. 7 shows the logarithmic homologues of the curves for stucco No. 1

and meerschaum No. 3, both for hydrogen and air. By tracing the log curves

for stucco No. 1, together with the axes, on a piece of tracing paper, and then

moving the tracing (so that the axes remain parallel to their original

direction) until the traced curves fit on to the curves for meerschaum No. 3,

it is found that the fit is perfect, a portion of the traced curve e'f (stucco)

coinciding with a portion of ab, while at the same time a portion of the traced

284

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

i

P. 1

50

S -2

li Hydrogen.

'/' Air.

-1012

Log. Pressure.

Fig. 7.

curve g'h' coincides with a portion of cd. The effect of the superposition is

shown in the figure, e'b and g'd being the portions of the curves which overlap.

0' is the new position of 0.

It will at once be seen that O'M is the logarithm of the ratio of cor-

responding abscissae, while O'N is the logarithm of the corresponding

ordinates.

In this particular case

O'N = -7 = log 5 and O'M = 77 = log 5'9.

These numbers differ somewhat from those given by Tables X. and XL,

and the difference is very suggestive. The absolute agreement of the curves

shows that the difference is not owing to experimental inaccuracy, and it will

be seen on comparing the results next given that the difference (5 and 5'9) is

owing to a difference in the temperature in the two instruments. If the

temperatures had been the same we should have had the same ratio for the

corresponding ordinates as for the abscissae ; but a difference in the tem-

perature would alter all the ordinates in a certain ratio without affecting the

abscissa?.

The difference O'N- 0'M= '07 = logl'175 gives the ratio in which the

differences of pressure are affected by a difference in temperature. This,

according to the law that the results are proportional to the square roots of

the differences of temperature, would be equivalent to a- difference of 21 in

the temperature of the water. This difference did not exist, hence there

must have been a difference, owing to the greater thickness or to the different

nature of the meerschaum plate.

The size of the woodcut does not admit the points indicating the actual

experiments being shown, but these are shown in figures 8 and 9, pages 286 A

and 286 B.

33]

IN THE GASEOUS STATE.

285

TABLE XII. Thermal transpiration of air by stucco plate No. 2 ('25 inch or

6 35 millims. thick). Temperature of steam, 212 F. or 100 C. ; tem-

perature of water, 70 F. or 21 C.

Mean pressure by

vacuum gauge

Difference of pressure

by siphon gauge

Batio

of mean

pressure to

difference

of pressure

Log of

mean

pressure

Log of

difference

of pressure

July 17

July 18

inches

millims.

inch

inch

millim.

30-25

768-3

0160

...

406

1892

2-480-1

1-204-3

30-05

763-3

0162

411

1855

2-477

1-209

28-05

712-4

0166

422

1710

2-448

1-220

27-25

692-1

0170

432

1600

2-435

1-230

25-85

656-6

0176

447

1470

2-412

1-245

24-90

632-4

0180

457

1383

2-396

1-255

23-15

588

0196

498

1181

2-364

1-292

22-05

560

...

0194

492

1137

2-343

1-287

20-30

515-6

0208

...

528

976

2-307

1-318

19-20

487-4

0204

518

946

2-283

1-309

18-00

457-2

0230

584

784

2-255

1-361

16-10

408-94

0240

610

670

2-207

1-380

15-8

401-32

0230

584

680

2-199

1-361

14-0

355-6

0254

...

645

551

2-146

1-404

13-80

350-52

0244

620

565

2-140

1-387

12-45

316-2

0266

676

453

2-095

1-425

11-85

301-00

0256

650

462

2-073

1-408

10-82

274-8

0276

701

391

2-034

1-440

10-05

255-2

...

0262

660

383

2-002

1-418

9-80

248-9

0282

...

716

348

1-991

1-450

9-10

231-1

0272

691

334

1-959

1-434

8-75

222-1

0284

721

308

1-942

1-453

8-10

205-7

0280

711

290

1-908

1-447

7-65

194-3

0290

736

264

1-883

1-462

7-15

181-6

0284

721

252

1-853

1-453

6-72

170-7

0294

746

229

1-817

1-468

6-20

157-5

...

0288

731

215

1-791

1-459

5-50

139-7

0290

746

190

1-740

1-462

5-25

133-3

. .

0286

726

183

1-720

1-456

4-40

111-7

0280

. .

711

157

1-643

1-447

4-30

109-2

*

0276

701

156

1-633

1-440

3-40

86-4

0266

676

128

1-531

1-422

3-35

85-1

0264

671

127

1-525

1-421

2-70

68-6

0242

615

112

1-431

1-381

2-40

60-96

0226

574

106

1-380

1-354

2-00

50-8

0214

543

93

1-301

1-330

1-45

36-8

0182

462

80

1-161

1-266

1-22

31-00

0176

447

70

1-086

1-245

80

20-82

0138

...

350

58

0-903

1-139

50

12-70

.

0108

274

48

0-699

1-033

38

9-65

0088

223

42

0-580

0-944

225

5-71

0050

...

127

45

0-352

0-699

286

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

TABLE XIII. Thermal transpiration of hydrogen by stucco plate No. 2

(25 inch or 6'35 millims. thick). Temperature of steam, 212 F. or

100 C. ; temperature of water, 70 F. or 21 C.

Mean pressure by

vacuum gauge

Difference of pressure

by siphon gauge

Ratio

of mean

pressure to

difference

of pressure

Log of

mean

pressure

Log of

difference

of pressure

July 19

July 20

inches

millims.

inch

inch

millims.

31-00

787-40

1072

2-723

290

2-491 - 1

2-030 - 3

30-55

775-50

1080

2-743

283

2-484

2-033

29-60

751-60

...

1084

2-753

273

2-456

2-035

28-10

713-70

1102

2-799

255

2-449

2-042

26-70

677-90

1122

2-850

237

2-426

2-050

25-50

647-50

1132

2-875

225

2-406

2-054

25-25

641-00

1130

2-870

223

2-402

2-053

24-15

613-40

...

1152

2-916

209

2-383

2-061

22-40

568-90

1180

2-997

190

2-350

2-072

22-05

560-00

...

me

2-977

188

2-343

2-070

21-10

535-90

...

1182

3-002

177

2-324

2-072

20-15

510-50

...

1186

3-012

170

2-304

2-074

20-00

508-00

1192

3-027

168

2-301

2-076

19-20

487-60

1190

3-023

160

2-283

2-075

17-15

435-60

1204

3-058

142

2-234

2-080

16-23

411-40

1208

3-068

134

2-209

2-082

16-00

406-40

1214

...

3-083

130

2-204

2-084

15-30

388-60

...

1214

126

2-185

2-084

14-60

370-80

1220

3-098

119

2-164

2-086

14-55

369-50

1220

* ..

jj

M

2-163

2-086

13-95

354-30

1220

M

114

2-144

2-086

13-20

335-20

1212

3-078

108

2-120

2-083

12-35

317-70

1216

3-088

101

2-091

2-085

11-95

281-80

1200

1200

3-048

100

2-077

2-070

10-70

271-80

1198

3-043

89

2-029

2-087

9-60

243-80

1176

2-987

80

1-982

2-070

8-65

219-70

1146

2-910

75

1-937

2-059

7-75

196-80

1120

2-844

69

1-889

2-049

6-30

160-00

1064

2-702

60

1-799

2-027

5-75

146-00

1000

2-540

56

1-759

2-000

5-10

129-50

0976

2-479

52

1-700

1-989

3-65

92-70

...

0854

2-169

42

1-562

1-931

3-40

86-30

0860

2-184

40

1-531

1-934

2-50

63-50

0704

1-788

35

1-398

1-847

1-60

40-00

...

0524

1-331

30

1-204

1-719

1-10

27-90

0420

1-066

26

1-041

1-623

35

8-88

0170

...

431

20

0-544

1-230

286 A

Stut&oWl.

9\ 4 * -el -7 -9

O Log. Pr

286 B-

Loq. X>iff. of Pr sea-rt.. . , f

$ I I I I Ijl

JDiff. i f -Plessf re- for- t/u '

SS

z

ir-fe

of-lfl

5S

V

c

*

\

Fig. 9.

33] IN THE GASEOUS STATE. 287

Stucco plate No. 2.

29. These facts will be better understood after examining the experiments

on a second stucco plate. The trial of this plate was owing to an accident to

the diffusiometer containing stucco plate No. 1. The diffusiometer was

thereupon refitted with another plate similar to No. 1 ; but the old tin plates

were replaced by new bright ones, and the new india-rubber rings were

somewhat thicker than the old ones.

In making the experiments contained in Tables XII. and XIII., there

was a slight change from the former plan, which had been to begin at the

higher pressures and thence proceed by successive exhaustions to the lower

pressures. This time one series of experiments was made as before, and

another in the inverse manner the diffusiometer being exhausted to com-

mence with and the air or hydrogen being allowed to enter between the

observations. Both series are given in the tables and are seen to agree very

closely.

In the case of hydrogen it was found as before, that although not great,

there was still a greater tendency to irregularity than with air, and as this

was evidently due to diffusion through the india-rubber during the very

considerable time (4 or .5 hours) which elapsed before the lower pressures

were reached, several independent experiments were made. The diffusiometer

being filled with pure hydrogen, was exhausted at once down to the particular

low pressure at which the reading was taken, so as not to allow time for

diffusion.

Differences of temperature brought to light by the log. curves.

30. The results with stucco plate No. 2 are smaller than with No. 1.

At first sight it was thought that this difference was entirely owing to No. 2

being somewhat coarser than No. 1, but when the logarithmic homologues of

the curves for this plate came to be compared with those for No. 1 and

meerschaum No. 3, after the manner described in Art. 28, it became apparent

that the difference in the results with plates No. 1 and 2 (stucco) was due to

two causes. Some of it was due, as had been supposed, to the greater

coarseness of No. 2, but a large part could only be explained on the assumption

that from some cause or another the difference of temperature with No. 2 was

less than with No. 1.

In fig. 10 it is seen that in order to bring the log. curves for stucco No. 2

into coincidence with the curves for stucco No. 1, it was necessary to increase

the abscissae of the former by '048 log I'll 7 : while the ordinates had to be

increased by '112. The difference in the abscissae, as shown in Art. 28,

represents the difference due to the coarseness of the plate ; thus the

openings in No. 2 are I'll? times as broad as the openings in No. 1.

And the difference between the differences of the ordinates and the abscissae

288

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

or

[33

Log. Pressure.

Fig. 10.

= -112 "048 = '064 = log T16 is the logarithm of the effect of a difference

of temperature, and to produce this effect the temperature of the water

would have had to be lowered 15. There was some difference, from 5 to 7,

leaving from 8 to 10 to be expressed as due to the bright tin plates and

thicker rings.

Comparison of the logarithmic homologues.

31. In figures 8 and 9 the curves for meerschaum are drawn in the

same position with reference to the axis, OM, ON. But while in figure 8

the curves for stucco No. 1 and No. 2 are shown in the position as plotted

from the columns of logarithms in the Tables VI., VII., XII. and XIII., in

figure 9 these curves hav^ all been shifted until they coincide with the

curves for meerschaum No. 3, in each case the two curves for air and hydrogen

being shifted together. The axes are also shown as shifted with each pair of

curves. The fitting of these curves is very remarkable ; nor is it only the

curves, for the points indicating the results are shown, and these all fall in so

truly that it was hardly necessary to draw a line until the points of low

pressure are reached. There is a slight deviation of that part of the curve for

hydrogen, stucco No. 1, which represents the pressures below 1 inch ; but this

has been already explained as being due to the infusion of air through the

india-rubber. In order to fully appreciate the force of this agreement, it

must be borne in mind that it is not merely the portions of the curves that

overlap that agree in direction, but the distance between the curves for

hydrogen and air which have been shifted in pairs.

Nothing could prove more forcibly than this, that the different results

obtained with different plates are quite independent of the nature of the gas

so long as the densities are in the ratio of the fineness of the plates.

So far, therefore, as thermal transpiration is concerned, we have an absolute

proof of Law V., Art. 9.

33]

IN THE GASEOUS STATE.

289

The relative coarseness of the plates.

The shifts in fig. 9 to bring the curves into coincidence being the

logarithms of the corresponding pressures, it follows from Law V. that these

shifts are the logarithms of the relative coarseness of the plates. Hence for

the mean (after some law) diameters of the apertures we have :

Plate. Coarseness.

Meerschaum No. 3 1

Stucco No. 1 5

Stucco No. 2 5-6

Further comparison of the results with the laws of Art. 9.

32. So far as the manner of variation of the differences of pressures with

the density of the gas, this is completely shown by the shapes of the curves

in figs. 4, 5, and 6, and is strictly according to Laws II. and III.

The agreement of the log. curves has been shown to confirm Law V. It

only remains, therefore, to notice the laws of variation at the greatest and

smallest pressures, to see how far these conform to the limits given in

Laws III. and IV.

According to Law III., when the density of the gas is sufficient, the

differences of pressure should be inversely proportional to the density.

Hence, according to this law, the product of the pressure into the

difference of pressure should approximate to a constant quantity as the

density increases.

In the case of stucco No. 2, we have, adding the two tables of logarithms,

subtracting '684 1 = log '483, and taking out the numbers

Pressure

Pressure x difference

of pressure -f- -483

30-25

1

30-05

1-004

28-05

964

27-25

957

25-85

940

24-90

927

23-15

933

This sufficiently shows that the approximation is very close and according

to Law III.

Coming now to the lower pressures, it will at once be seen that in all

cases there is a tendency towards constancy. This is best seen in fig. 9,

where the curves not only converge towards the left but turn towards the

horizontal.

o. R.

290 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

It is clear, however, from these curves, that the limit had not been

reached, nor is it possible to say simply from the shape of the curves how far

it might be off.

The following comparison, however, will show that the indication is in

favour of Law IV., viz. : that the ultimate ratio which the difference of

pressure bears to the mean pressure should be as the ratio which the differ-

ence of the square roots of the absolute temperature bears to the square root

of the mean absolute temperature. According to this, we should have in the

case of the meerschaum plate the ratio of the difference of pressure to the

mean pressure equal to

V212 + 461 - \/63 + 461 = 1

V137-5 + 461 ~8'

whereas, supposing that there was a difference of 20 between the surfaces of

the meerschaum and the opposite tin plates

V137-5 +461 ~ 11 '

between which values it is probable that the actual ratio lies.

The highest ratio of the difference of pressure to the mean pressure

obtained is 1 to 13, and this may well be considered as an approximation

to 1 to 11.

Thus, not only in their general features, but in the approximation towards

definite limits, the experimental results show a close agreement with the laws

as deduced from the theory.

SECTION III. EXPERIMENTS RESPECTING THE RATE OF TRANSPIRATION.

33. The experiments to be described in this section, besides being

necessary for the verification of the Laws V., VI., and VII., Art. 9, were

necessary to complete the verification of Law I. In the last section no direct

notice was taken of the rate of thermal transpiration when unprevented by

the difference of pressure on the two sides of the plate, and for this reason.

Although the thermal differences of pressure indicate in a general way

the manner in which transpiration would have taken place had the pressure

been equal, yet in order to examine the results strictly, as regards the various

rates of thermal transpiration to which they correspond, it is necessary to

know the exact law of transpiration for gases under pressure. The com-

parative rates of transpiration for different gases and the rates of transpiration

of each gas for different pressures are not sufficient. So far, the laws

established by Graham are all that can be desired, but these laws say nothing

about the variation in the rate of transpiration consequent on a large variation

in the density of the gas. Thus, Graham has shown that, through a fine

33] IN THE GASEOUS STATE. 291

graphite plate, the time of transpiration of a constant volume (measured at

the mean pressure) will be exactly proportional to the difference of pressure,

and will diminish slightly with the density, but his experiments were not

carried to pressures many times less than the pressure of the atmosphere ;

whereas, for the purpose of this investigation, it was necessary to compare

results at pressures as low as '01 of an atmosphere. Nor is this the only

point in which Graham's results appeared insufficient for the present com-

parison. Graham had found that the law of transpiration for a fine graphite

plate differed essentially from the law for a stucco plate ; his experiments

having been made in both cases at pressures not many times less than the

pressure of the atmosphere. Thus, for the stucco plate, the comparative

times of transpiration of air and hydrogen were as 2'8 to 1, while for the

graphite plate they were as 3*8 to 1. He had also shown that for plates of

intermediate coarseness an intermediate ratio would maintain ; but he had

given no law that would enable us to predict the result with any particular

plate.

In order, therefore, to effect my comparison, it was necessary, by actual

experiment, to ascertain the rates of transpiration through my particular

plates with the same gases as those used for thermal transpiration, and at

similar pressures. It was this consideration which mainly determined the

manner of making the experiments.

The apparatus.

34. The thermo-diffusiometer, without the streams of steam and water,

after having undergone certain slight modifications, lent itself very well to

this part of the investigation.

By means of extra branches from the tube KK, fig. 3, two 8 oz. flasks

were connected with the chambers, one on each side of the porous plate, the

object of these flasks being simply to enlarge the capacity of the chambers.

The branch to the flask on the right was outside the tap P, so that by

closing this tap the flask would be cut off from the instrument, and the action

of the pump would be confined to that one flask.

In this condition the mercury pump had a definite capacity about

6 fluid oz., the capacity of the flasks was definite about 8 fluid oz. each, and

besides these there were the tubes and chambers in the diffusiometer also of

plotted, the mean pressures being taken as abscissae. The points were joined

so as to form curves, and then finding points on the curve whose ordinates

corresponded to a particular number in the first column, the abscissae gave

the numbers required for the third column in Tables X. and XL In this

way the numbers in the third column are rather more uniform than they

would have been had they been the results of actual observation.

Tables X. and XL show that within the limits of accuracy of the

experiments the pressures in the stucco correspond with pressures in the

meerschaum six times as great. This is exactly according to Law V., Art. 7,

from which it appears that the numerical relation between the corresponding

282

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

TABLE X. Showing the pressures of air for which the ratio of the difference

of pressure to the mean pressure is the same for stucco No. 1 and

meerschaum No. 3.

Ratio of mean

pressure to

difference

of pressure

Meerschaum

No. 3.

Pressure

Stucco No. 1.

Pressure

Ratio of

corresponding

pressures

inches

inches

141

31-0

5-1

6-08

138

29-5

5-0

5-90

132

28-5

4-75

6-00

129

27-5

4-6

6-00

116

24-5

4-0

6-1

108

23-0

3-7

6-2

104

21-50

3-5

6-1

94

20-00

3-0

6-6

99

19-50

3-25

6-0

85

18-00

2-6

6-9

90

17-00

2-8

6-05

72

12-5

1-9

6-5

70

11-5

1-8

6-3

57

8-25

1-15

7-0

59

7-8

1-25

6-2

44

5-25

0-5

10-5

44

4-70

0-5

9-0

37

3-40

0-35

9-9

38

3-10

0-35

9-0

29

2-10

0-24

8-5

32

2-00

0-301

6-6

27

1-40

0-20

7-0

,27

1-32

0-20

6-0

pressures is the relation between the diameters of the interstices of the

meerschaum and stucco plates. This fact also is confirmed, for not only does

it appear that the ratio is independent of the mean density of the gas, but it

is the same for hydrogen as it is for air, showing that the relation depends

only on the nature of the plates.

Logarithmic homologues of the curves in figs. 5 and 6.

28. It appeared, however, that as a method of obtaining the corresponding

pressures the comparison of the ratios was not entirely satisfactory, for it

involved the assumption that the ratio of corresponding differences of pressure

should be exactly the same as the ratio of corresponding mean pressures ;

whereas this would only be the case if the differences of temperature were

exactly the same for both plates. It seemed desirable therefore to find a

means of comparing the curves for the two plates on the assumption that the

corresponding abscissae might bear one ratio and the corresponding ordinates

another, or if 1 and 2 are corresponding points, # 2 = ax^ while y 2 = by^.

33]

IN THE GASEOUS STATE.

283

TABLE XI. Showing the pressures of hydrogen at which the ratio of the

difference of pressure to the mean pressure is the same for meerschaum

No. 3 and stucco No. 1.

Katio of mean

pressure to

difference

of pressure

Corresponding pressures

Ratio of

corresponding

pressures

Meerschaum

No. 3

Stucco

No. 1

inches

inches

44

35

5-8

6-0

43

34

5-5

6-2

40

32

5-0

6-4

39

30

4-8

6-2

38

29-5

4-6

6-4

37

29-5

4-4

6-7

36

27

4-2

6-4

32

22

3-4

6-4

29-5

18-5

2-9

6-3

31

18

3-2

5-6

30

18

3-0

6-0

25

12

2-0

6-0

24-6

11-40

1-9

6-0

25

11-50

2-0

5-25

21

7-70

8-0

9-7

24

7-60

1-7

4-5

A graphic method of doing this simply and perfectly was found by com-

paring not the curves themselves, but what may be called their logarithmic

homologues.

Instead of plotting, as in figs. 4 and 5, the mean pressures and differences

of pressure as the abscissae and ordinates of the points on the curve, the

logarithms of these quantities are plotted. Thus, aci=loga; 1 , y 1 ' = \ogy l ,

where ac^ may be taken to be a point on any one of the curves already

plotted, and aj/y/ the corresponding point on the logarithmic homologue. It

is thus seen that if for two curves (1) and (2), # 2 = cw; 1 and y z = byi, then

a; 2 ' = a?/ -f- log a and y? = y^ + log b ; or the logarithmic homologues will all

be similar curves but differently placed with regard to the axes, such that

the one curve may be brought into coincidence with the other by a shift of

which the co-ordinates are log a, and log b.

Fig. 7 shows the logarithmic homologues of the curves for stucco No. 1

and meerschaum No. 3, both for hydrogen and air. By tracing the log curves

for stucco No. 1, together with the axes, on a piece of tracing paper, and then

moving the tracing (so that the axes remain parallel to their original

direction) until the traced curves fit on to the curves for meerschaum No. 3,

it is found that the fit is perfect, a portion of the traced curve e'f (stucco)

coinciding with a portion of ab, while at the same time a portion of the traced

284

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

i

P. 1

50

S -2

li Hydrogen.

'/' Air.

-1012

Log. Pressure.

Fig. 7.

curve g'h' coincides with a portion of cd. The effect of the superposition is

shown in the figure, e'b and g'd being the portions of the curves which overlap.

0' is the new position of 0.

It will at once be seen that O'M is the logarithm of the ratio of cor-

responding abscissae, while O'N is the logarithm of the corresponding

ordinates.

In this particular case

O'N = -7 = log 5 and O'M = 77 = log 5'9.

These numbers differ somewhat from those given by Tables X. and XL,

and the difference is very suggestive. The absolute agreement of the curves

shows that the difference is not owing to experimental inaccuracy, and it will

be seen on comparing the results next given that the difference (5 and 5'9) is

owing to a difference in the temperature in the two instruments. If the

temperatures had been the same we should have had the same ratio for the

corresponding ordinates as for the abscissae ; but a difference in the tem-

perature would alter all the ordinates in a certain ratio without affecting the

abscissa?.

The difference O'N- 0'M= '07 = logl'175 gives the ratio in which the

differences of pressure are affected by a difference in temperature. This,

according to the law that the results are proportional to the square roots of

the differences of temperature, would be equivalent to a- difference of 21 in

the temperature of the water. This difference did not exist, hence there

must have been a difference, owing to the greater thickness or to the different

nature of the meerschaum plate.

The size of the woodcut does not admit the points indicating the actual

experiments being shown, but these are shown in figures 8 and 9, pages 286 A

and 286 B.

33]

IN THE GASEOUS STATE.

285

TABLE XII. Thermal transpiration of air by stucco plate No. 2 ('25 inch or

6 35 millims. thick). Temperature of steam, 212 F. or 100 C. ; tem-

perature of water, 70 F. or 21 C.

Mean pressure by

vacuum gauge

Difference of pressure

by siphon gauge

Batio

of mean

pressure to

difference

of pressure

Log of

mean

pressure

Log of

difference

of pressure

July 17

July 18

inches

millims.

inch

inch

millim.

30-25

768-3

0160

...

406

1892

2-480-1

1-204-3

30-05

763-3

0162

411

1855

2-477

1-209

28-05

712-4

0166

422

1710

2-448

1-220

27-25

692-1

0170

432

1600

2-435

1-230

25-85

656-6

0176

447

1470

2-412

1-245

24-90

632-4

0180

457

1383

2-396

1-255

23-15

588

0196

498

1181

2-364

1-292

22-05

560

...

0194

492

1137

2-343

1-287

20-30

515-6

0208

...

528

976

2-307

1-318

19-20

487-4

0204

518

946

2-283

1-309

18-00

457-2

0230

584

784

2-255

1-361

16-10

408-94

0240

610

670

2-207

1-380

15-8

401-32

0230

584

680

2-199

1-361

14-0

355-6

0254

...

645

551

2-146

1-404

13-80

350-52

0244

620

565

2-140

1-387

12-45

316-2

0266

676

453

2-095

1-425

11-85

301-00

0256

650

462

2-073

1-408

10-82

274-8

0276

701

391

2-034

1-440

10-05

255-2

...

0262

660

383

2-002

1-418

9-80

248-9

0282

...

716

348

1-991

1-450

9-10

231-1

0272

691

334

1-959

1-434

8-75

222-1

0284

721

308

1-942

1-453

8-10

205-7

0280

711

290

1-908

1-447

7-65

194-3

0290

736

264

1-883

1-462

7-15

181-6

0284

721

252

1-853

1-453

6-72

170-7

0294

746

229

1-817

1-468

6-20

157-5

...

0288

731

215

1-791

1-459

5-50

139-7

0290

746

190

1-740

1-462

5-25

133-3

. .

0286

726

183

1-720

1-456

4-40

111-7

0280

. .

711

157

1-643

1-447

4-30

109-2

*

0276

701

156

1-633

1-440

3-40

86-4

0266

676

128

1-531

1-422

3-35

85-1

0264

671

127

1-525

1-421

2-70

68-6

0242

615

112

1-431

1-381

2-40

60-96

0226

574

106

1-380

1-354

2-00

50-8

0214

543

93

1-301

1-330

1-45

36-8

0182

462

80

1-161

1-266

1-22

31-00

0176

447

70

1-086

1-245

80

20-82

0138

...

350

58

0-903

1-139

50

12-70

.

0108

274

48

0-699

1-033

38

9-65

0088

223

42

0-580

0-944

225

5-71

0050

...

127

45

0-352

0-699

286

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

TABLE XIII. Thermal transpiration of hydrogen by stucco plate No. 2

(25 inch or 6'35 millims. thick). Temperature of steam, 212 F. or

100 C. ; temperature of water, 70 F. or 21 C.

Mean pressure by

vacuum gauge

Difference of pressure

by siphon gauge

Ratio

of mean

pressure to

difference

of pressure

Log of

mean

pressure

Log of

difference

of pressure

July 19

July 20

inches

millims.

inch

inch

millims.

31-00

787-40

1072

2-723

290

2-491 - 1

2-030 - 3

30-55

775-50

1080

2-743

283

2-484

2-033

29-60

751-60

...

1084

2-753

273

2-456

2-035

28-10

713-70

1102

2-799

255

2-449

2-042

26-70

677-90

1122

2-850

237

2-426

2-050

25-50

647-50

1132

2-875

225

2-406

2-054

25-25

641-00

1130

2-870

223

2-402

2-053

24-15

613-40

...

1152

2-916

209

2-383

2-061

22-40

568-90

1180

2-997

190

2-350

2-072

22-05

560-00

...

me

2-977

188

2-343

2-070

21-10

535-90

...

1182

3-002

177

2-324

2-072

20-15

510-50

...

1186

3-012

170

2-304

2-074

20-00

508-00

1192

3-027

168

2-301

2-076

19-20

487-60

1190

3-023

160

2-283

2-075

17-15

435-60

1204

3-058

142

2-234

2-080

16-23

411-40

1208

3-068

134

2-209

2-082

16-00

406-40

1214

...

3-083

130

2-204

2-084

15-30

388-60

...

1214

126

2-185

2-084

14-60

370-80

1220

3-098

119

2-164

2-086

14-55

369-50

1220

* ..

jj

M

2-163

2-086

13-95

354-30

1220

M

114

2-144

2-086

13-20

335-20

1212

3-078

108

2-120

2-083

12-35

317-70

1216

3-088

101

2-091

2-085

11-95

281-80

1200

1200

3-048

100

2-077

2-070

10-70

271-80

1198

3-043

89

2-029

2-087

9-60

243-80

1176

2-987

80

1-982

2-070

8-65

219-70

1146

2-910

75

1-937

2-059

7-75

196-80

1120

2-844

69

1-889

2-049

6-30

160-00

1064

2-702

60

1-799

2-027

5-75

146-00

1000

2-540

56

1-759

2-000

5-10

129-50

0976

2-479

52

1-700

1-989

3-65

92-70

...

0854

2-169

42

1-562

1-931

3-40

86-30

0860

2-184

40

1-531

1-934

2-50

63-50

0704

1-788

35

1-398

1-847

1-60

40-00

...

0524

1-331

30

1-204

1-719

1-10

27-90

0420

1-066

26

1-041

1-623

35

8-88

0170

...

431

20

0-544

1-230

286 A

Stut&oWl.

9\ 4 * -el -7 -9

O Log. Pr

286 B-

Loq. X>iff. of Pr sea-rt.. . , f

$ I I I I Ijl

JDiff. i f -Plessf re- for- t/u '

SS

z

ir-fe

of-lfl

5S

V

c

*

\

Fig. 9.

33] IN THE GASEOUS STATE. 287

Stucco plate No. 2.

29. These facts will be better understood after examining the experiments

on a second stucco plate. The trial of this plate was owing to an accident to

the diffusiometer containing stucco plate No. 1. The diffusiometer was

thereupon refitted with another plate similar to No. 1 ; but the old tin plates

were replaced by new bright ones, and the new india-rubber rings were

somewhat thicker than the old ones.

In making the experiments contained in Tables XII. and XIII., there

was a slight change from the former plan, which had been to begin at the

higher pressures and thence proceed by successive exhaustions to the lower

pressures. This time one series of experiments was made as before, and

another in the inverse manner the diffusiometer being exhausted to com-

mence with and the air or hydrogen being allowed to enter between the

observations. Both series are given in the tables and are seen to agree very

closely.

In the case of hydrogen it was found as before, that although not great,

there was still a greater tendency to irregularity than with air, and as this

was evidently due to diffusion through the india-rubber during the very

considerable time (4 or .5 hours) which elapsed before the lower pressures

were reached, several independent experiments were made. The diffusiometer

being filled with pure hydrogen, was exhausted at once down to the particular

low pressure at which the reading was taken, so as not to allow time for

diffusion.

Differences of temperature brought to light by the log. curves.

30. The results with stucco plate No. 2 are smaller than with No. 1.

At first sight it was thought that this difference was entirely owing to No. 2

being somewhat coarser than No. 1, but when the logarithmic homologues of

the curves for this plate came to be compared with those for No. 1 and

meerschaum No. 3, after the manner described in Art. 28, it became apparent

that the difference in the results with plates No. 1 and 2 (stucco) was due to

two causes. Some of it was due, as had been supposed, to the greater

coarseness of No. 2, but a large part could only be explained on the assumption

that from some cause or another the difference of temperature with No. 2 was

less than with No. 1.

In fig. 10 it is seen that in order to bring the log. curves for stucco No. 2

into coincidence with the curves for stucco No. 1, it was necessary to increase

the abscissae of the former by '048 log I'll 7 : while the ordinates had to be

increased by '112. The difference in the abscissae, as shown in Art. 28,

represents the difference due to the coarseness of the plate ; thus the

openings in No. 2 are I'll? times as broad as the openings in No. 1.

And the difference between the differences of the ordinates and the abscissae

288

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

or

[33

Log. Pressure.

Fig. 10.

= -112 "048 = '064 = log T16 is the logarithm of the effect of a difference

of temperature, and to produce this effect the temperature of the water

would have had to be lowered 15. There was some difference, from 5 to 7,

leaving from 8 to 10 to be expressed as due to the bright tin plates and

thicker rings.

Comparison of the logarithmic homologues.

31. In figures 8 and 9 the curves for meerschaum are drawn in the

same position with reference to the axis, OM, ON. But while in figure 8

the curves for stucco No. 1 and No. 2 are shown in the position as plotted

from the columns of logarithms in the Tables VI., VII., XII. and XIII., in

figure 9 these curves hav^ all been shifted until they coincide with the

curves for meerschaum No. 3, in each case the two curves for air and hydrogen

being shifted together. The axes are also shown as shifted with each pair of

curves. The fitting of these curves is very remarkable ; nor is it only the

curves, for the points indicating the results are shown, and these all fall in so

truly that it was hardly necessary to draw a line until the points of low

pressure are reached. There is a slight deviation of that part of the curve for

hydrogen, stucco No. 1, which represents the pressures below 1 inch ; but this

has been already explained as being due to the infusion of air through the

india-rubber. In order to fully appreciate the force of this agreement, it

must be borne in mind that it is not merely the portions of the curves that

overlap that agree in direction, but the distance between the curves for

hydrogen and air which have been shifted in pairs.

Nothing could prove more forcibly than this, that the different results

obtained with different plates are quite independent of the nature of the gas

so long as the densities are in the ratio of the fineness of the plates.

So far, therefore, as thermal transpiration is concerned, we have an absolute

proof of Law V., Art. 9.

33]

IN THE GASEOUS STATE.

289

The relative coarseness of the plates.

The shifts in fig. 9 to bring the curves into coincidence being the

logarithms of the corresponding pressures, it follows from Law V. that these

shifts are the logarithms of the relative coarseness of the plates. Hence for

the mean (after some law) diameters of the apertures we have :

Plate. Coarseness.

Meerschaum No. 3 1

Stucco No. 1 5

Stucco No. 2 5-6

Further comparison of the results with the laws of Art. 9.

32. So far as the manner of variation of the differences of pressures with

the density of the gas, this is completely shown by the shapes of the curves

in figs. 4, 5, and 6, and is strictly according to Laws II. and III.

The agreement of the log. curves has been shown to confirm Law V. It

only remains, therefore, to notice the laws of variation at the greatest and

smallest pressures, to see how far these conform to the limits given in

Laws III. and IV.

According to Law III., when the density of the gas is sufficient, the

differences of pressure should be inversely proportional to the density.

Hence, according to this law, the product of the pressure into the

difference of pressure should approximate to a constant quantity as the

density increases.

In the case of stucco No. 2, we have, adding the two tables of logarithms,

subtracting '684 1 = log '483, and taking out the numbers

Pressure

Pressure x difference

of pressure -f- -483

30-25

1

30-05

1-004

28-05

964

27-25

957

25-85

940

24-90

927

23-15

933

This sufficiently shows that the approximation is very close and according

to Law III.

Coming now to the lower pressures, it will at once be seen that in all

cases there is a tendency towards constancy. This is best seen in fig. 9,

where the curves not only converge towards the left but turn towards the

horizontal.

o. R.

290 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

It is clear, however, from these curves, that the limit had not been

reached, nor is it possible to say simply from the shape of the curves how far

it might be off.

The following comparison, however, will show that the indication is in

favour of Law IV., viz. : that the ultimate ratio which the difference of

pressure bears to the mean pressure should be as the ratio which the differ-

ence of the square roots of the absolute temperature bears to the square root

of the mean absolute temperature. According to this, we should have in the

case of the meerschaum plate the ratio of the difference of pressure to the

mean pressure equal to

V212 + 461 - \/63 + 461 = 1

V137-5 + 461 ~8'

whereas, supposing that there was a difference of 20 between the surfaces of

the meerschaum and the opposite tin plates

V137-5 +461 ~ 11 '

between which values it is probable that the actual ratio lies.

The highest ratio of the difference of pressure to the mean pressure

obtained is 1 to 13, and this may well be considered as an approximation

to 1 to 11.

Thus, not only in their general features, but in the approximation towards

definite limits, the experimental results show a close agreement with the laws

as deduced from the theory.

SECTION III. EXPERIMENTS RESPECTING THE RATE OF TRANSPIRATION.

33. The experiments to be described in this section, besides being

necessary for the verification of the Laws V., VI., and VII., Art. 9, were

necessary to complete the verification of Law I. In the last section no direct

notice was taken of the rate of thermal transpiration when unprevented by

the difference of pressure on the two sides of the plate, and for this reason.

Although the thermal differences of pressure indicate in a general way

the manner in which transpiration would have taken place had the pressure

been equal, yet in order to examine the results strictly, as regards the various

rates of thermal transpiration to which they correspond, it is necessary to

know the exact law of transpiration for gases under pressure. The com-

parative rates of transpiration for different gases and the rates of transpiration

of each gas for different pressures are not sufficient. So far, the laws

established by Graham are all that can be desired, but these laws say nothing

about the variation in the rate of transpiration consequent on a large variation

in the density of the gas. Thus, Graham has shown that, through a fine

33] IN THE GASEOUS STATE. 291

graphite plate, the time of transpiration of a constant volume (measured at

the mean pressure) will be exactly proportional to the difference of pressure,

and will diminish slightly with the density, but his experiments were not

carried to pressures many times less than the pressure of the atmosphere ;

whereas, for the purpose of this investigation, it was necessary to compare

results at pressures as low as '01 of an atmosphere. Nor is this the only

point in which Graham's results appeared insufficient for the present com-

parison. Graham had found that the law of transpiration for a fine graphite

plate differed essentially from the law for a stucco plate ; his experiments

having been made in both cases at pressures not many times less than the

pressure of the atmosphere. Thus, for the stucco plate, the comparative

times of transpiration of air and hydrogen were as 2'8 to 1, while for the

graphite plate they were as 3*8 to 1. He had also shown that for plates of

intermediate coarseness an intermediate ratio would maintain ; but he had

given no law that would enable us to predict the result with any particular

plate.

In order, therefore, to effect my comparison, it was necessary, by actual

experiment, to ascertain the rates of transpiration through my particular

plates with the same gases as those used for thermal transpiration, and at

similar pressures. It was this consideration which mainly determined the

manner of making the experiments.

The apparatus.

34. The thermo-diffusiometer, without the streams of steam and water,

after having undergone certain slight modifications, lent itself very well to

this part of the investigation.

By means of extra branches from the tube KK, fig. 3, two 8 oz. flasks

were connected with the chambers, one on each side of the porous plate, the

object of these flasks being simply to enlarge the capacity of the chambers.

The branch to the flask on the right was outside the tap P, so that by

closing this tap the flask would be cut off from the instrument, and the action

of the pump would be confined to that one flask.

In this condition the mercury pump had a definite capacity about

6 fluid oz., the capacity of the flasks was definite about 8 fluid oz. each, and

besides these there were the tubes and chambers in the diffusiometer also of

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