1384

THE COLLIERY GUARDIAN,

June 27, 1913.

i-

flight between the planes, then the actually transmitted
momentum will be i n V G x l and this, on account

6	K

of the elastic rebound of the molecules, will produce a
transmitted pressure of 2 x m V Gr X y. But, as has
been said, experiment shows that in oxygen gas for two
planes of unit area = 1 cm.2, when V = 1 cm. per second,
and the distance apart of the planes = 1 cm., =
0’000187 dynes, whence then 0000187 dynes =

2 x j nm V G X where V = 1, nm = p = 000143,
G = 46,100 cm. and k is the average number of collisions
during an average flight of a molecule through 1 cm. of
space, i) was measured by means of a torsion disc,
suspended with its surface near to a swiftly-moving disc.
It varies, of course, with different gases.

But if k be the number of collisions during a flight
of 1 cm., and 0 be the number of collisions per second,
and the mean average speed be Q, and V the velocity of
the moving plane be 1 cm. per second, then

q __q ___n m Q G_____ pQ G-

3 y 3^ ’

p being, as previously said, the density of the gas at
0 deg. Cent, and 760 mm. of mercury.

mi, *	0*00143 x 42,500 x 46,100

Therefore 0= ----------3 x 0.000187----

= 5,000,000,000.

This huge number, usually written 5 X 109, represents
the number of times an average molecule of oxygen
collides in a second.

Its average, or mean free path, is obviously found by
dividing the mean velocity by the number of collisions,
and thus X = ?_ = 0’0000085 cm. Certain corrections
C

have, however, to be applied to this formula due to the
fact that when we are calculating the mean free path it
is not O, the absolute velocity of the molecules, that must
be taken into consideration, but the relative velocity
with regard to one another. And as they are all in
motion it can be shown that the true value of the free
path is three-fourths of what it would be at rest. On
the other hand, there are corrections to be put in on
account of the law of the distribution of velocities.
Whence it results that the proper value of X

= 1-17 x ® = 117

C	5 x 109

= 000000995 cm.

Having thus determined the mean length of the free
path of a molecule, it is next to be seen whether this
value can be used to calculate its diameter.

If a molecule A of diameter a moves towards a
molecule B of the same diameter, it is obvious that it
will strike it, if the centre of A goes within a distance
= <r of the centre of B. So that we may treat the
molecule A as a point, provided we consider B as
having a radius = or instead of a diameter of that size.
Now in a centimeter of volume, if there are n spheres of
radius = <r, the total side sections they would present
to a point flying through a distance of 1 cm. in them
would be n x ttct2. This is the total “obstructive value”
of a cubic centimetre of gas. Whence the probability
of a particle striking one or other of them, in its
passage through 1 cm. of space, is the ratio of n X ttct2
to a cubic centimetre—that is, to 1.

Whence, then, the probability of its striking one of
them in a flight of one second through O centimetres of
space is O .wtto-2, and this, therefore, is 0, the probable
number of collisions that a molecule will have in one
second. But we have already found the value of C to
, O
be -

X

Whence, then, O. n^rcr2 = -»

X
or	^=—. (1)

nTFA

Refinements have been made on this value, but not so
as to affect its amount materially, and we may therefore
adopt it.

This equation, then, gives us a relation between or
the diameter of a molecule, and n the number of
molecules in a cubic centimetre of oxygen, and X the
mean free path. We must now seek for
relations between these quantities.

The volume of a spherical molecule of

^diameter = a is 77

volume of all the n molecules in a cubic centimetre =
= W g n it cr3 (2). Combining equations (1) and (2)
we obtain <r = 6 W X (3); <r being the diameter of
a molecule, X the mean free path, and W the actual' floating coal dust have a diameter of about 20U00

some more

which the

Hence the

volume of cubic space occupied by all the molecules in
a cubic centimetre of gas at standard temperature and
pressure.

If by pressure and freezing we compress the gas into
a liquid form—or the densest form which experiment
permits—the volume of the n molecules will, on the
assumption that molecules are hard elastic bodies, still
remain unchanged. Now when oxygen is compressed as
far as possible, it becomes a liquid rather denser than
water. Assuming, however, that it is compressed till it is
as dense as water, the volume that a gaseous cubic centi-
metre of it would assume when liquid = p = 0’00143
cm.3 This, however, we may be sure is larger than the
volume into which the gas could be squeezed if one could
compress it till the spherical form of the molecules was
lost, and the whole was in a continuous mass, with no
interstices.

For in a solid form oxygen can expand with heat, and
in a liquid form the molecules can glide about over one
another, showing that there are interstices between
them or even space around them. Taking, then, the
ultimate volume (according to the views of Clausius) at
two-fifths of the volume into which experimental pres-
sure can compress a gas, we shall have

W = f X 0 00143 cm.3
Inserting this value into (3) we obtain

tr = 6 x f x 0 00143 x 9’95 x IO-6

= 3’4 x 10“8 cm.,
and thus we obtain the diameter of a molecule.

Of course, this does not tell us what a molecule is, nor
what it is like; we can only say that it behaves as
though it were a hard, elastic spherical ball of the above
diameter. In reality, it may be a “centre of force,” or
a vortex ring in the ether.

We are now in a position to hazard a conjecture as to
the number of molecules in a cubic centimetre of oxygen.
For by (1) n = —10-8)a x 0.95 x 10-6

= 2-76 x 1019.

This enormous number—27 millions of millions of
millions in the small space of a cubic centimetre, or
J^th of a cubic inch — makes some call on the
imaginative faculty. When, however, we reflect that
liquid oxygen is at least seven or eight hundred times
as dense as oxygen in a gaseous form, we must, in
order to get a conception of the number of molecules
in liquid matter, raise this figure to 20 thousand
millions of millions of millions to the cubic centimetre.

On this theory, then, the physicists leave plenty of
molecules for the physiologist to construct with a com-
plicated form his most tiny germs. Yet the molecular
theory shows that when Nature works on a very small
scale, her power of producing complex and varied forms
is gradually diminished.

The distance apart of the molecules of oxygen in a
gaseous form obviously depends on the number which
there are in a cubic centimetre. Let d be the distance
from centre to centre, and arrange the molecules
uniformly throughout the cubic centimetre. Then the
volume of a sphere whose diameter is d = | it d3.

g

Hence n | ir & = 1 cm.’—that is, d* = x 2.7 x 10m;
whence d = 41 X 10 “ 8 cm. Thus, then, since a =
3’4 x 10"8 cm., the molecules of oxygen in a gaseous
form are apart from centre to centre a distance of about
12 times their diameter. The mass of a molecule of
oxygen is, of course, known if we know the mass of a
cubic centimetre of gas and the number of molecules
in it. Thus

™ = 5 x 10-28 grammes-
% Ci /O X Lv

Whence, then, we see that a cubic centimetre of
oxygen gas must be imagined as containing 2’76 X 1019
molecules, each of 3’4 X 10“8 cm. diameter and a mass
of 5 X 10“23 grammes, whose average distance apart
from centre to centre is 41 X 10“8 cm., and which are

Hull Goal Export!.—The official return of the exports of
coal from Hull for the week ending Tuesday, June 17,1913,
is as follows:—Antwerp, 490 tons; Amsterdam, 730; Abo,
193; Algiers, 1,979; Alderney, 187; Bombay, 501; Bremen,
1,550; Copenhagen, 461; Christiania, 820; Cronstadt,
23,517; Drontheim, 419; Degerhamn, 762; Gothenburg,
604; Ghent, 545; Harlingen, 1,459; Hamburg, 7,223;
Halmstad, 640; Harburg, 4,719; Leghorn, 303; Marseilles,
497; Neustadt, 880; Norrsunde, 1,061; Naples, 970;
Oxelosund, 2,120; Oport, 1,352; Oscarshamn, 1,279; Port
Kunda, 1,364; Palermo, 2,592; Rotterdam, 8,466; Reval,
5,510; Riga, 5,842; Rouen, 4,708; Stockholm, 846; St.
Petersburg, 6,638; St. Malo, 647; Stettin, 3,080; Trieste,
335; Tuborg, 1,590; Venice, 502; total, 97,381 tons.
Corresponding period last year, 92,484 tons.

West Riding Miners’ Permanent Relief Fund.—In the
last five years the membership of the Yorkshire Miners’
Permanent Relief Society has decreased from 25,000 to
16,000. There has consequently been a serious depletion of
the accumulated funds in order to meet liabilities, and the
future of the society has been a matter of anxiety for some
time. On the 18th inst. a special general meeting of the
society was held at Barnsley to consider a new scheme of
benefits. Since the annual meeting in March members of
the society had been asked to vote on two schemes of
reconstruction submitted by the board of management, and
the result of the ballot was announced at the meeting. No. 1
scheme, which the board recommended for adoption, pro-
posed that the society become a Fatal Accident Society, the
contribution to be l|d. per week, with provision for pay-
ment of £5 on the death of a married man, and weekly allow-
ance for widows and children; .£3 in the case of an unmarried
man; and .£10 on the death of a widow. No. 2 scheme
(which was finally adopted) provides for an increased con-
tribution, from 8d. to 9d. per fortnight, and a continuance
moving about with a mean average velocity of 425 m. ’ of benefit in case of fatal and minor accident and permanent

per second and a mean free path of 9*95 x 10“6 cm.,
each of which has 5 X 109 collisions per second.

These results have been obtained by knowing—

The mass of 1 cm.8 of oxygen gas... == 000143 grammes.

The atmospheric pressure on 1 cm. 1,013,200 dynes.

The density of liquid oxygen (say) £	wa^er-

The viscous pull exercised by a
plane 1 sq. cm. moving with a
velocity of 1 cm. per second, on
another plane of 1 sq.cm, at 1 cm.

distance in oxygen ............= 0’000187 dynes.

Hence then the diameter of a molecule is less than
T^L_th part of a wave-length of light. It has been
calculated that if oil be put on water it is capable
of spreading until a layer of single molecules remains.
No doubt the molecules float in juxtaposition like a
mass of bottle corks on water. Particles of very fine
of an

(1)

(2)

(3)

(4)

inch, or 1*25 X 10 “4 cm. This is about 4,000 times as big
as the diameter of an oxygen molecule. So that an
oxygen molecule is in size to the smallest floating
particle of coaldust, as a hailstone is to a balloon.

Within the space of such a particle of coaldust about
30 millions of gaseous molecules could move about
freely. If compressed into a solid state about 20
thousands of millions of oxygen molecules could occupy
this space. In fact, such a particle of dust probably
contains about 50 thousand million molecules.

So incredibly small are the molecules of which bodies
are composed, and so incredibly rapid are their col-
lisions when in a gaseous form.

The speed of the molecules is slow. This is fortunate.
The mean speed only about equals the speed of a cannon
ball. If the speed of a molecule reached 10,000 metres
per second, it would fly off from the earth and never
come back. But the law of distribution of velocities
shows us that such an enormous proportion of the mole-
cules have ordinary speeds that even in the lightest
gases not one in 10,000 millions could escape, even if it
could get through the mass of its jostling neighbours.
Here at all events the law of the prevalence of medio-
crity is useful.

The above numbers are not free from controversy,
being founded on a number of assumptions which are
not yet rigidly established.

But they sufficiently compare with the results of other
modes of calculation to warrant a belief in their
approximate truth.

When the kinetic theory of gases is first presented to
the mind, it appears extravagant, and even comical. I
remember the astonishment of a mine manager—a man
who managed more men than there are leaves in a
mathematical book, when this theory was first pro-
pounded to him. It actually took away his breath.
“ What! ” he said, “ are these gases in a mine buzzing
about like flies in a bottle ? ” He thought that fun was
being made of him. But no reasonable doubt of the
truth of the theory now exists, and it affords an eternal
monument to the ingenuity of those master-minds by
whom it has been designed and elaborated.

disablement, on a reduced scale. The chairman (Mr. C. H.
Cobbold) said only 51 agencies had returned the result of
their voting. In favour of No. 1 scheme 18 agencies voted,
and in favour of No. 2,33 agencies. According to the voting
power of the agencies, 35 voted for No. 1 scheme, and 49 for
No. 2 scheme, but if the vote of the honorary members and
members of the board of management had been allowed
there would have been a majority of 57 to 49 in favour of
No. 1 scheme. Inasmuch as in the alteration of rules they
required a two-thirds majority, he submitted} they were
not bound to act on the result of the ballot. It had to be
remembered that the society was taking from its capital
about £4,000 per year, which meant a loss of .£160 per year,
and was not reducing the liabilities. Under the present
conditions he thought the capital would not last more than
eight years. On a show of hands the delegates rejected
No. 1 scheme, and No. 2 scheme was then agreed to without
opposition.