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390 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

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pressure and temperature the value of NMif is the same for all gases. But

we found in Prop. VI. that when two sets of particles communicate agitation
to one another, the value of Mif is the same in each. From this it appears
that N, the number of particles in unit of volume, is the same for all gases
at the same pressure and temperature. This result agrees with the chemical law,
that equal volumes of gases are chemically equivalent.

We have next to determine the value of I, the mean length of the path
of a particle between consecutive collisions. The most direct method of doing
this depends upon the fact, that when different strata of a gas slide upon
one another with different velocities, they act upon one another with a tan-
gential force tending to prevent this sliding, and similar in its results to the
friction between two solid surfaces sliding over each other in the same way.
The explanation of gaseous friction, according to our hypothesis, is, that particles
having the mean velocity of translation belonging to one layer of the gas, pass
out of it into another layer having a different velocity of translation ; and
by striking against the particles of the second layer, exert upon it a tangential
force which constitutes the internal friction of the gas. The whole friction
between two portions of gas separated by a plane surface, depends upon the
total action between all the layers on the one side of that surface upon all the
layers on the other side.

Prop. XIII. To find the internal friction in a system of moving particles.

Let the system be divided into layers parallel to the plane of xy, and
let the motion of translation of each layer be u in the direction of x, and
let u = A+Bz. We have to consider the mutual action between the layers on
the positive and negative sides of the plane xy. Let us first determine the
action between two layers dz and dz\ at distances z and — z' on opposite sides
of this plane, each unit of area. The number of particles which, starting from
dz in unit of time, reach a distance between nl and (n-{-dn)l is by (19),

N J e"** dz dn.

The number of these which have the ends of their paths in the layer dz' is

N — -jt e"" dz dz' dn.

The mean velocity in the direction of x which each of these has before impact
is A + Bz, and after impact A+Bz'; and its mass is M, so that a mean

ILLUSTRATIONS OF THE DYNAAIICAL THEORY OF GASES. 391

momentum =MB{z-z) is communicated by each particle. The whole action due
to these collisions is therefore

NMB ^, (z - z) e-** dz dz dn.

We must first integrate with respect to z' between z' = and z' = z — nl; this
gives

^NMB 2^ (nH' -z')e-''dz dn

for the action between the layer dz and all the layers below the plane xy.
Then integrate from z = to z = nl,

^MNBlm'e-'' dn.
Integrate from n = to n = oo , and we find the whole friction between unit
of area above and below the plane to be

where /x is the ordinary coefficient of internal friction,

-i'^^-iTlS" • ^^^>'

where p is the density, I the mean length of path of a particle, and v the

... 2a ^ lYk

mean velocity v = -j= = 2 J — ,

'=I^V.T (^^)-

Now Professor Stokes finds by experiments on air.

J:

'^ = •116.

If we suppose n/^ = 930 feet per second for air at 60°, and therefore the mean
velocity 1^ = 1505 feet per second, then the value of I, the mean distance
travelled over by a particle between consecutive collisions, =4 47^000 ^^ ^^ ^^
inch, and each particle makes 8,077,200,000 collisions per second.

A remarkable result here presented to us in equation (24), is that if this
explanation of gaseous friction be true, the coefficient of friction is independent
of the density. Such a consequence of a mathematical theory is very startling,
and the only experiment I have met with on the subject does not seem to
confirm it. We must next compare our theory with what is known of the
difiusion of gases, and the conduction of heat through a gas.

392 ILLUSTRATIONS OF THE DYNAMICAI. THEORY OF GASES.

PART II.
* On the Process of Diffusion of two or more kinds of moving particles

AMONG one AI^OTHER.

We have shewn, in the first part of this paper, that the motions of a
system of many small elastic particles are of two kinds : one, a general motion
of translation of the whole system, which may be called the motion in mass;
and the other a motion of agitation, or molecular motion, in virtue of which
velocities in all directions are distributed among the particles according to a
certain law. In the cases we are considering, the collisions are so frequent that
the law of distribution of the molecular velocities, if disturbed in any way,
will be re-established in an inappreciably short time; so that the motion will
always consist of this definite motion of agitation, combined with the general
motion of translation.

When two gases are in communication, streams of the two gases might
run freely in opposite directions, if it were not for the collisions which take
place between the particles. The rate at which they actually interpenetrate each
other must be investigated. The diffusion is due partly to the spreading of the
particles by the molecular agitation, and partly to the actual motion of the
two opposite currents in mass, produced by the pressure behind, and resisted

* [The methods and results of this paper have been criticised by Clausius in a memoir published
in PoggendorflTs Anncden, VoL cxv., and in the Philosophical Magazine, Vol xxiiL His main objec-
tion is that the various circumstances of the strata, discussed in the paper, have not been sufficiently
represented in the equations. In particular, if there be a series of strata at different temperatures
perpendicular to the axis of x, then the proportion of molecules whose directions form with the
axis of X angles whose cosines lie between /a and /i + <?/x is not \dfj. sa has been assumed by Maxwell
throughout his work, but \Hdfi. where £f is a factor to be determined. In discussing the steady
conduction of heat through a gas Clausius assumes that, in addition to the velocity attributed to
the molecule according to Maxwell's theory, we must also suppose a velocity normal to the stratum
and depending on the temperature of the stratum. On this assumption the factor H is iuA'estigated
along with other modifications, and an expression for the assumed velocity is determined from the
consideration that when the flow of heat is steady there is no movement of the mass. Clausius
combining his own results with those of Maxwell points out that the expression contained in (28)
of the paper involves as a result the motion of the gas. He also disputes the accuracy of ex-
pression (59) for the Conduction of Heat. In the introduction to the memoir published in the
Phil Trans., 1866, it will be found that Maxwell expresses dissatisfaction with his former theory
of the Diffusion of Gases, and admits the force of the objections made by Clausius to his expression
for the Conduction of Heat. Ed.l

ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GABES. 393

by the collisions of the opposite stream. When the densities are equal, the
diffusions due to these two causes respectively are as 2 to 3.

Prop. XIV. In a system of particles whose density, velocity, &c. are
functions of x, to find the quantity of matter transferred across the plane of yz,
due to the motion of agitation alone.

If the number of particles, their velocity, or their length of path is greater
on one side of this plane than on the other, then more particles will cross the
plane in one direction than in the other ; and there will be a transference of
matter across the plane, the amount of which may be calculated.

Let there be taken a stratum whose thickness is dx, and
area unity, at a distance x from the origin. The number of
collisions taking place in this stratum in unit of time will be

Njdx. '^^

The proportion of these which reach a distance between nl and {n-^dn)l before

they strike another particle is

e"" dji.

The proportion of these which pass through the plane yz is

nl + x

2nl

when X is between —nl and 0,

and ^r-T- when x is between and + nl ;

2nl

the sign being negative in the latter case, because the particles cross the plane
in the negative direction. The mass of each particle is M ; so that the quantity
of matter which is projected from the stratum dx, crosses the plane yz in. a.
positive direction, and strikes other particles at distances between nl and

(n + dn) I is

MNvlxTnl) J _„, ,^-s
2^^ -dxe ""dn (26),

where x must be between ±nl, and the upper or lower sign is to be taken
according as x is positive or negative.

In integrating this expression, we must remember that N, v, and I are
functions of x, not vanishing with x, and of which the variations are very
small between the limits x= —nl and x= +nl.

VOL. L 50

394 ILLUSTBATIONS OF THE DYNAMICAL THEORY OF GASES.

As we may have occasion to perform similar integrations, we may state
here, to save trouble, that if U and r are functions of x not vanishing with x,
whose variations are very small between the limits x= +r and x= —r,

/>^^^ = sf2^(^'"") (^^)-

When m is an odd number, the upper sign only is to be considered;
when m is even or zero, the upper sign is to be taken with positive values
of X, and the lower with negative values. Applying this to the case before us,

We have now to integrate

n being taken from to oo . We thus find for the quantity of matter trans-
ferred across unit of area by the motion of agitation in unit of time,

«=-*s('"'^) (^^)'

where p = MN is the density, v the mean velocity of agitation, and I the mean
length of path.

Prop. XV. The quantity transferred, in consequence of a mean motion of
translation V, would obviously be

Q^Vp (29).

Prop. XVI. To find the resultant dynamical effect of all the collisions
which take place in a given stratum.

Suppose the density and velocity of the particles to be functions of x,
then more particles will be thrown into the given stratum from that side
on which the density is greatest ; and those particles which have greatest
velocity will have the greatest effect, so that the stratum will not be generally

ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 395

in equilibrium, and the dynamical measure of the force exerted on the stratum
will be the resultant momentum of all the particles which lodge in it during
unit of time. We shall first take the case in which there is no mean motion
of translation, and then consider the effect of such motion separately.

Let a stratum whose thickness is a (a small quantity
compared with I), and area unity, be taken at the origin,
perpendicular to the axis of x ; and let another stratum, of
thickness dx, and area unity, be taken at a distance x from
the first.

If M^ be the mass of a particle, N the number in unit of volume, v the
velocity of agitation,- I the mean length of path, then the number of collisions
which take place in the stratum dx is

Njdx,

The proportion of these which reach a distance between n/ and (n + dn) I is

e"" dn.

The proportion of these which have the extremities of their paths in the
stratum a is

a

2nl'

The velocity of these particles, resolved in the direction of x, is

vx
^nl'

and the mass is M ; so that multiplying all these terms together, we get

NMv'ax _„ , J /„-.x

-2^?^' ''^''" <3°>

for the momentum of the particles fulfilling the above conditions.

To get the whole momentum, we must first integrate with respect to x
from x= —nl to x = + nl, remembering that I may be a function of x, and is a
very small quantity. The result is

50-2

396 ILLUSTRATIONS OF THE DYNAMICAL ISHEORY OF GASES.

Integrating with respect to n from n = to n = co , the result is

-4A^>^^ ^^^>

as the -whole resultant force on the stratum a arising from these collisions,
jyjow =p by Prop. XII., and therefore we may write the equation

dp
the ordinary hydrodynamical equation.

-1=^" (^^)'

Prop. XVII. To Jind the resultant effect of the collisions upon each of
several different systems of particles mixed together.

Let M^, Mj, &c. be the masses of the different kinds of particles, N„
N,, &c. the number of each kind in unit of volume, v^, v^, &c. their velocities
of agitation, Z,, l^ their mean paths, p^, p^, &c. the pressures due to each
system of particles ; then

J = Ap^ + Bp^ + &c.
\=Cp, + Dp, + kc.

(33).

The number of collisions of the first kind of particles with each other in unit
of time will be

N{OiAp^.

The number of collisions between particles of the first and second kinds will be

N{o^Bp^, or N^vJJp^y because v^B=v*C.

The number of colHsions between particles of the second kind will be
N^vJ)pi, and so on, if there are more kinds of particles.

Let us now consider a thin stratum of the mixture whose volume is unity.

The resultant momentum of the particles of the first kind which lodge in
it during unit of time is

dx '

ILLU8TRA.TI0NS OF THE DYNAMICAL THEORY OF GASES. 397

The proportion of these which strike particles of the first kind is

The whole momentum of these remains among the particles of the first kind.
The proportion wliich strike particles of the second kind is

BpA.

The momentum of these is divided between the striking particles in the ratio

M

of their masses ; so that p^ — W of the whole goes to particles of the first

M

kind, and -^t^ — ^^, to particles of the second kind.

Jtf 1 + M, ^

The effect of these collisions is therefore to produce a force

on particles of the first system, and

on particles of the second system.

The effect of the collisions of those particles of the second system whic^i
strike into the stratum, is to produce a force

on the first system, and

on the second.

The whole effect of these collisions is therefore to produce a resultant force

- 1 (^M.^M ^) - 1 W.^/^c (3.)

on the first system,

on the second, and so on.

398 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

Prop. XVIII. To find the mechanical effect of a difference in the mean
velocity of translation of two systems of moving particles.

Let F,, Fj be the mean velocities of translation of the two systems

MM
respectively, then ^ ' ' ( Fj — Fj) is the mean momentum lost by a particle

of the first, and gained by a particle of the second at collision. The number
of such collisions in unit of volume is

NjBp^v,, or N^Cp^v,;
therefore the whole effect of the collisions is to produce a force

= -^'^''="-]^^. ('"■-'"•) (*«)

on the first system, and an equal and opposite force

= +^=C'p.t..-^^^ (F.- V,) (37)

on unit of volume of the second system.

Prop. XIX. To find the law of diffusion in the case of two gases diffu^ng
into each other through a plug made of a porous material, as in the case of
the experiments of Graham.

The pressure on each side of the plug being equal, it was found by Graham
that the quantities of the gases which passed in opposite directions through the
plug in the same time were directly as the square roots of their specific gravities.

We may suppose the action of the porous material to be similar to that
of a number of particles fixed in space, and obstructing the motion of the
particles of the moving systems. If Z, is the mean distance a particle of the
first kind would have to go before striking a fixed particle, and L^ the distance
for a particle of the second kind, then the mean paths of particles of each
kind will be given by the equations

J = ^^, + -Bp, + i, l = Cp, + Z>^, + -i (38).

The mechanical effect upon the plug of the pressures of the gases on each side,
and of the percolation of the gases through it, may be found by Props. XVII.
and XVIII. to be

M,N,v,V, ^ MJs[,v,V, dp, I dp, k^^ ,3^.

L, Zj dx Li dx L.i

ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 399

and this must be zero, if the pressures are equal on each side of the plug.
Now if Q,, Qj be the quantities transferred through the plug by the mean
motion of translation, ^, = PiV, = J/jiV, F, ; and since by Graham's law

we shall have

M^N{Ui Fi = - MJSf^i\ F, = Z7 suppose ;

and since the pressures on the two sides are equal, -p= ~~j^> ^^^ ^^® ^^^7

way in which the equation of equilibrium of the plug can generally subsist is
when L^ = L^ and l^ = ly This implies that A = C and B = D. Now we know

that ViB = v*C. Let K=^ —., then we shall have

A = C=^Kv,\ B = D = ^Kv^ (40),

and i=i=K{v,p, + i\p,)^-j^ (41).

The diffusion is due partly to the motion of translation, and partly to that of
agitation. Let us find the part due to the motion of translation-

The equation of motion of one of the gases through the plug is found by
adding the forces due to pressures to those due to resistances, and equating
these to the moving force, which in the case of slow motions may be neglected
altogether. The result for jthe first is

dx

(^M+^M^^j + fcpA^li,,

+ ^-^'''*'' -^k (^■- ^=)+ -i-' = '> (*2).

Making use of the simplifications we have just discovered, this becomes

^ ^^ {v,% + v:p:) + K -^, (p,v, +p,v,) U + yU (43),

whence l^= -^ ia(v,^p,^v,%)

A^iVj {p^V^ +i?aVi) + f~

400 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

whence the rate of diffusion due to the motion of translation may be found ; for

(?. = J, andft=-J (45).

To find the difiusion due to the motion of agitation, we must find the
value of q^.

L d p.

V, dx 1+ KL (v,p^ + v^p,) '

^' - .t1I^i+^^^(^^-^^^» ('')•

SimHarly, q,= + l^^{l+KLi^{p,+p:)} (47).

The whole diffusions are Q^ + q, and Q, + q,. The values of q, and q, have a

term not following Graham's law of the square roots of the specific gravities,

but following the law of equal volumes. The closer the material of the plug,
the less will this term affect the result.

Our assumptions that the porous plug acts like a system of fixed particles,
and that Graham's law is fulfilled more accurately the more compact the
material of the plug, are scarcely sufficiently well verified for the foundation of
a theory of gases ^ and even if we admit the original assumption that they are
systems of moving elastic particles, we have not very good evidence as yet for
the relation among the quantities A, B, C, and D.

Prop. XX. To find the rate of diffusion between two vessels connected hy a
tube.

When diffusion takes place through a large opening, such as a tube con-
necting two vessels, the question is simplified by the absence- of the porous
diffusion plug; and since the pressure is constant throughout the apparatus, the
volumes of the two gases passing opposite ways through the tube at the same
time must be equal Now the quantity of gas which passes through the tube
is due partly to the motion of agitation as in Prop. XIV., and partly to the
mean motion of translation as in Prop. XV.

ILLUSTRATIONS OF THE DYNAMICAL THEORY OF OASES. 401

Let US suppose the volumes of the two vessels to be a and h, and the
length of the tube between them c, and its trans-
verse section s. Let a be filled with the first gas, /^ * ^ /^
and h with the second at the commencement of
the experiment, and let the pressure throughout
the apparatus be P.

Let a volume y of the first gas pass from a to 6, and a volume y of the
second pass from h to a \ then if p, and p^ represent the pressures in a. due
to the first and second kinds of gas, and p\ and p\ the same in the vessel h,

r>='±^:yp r)=y-P r>'=y-P V'^—^P {i%\

Since there is still equilibrium,

which gives y = y and p^ +^, = P =p\ ■\-p„ (49).

The rate of diffusion will be +-^ for the one gas, and —-— for the other,
measured in volume of gas at pressure P.

Now the rate of diflfusion of the first gas will be

dji_^iji,±pj,_^-±yp^'^^^^

dt~' p -' — p — (50)'

and that of the second,

-di=' p (='i)-

We have also the equation, derived from Props. XVI. and XVIL,

^ {Ap,l, (M, + if,) + BplM, - CpJ^M} + Bp,p,vM{ F. - F,) = (52).

From these three equations we can eliminate F, and V., and find -^ in

ift

terms of p and -j- , so that we may w^rite

S=/(^"S) (-)•

VOL. I. 51

402 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

Since the capacity of the tube is small compared with that of the vessels,

we may consider -^ constant through the whole length of the tube. "We may

then solve the differential equation in p and x; and then making p=Pi when
x = 0, and p=Pi when x = c, and substituting for p^ and p\ their values in
terms of y, we shall have a differential equation in y and t, which being solved,
will give the amount of gas diffused in a given time.

The solution of these equations would be difficult unless we assume rela-
tions among the quantities Ay B, C, D, which are not yet sufficiently estab-
lished in the case of gases of different density. Let us suppose that in a
particular case the two gases have the same density, and that the four quan-
tities A, B, Cy D are all equal.

The volume diffused, owing to the motion of agitation of the particles, is
then

3 P dx ''''
and that due to the motion of translation, or the interpenetration of the two

gases in opposite streams, is

5 dp kl
P dx V '

The values of v are distributed according to the law of Prop. IV., so that

the mean value oi v is -i^ , and that of - is -7=- , that of k being \a^. The
VTT V Vira

diffusions due to these two causes are therefore in the ratio of 2 to 3, and

their sum is

dy _ ^ J2k si dp , .

di-~^s]~^Pdx ^^^^•

If we suppose -^ constant throughout the tube, or, in other words, if we
regard the motion as steady for a short time, then -r- will be constant and
equal to — — —\ or substituting from (48),

ah ,, ~t^

(a+6)^

whence y = — /(I— e"" "*** ) (56).

ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 403

By choosing pairs of gases of equal density, and ascertaining the amount
of diffusion in a given time, we might determine the value of I in this expres-
sion. The diffusion of nitrogen into carbonic oxide or of deutoxide of nitrogen
into carbonic acid, would be suitable cases for experiment. The only existing
experiment which approximately fulfils the conditions is one by Graham, quoted
by Herapath from Brande's Quarterly Journal of Science, Vol. xviiL p. 7Q.

A tube 9 inches long and 0*9 inch diameter, communicated with the
atmosphere by a tube 2 inches long and 0'12 inch diameter; 152 parts of
olefiant gas being placed in the tube, the quantity remaining after four hours
was 9 9. parts.

In this case there is not much difference of specific gravity between the

and we have a = 9 x (0'9)'' - cubic inches, 2^=00, c = 2 inches, and

(0*12)' - square inches;

^^ log. 10.^. log.. (^^) (57);

.-. ^ = 0-00000256 inch =39^000 i"ch (58).

Prop. XXI. To Jind the amount of energy which crosses unit of area in
unit of time when the velocity of agitation is greater on one side of the area
than on the other.

The energy of a single particle is composed of two parts, — the vis viva
of the centre of gravity, and the vis viva of the various motions of rotation
round that centre, or, if the particle be capable of internal motions, the vis
viva of these. We shall suppose that the whole vis viva bears a constant
proportion to that due to the motion of the centre of gravity, or

where )8 is a coefficient, the experimental value of which is 1*634. Substituting
E for Ji" in Prop. XIV., we get for the transference of energy across unit
of area in unit of time,

51—2

404 ILLUSTRATIONS OF THE DYNAMICAI, THEORY OF GASES.

where J is the mechanical equivalent of heat in foot-pounds, and q[ is the
transfer of heat in thermal units.

Now MN=p, and l = -i-, so that MNl = -. ;
'^^ Ap A

••••^^=-*'^l (-)■

Also, if T is the absolute temperature,

1 dT^2dv^^
T dx~ V dx'

.■.Jq= -ify.lv ^"^ (60),

where p must be measured in dynamical units of force.

Let J =772 foot-pounds, _p = 2116 pounds to square foot, ^ = 4:ooVoo i^^^^'
v=1505 feet per second, T=522 or 62" Fahrenheit; then

2=;« (">'

where q is the flow of heat in thermal units per square foot of area ; and T'
and T are the temperatures at the two sides of a stratum of air x inches thick.

In Prof. Rankine's work on the Steam-engine, p. 259, values of the thennal
resistance, or the reciprocal of the conductivity, are given for various substances
as computed from a Table of conductivities deduced by M. Peclet from experi-
ments by M. Despretz : —

Resistance.

Gold, Platinum, Silver 0-0036

Copper 0-0040

Iron 0-0096