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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



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