Chapter XIX: Part 19

A further property based on experience is that the motions set up in a mixture by diffusion are very slow compared with those set up by mechanical actions, such as differences of pressure. Thus, if two gases at equal temperature and pressure be allowed to mix by diffusion, the heavier gas being below the lighter, the process will take a long time; on the other hand, if two gases, or parts of the same gas, at different pressures be connected, equalization of pressure will take place almost immediately. It follows from this property that the forces required to overcome the "inertia" of the fluids in the motions due to diffusion are quite imperceptible. At any stage of the process, therefore, any one of the diffusing fluids may be regarded as in equilibrium under the action of its own partial pressure, the external forces to which it is subjected and the resistance to diffusion of the other fluids.

5. _Slow Diffusion of two Gases. Relation between the Coefficients of Resistance and of Diffusion._--We now suppose the diffusing substances to be two gases which obey Boyle's law, and that diffusion takes place in a closed cylinder or tube of unit sectional area at constant temperature, the surfaces of equal density being perpendicular to the axis of the cylinder, so that the direction of diffusion is along the length of the cylinder, and we suppose no external forces, such as gravity, to act on the system.

The densities of the gases are denoted by [rho]1, [rho]2, their
velocities of diffusion by u1, u2, and if their partial pressures are
p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where
k1, k2 are constants for the two gases, the temperature being constant.
The axis of the cylinder is taken as the axis of x.

From the considerations of the preceding section, the effects of
inertia of the diffusing gases may be neglected, and at any instant of
the process either of the gases is to be treated as kept in
equilibrium by its partial pressure and the resistance to diffusion
produced by the other gas. Calling this resistance per unit volume R,
and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of
resistance, the equations of equilibrium give

dp1 dp2
--- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1).
dx dx

These involve

dp1 dp2
--- + --- = 0 or p1 + p2 = P (2)
dx dx

where P is the total pressure of the mixture, and is everywhere
constant, consistently with the conditions of mechanical equilibrium.

Now dp1/dx is the pressure-gradient of the first gas, and is, by
Boyle's law, equal to k1 times the corresponding density-gradient.
Again [rho]1u1 is the mass of gas flowing across any section per unit
time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux
of partial pressure produced by the motion of the gas. Since the total
pressure is everywhere constant, and the ends of the cylinder are
supposed fixed, the fluxes of partial pressure due to the two gases
are equal and opposite, so that

p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0 (3).

From (2) (3) we find by elementary algebra

u1/p2 = - u2/p1 = (u1 - u2)/(p1 + p2) = (u1 - u2)/P,

and therefore

p2u1 = - p2u2 = p1p2(u1 - u2)/P = k1k2[rho]1[rho]2(u1 - u2)/P

Hence equations (1) (2) gives

dp1 CP dp2 CP
--- + ---- (p1u1) = 0, and --- + ---- (p2u2) = 0;
dx k1k2 dx k1k2

whence also substituting p1 = k1[rho]1, p2 = k2[rho]2, and by
transposing

k1k2 d[rho]1 k1k2 d[rho]2
[rho]1u1 = - ---- -------, and [rho]2u2 = - ---- -------.
CP dx CP dx

We may now define the "coefficient of diffusion" of either gas as the
ratio of the rate of flow of that gas to its density-gradient. With
this definition, the coefficients of diffusion of both the gases in a
mixture are equal, each being equal to k1k2/CP. The ratios of the
fluxes of partial pressure to the corresponding pressure-gradients are
also equal to the same coefficient. Calling this coefficient K, we
also observe that the equations of continuity for the two gases are

d[rho]1 d([rho]1u1) d[rho]2 d([rho]2u2)
------- + ----------- = 0, and ------- + ----------- = 0,
dt dx dt dx

leading to the equations of diffusion

d[rho]1 d / d[rho]1\ d[rho]2 d / d[rho]2\
------- = -- ( K ------- ) , and ------- = -- ( K ------- ),
dt dx \ dx / dt dx \ dx /

exactly as in the case of diffusion through a solid.

If we attempt to treat diffusion in liquids by a similar method, it is, in the first place, necessary to define the "partial pressure" of the components occurring in a liquid mixture. This leads to the conception of "osmotic pressure," which is dealt with in the article SOLUTION. For dilute solutions at constant temperature, the assumption that the osmotic pressure is proportional to the density, leads to results agreeing fairly closely with experience, and this fact may be represented by the statement that a substance occurring in a dilute solution behaves like a perfect gas.

6. _Relation of the Coefficient of Diffusion to the Units of Length and Time._--We may write the equation defining K in the form

I d[rho]
u = -K X ----- ------.
[rho] dx

Here -d[rho]/[rho]dx represents the "percentage rate" at which the density decreases with the distance x; and we thus see that the coefficient of diffusion represents the ratio of the velocity of flow to the percentage rate at which the density decreases with the distance measured in the direction of flow. This percentage rate being of the nature of a number divided by a length, and the velocity being of the nature of a length divided by a time, we may state that K is of two dimensions in length and - 1 in time, i.e. dimensions L^2/T.

_Example 1._ Taking K = 0.1423 for carbon dioxide and air (at
temperature 0 deg. C. and pressure 76 cm. of mercury) referred to a
centimetre and a second as units, we may interpret the result as
follows:--Supposing in a mixture of carbon dioxide and air, the
density of the carbon dioxide decreases by, say, 1, 2 or 3% of itself
in a distance of 1 cm., then the corresponding velocities of the
diffusing carbon dioxide will be respectively 0.01, 0.02 and 0.03
times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. per second
in the three cases.

_Example 2._ If we wished to take a foot and a second as our units, we
should have to divide the value of the coefficient of diffusion in
Example 1 by the square of the number of centimetres in 1 ft., that
is, roughly speaking, by 900, giving the new value of K = 0.00016
roughly.

7. _Numerical Values of the Coefficient of Diffusion._--The table on p. 258 gives the values of the coefficient of diffusion of several of the principal pairs of gases at a pressure of 76 cm. of mercury, and also of a number of other substances. In the gases the centimetre and second are taken as fundamental units, in other cases the centimetre and day.

8. _Irreversible Changes accompanying Diffusion._--The diffusion of two gases at constant pressure and temperature is a good example of an "irreversible process." The gases always tend to mix, never to separate. In order to separate the gases a change must be effected in the external conditions to which the mixture is subjected, either by liquefying one of the gases, or by separating them by diffusion through a membrane, or by bringing other outside influences to bear on them. In the case of liquids, electrolysis affords a means of separating the constituents of a mixture. Every such method involves some change taking place outside the mixture, and this change may be regarded as a "compensating transformation." We thus have an instance of the property that every irreversible change leaves an indelible imprint somewhere or other on the progress of events in the universe. That the process of diffusion obeys the laws of irreversible thermodynamics (if these laws are properly stated) is proved by the fact that the compensating transformations required to separate mixed gases do not essentially involve anything but transformation of energy. The process of allowing gases to mix by diffusion, and then separating them by a compensating transformation, thus constitutes an irreversible cycle, the outside effects of which are that energy somewhere or other must be less capable of transformation than it was before the change. We express this fact by stating that an irreversible process essentially implies a loss of availability. To measure this loss we make use of the laws of thermodynamics, and in particular of Lord Kelvin's statement that "It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects."

+-------------------------------------------+------------+---------------------+--------------+
| Substances. | Temp. | K. | Author. |
+-------------------------------------------+------------+---------------------+--------------+
| Carbon dioxide and air | 0 deg.C.| 0.1423 cm^2/sec. | J. Loschmidt.|
| " " hydrogen | 0 deg.C.| 0.5558 " | " |
| " " oxygen | 0 deg.C.| 0.1409 " | " |
| " " carbon monoxide | 0 deg.C.| 0.1406 " | " |
| " " marsh gas (methane) | 0 deg.C.| 0.1586 " | " |
| " " nitrous oxide | 0 deg.C.| 0.0983 " | " |
| Hydrogen and oxygen | 0 deg.C.| 0.7214 " | " |
| " " carbon monoxide | 0 deg.C.| 0.6422 " | " |
| " " sulphur dioxide | 0 deg.C.| 0.4800 " | " |
| Oxygen and carbon monoxide | 0 deg.C.| 0.1802 " | " |
| Water and ammonia | 20 deg.C.| 1.250 " | G. Hufner. |
| " " | 5 deg.C.| 0.822 " | " |
| " common salt (density 1.0269) | | 0.355 cm^2/hour. | J. Graham. |
| " " " " |14.33 deg.C.| 1.020, 0.996, 0.972,| " |
| | | 0.932 cm^2/day. | F. Heimbrodt.|
| " zinc sulphate (0.312 gm/cm^3) | | 0.1162 " | W. Seitz. |
| " zinc sulphate (normal) | | 0.2355 " | " |
| " zinc acetate (double normal) | | 0.1195 " | " |
| " zinc formate (half normal) | | 0.4654 " | " |
| " cadmium sulphate (double normal)| | 0.2456 " | " |
| " glycerin (1/8n, 1/2n, |10.14 deg.C.| 0.356, 0.350, 0.342,| F. Heimbrodt.|
| 7/8n, 7/8n) | | 0.315 cm^2/day. | " |
| " urea " " |14.83 deg.C.| 0.973, 0.946, 0.926,| " |
| | | 0.883 cm^2/day. | " |
| " hydrochloric acid |14.30 deg.C.| 2.208, 2.331, | " |
| | | 2.480 cm^2/day | " |
| Gelatin 20% and ammonia | 17 deg.C.| 127.1 " | A. Hagenbach.|
| " " carbon dioxide | | 0.845 " | " |
| " " nitrous oxide | | 0.509 " | " |
| " " oxygen | | 0.230 " | " |
| " " hydrogen | | 0.0565 " | " |
+-------------------------------------------+------------+---------------------+--------------+

Let us now assume that we have any syste m such as the gases above
considered, and that it is in the presence of an indefinitely extended
medium which we shall call the "auxiliary medium." If heat be taken
from any part of the system, only part of this heat can be converted
into work by means of thermodynamic engines; and the rest will be
given to the auxiliary medium, and will constitute unavailable energy
or waste. To understand what this means, we may consider the case of a
condensing steam engine. Only part of the energy liberated by the
combustion of the coal is available for driving the engine, the rest
takes the form of heat imparted to the condenser. The colder the
condenser the more efficient is the engine, and the smaller is the
quantity of waste.

The amount of unavailable energy associated with any given
transformation is proportional to the absolute temperature of the
auxiliary medium. When divided by that temperature the quotient is
called the change of "entropy" associated with the given change (see
THERMODYNAMICS). Thus if a body at temperature T receives a quantity
of heat Q, and if T0 is the temperature of the auxiliary medium, the
quantity of work which could be obtained from Q by means of ideal
thermodynamic engines would be Q(1 - T0/T), and the balance, which is
QT0/T, would take the form of unavailable or waste energy given to the
medium. The quotient of this, when divided by T0, is Q/T, and this
represents the quantity of entropy associated with Q units of heat at
temperature T.

Any irreversible change for which a compensating transformation of
energy exists represents, therefore, an increase of unavailable
energy, which is measurable in terms of entropy. The increase of
entropy is independent of the temperature of the auxiliary medium. It
thus affords a measure of the extent to which energy has run to waste
during the change. Moreover, when a body is heated, the increase of
entropy is the factor which determines how much of the energy imparted
to the body is unavailable for conversion into work under given
conditions. In all cases we have

increase of unavailable energy
------------------------------- = increase of entropy.
temperature of auxiliary medium

When diffusion takes place between two gases inside a closed vessel at
uniform pressure and temperature no energy in the form of heat or work
is received from without, and hence the entropy gained by the gases
from without is zero. But the irreversible processes inside the vessel
may involve a gain of entropy, and this can only be estimated by
examining by what means mixed gases can be separated, and, in
particular, under what conditions the process of mixing and separating
the gases could (theoretically) be made reversible.

9. _Evidence derived from Liquefaction of one or both of the Gases._--The gases in a mixture can often be separated by liquefying, or even solidifying, one or both of the components. In connexion with this property we have the important law according to which "The pressure of a vapour in equilibrium with its liquid depends only on the temperature and is independent of the pressures of any other gases or vapours which may be mixed with it." Thus if two closed vessels be taken containing some water and one be exhausted, the other containing air, and if the temperatures be equal, evaporation will go on until the pressure of the vapour in the exhausted vessel is equal to its _partial_ pressure in the other vessel, notwithstanding the fact that the _total_ pressure in the latter vessel is greater by the pressure of the air.

To separate mixed gases by liquefaction, they must be compressed and
cooled till one separates in the form of a liquid. If no changes are
to take place outside the system, the separate components must be
allowed to expand until the work of expansion is equal to the work of
compression, and the heat given out in compression is reabsorbed in
expansion. The process may be made as nearly reversible as we like by
performing the operations so slowly that the substances are
practically in a state of equilibrium at every stage. This is a
consequence of an important axiom in thermodynamics according to which
"any small change in the neighbourhood of a state of equilibrium is to
a first approximation reversible."

Suppose now that at any stage of the compression the partial pressures
of the two gases are p1 and p2, and that the volume is changed from V
to V - dV. The work of compression is (p1 + p2)dV, and this work will
be restored at the corresponding stage if each of the separated gases
increases in volume from V - dV to V. The ultimate state of the
separated gases will thus be one in which each gas occupies the volume
V originally occupied by the mixture.

We may now obtain an estimate of the amount of energy rendered
unavailable by diffusion. We suppose two gases occupying volumes V1
and V2 at equal pressure p to mix by diffusion, so that the final
volume is V1 + V2. Then if before mixing each gas had been allowed to
expand till its volume was V1 + V2, work would have been done in the
expansion, and the gases could still have been mixed by a reversal of
the process above described. In the actual diffusion this work of
expansion is lost, and represents energy rendered unavailable at the
temperature at which diffusion takes place. When divided by that
temperature the quotient gives the increase of entropy. Thus the
irreversible processes, and, in particular, the entropy changes
associated with diffusion of two gases at uniform pressure, are the
same as would take place if each of the gases in turn were to expand
by rushing into a vacuum, till it occupied the whole volume of the
mixture. A more rigorous proof involves considerations of the
thermodynamic potentials, following the methods of J. Willard Gibbs
(see ENERGETICS).

Another way in which two or more mixed gases can be separated is by
placing them in the presence of a liquid which can freely absorb one
of the gases, but in which the other gas or gases are insoluble. Here
again it is found by experience that when equilibrium exists at a
given temperature between the dissolved and undissolved portions of
the first gas, the partial pressure of that gas in the mixture depends
on the temperature alone, and is independent of the partial pressures
of the insoluble gases with which it is mixed, so that the conclusions
are the same as before.

10. _Diffusion through a Membrane or Partition. Theory of the semi-permeable Membrane._--It has been pointed out that diffusion of gases frequently takes place in the interior of solids; moreover, different gases behave differently with respect to the same solid at the same temperature. A membrane or partition formed of such a solid can therefore be used to effect a more or less complete separation of gases from a mixture. This method is employed commercially for extracting oxygen from the atmosphere, in particular for use in projection lanterns where a high degree of purity is not required. A similar method is often applied to liquids and solutions and is known as "dialysis."

In such cases as can be tested experimentally it has been found that a gas always tends to pass through a membrane from the side where its density, and therefore its partial pressure, is greater to the side where it is less; so that for equilibrium the partial pressures on the two sides must be equal. This result is unaffected by the presence of other gases on one or both sides of the membrane. For example, if different gases at the same pressure are separated by a partition through which one gas can pass more rapidly than the other, the diffusion will give rise to a difference of pressure on the two sides, which is capable of doing mechanical work in moving the partition. In evidence of this conclusion Max Planck quotes a test experiment made by him in the Physical Institute of the university of Munich in 1883, depending on the fact that platinum foil at white heat is permeable to hydrogen but impermeable to air, so that if a platinum tube filled with hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.

The details of the experiment may be quoted here:--"A glass tube of
about 5 mm. internal diameter, blown out to a bulb at the middle, was
provided with a stop-cock at one end. To the other a platinum tube 10
cm. long was fastened, and closed at the end. The whole tube was
exhausted by a mercury pump, filled with hydrogen at ordinary
atmospheric pressure, and then closed. The closed end of the platinum
portion was then heated in a horizontal position by a Bunsen burner.
The connexion between the glass and platinum tubes, having been made
by means of sealing-wax, had to be kept cool by a continuous current
of water to prevent the softening of the wax. After four hours the
tube was taken from the flame, cooled to the temperature of the room,
and the stop-cock opened under mercury. The mercury rose rapidly,
almost completely filling the tube, proving that the tube had been
very nearly exhausted."

In order that diffusion through a membrane may be reversible so far as a particular gas is concerned, the process must take place so slowly that equilibrium is set up at every stage (see S 9 above). In order to separate one gas from another consistently with this condition it is necessary that no diffusion of the latter gas should accompany the process. The name "semi-permeable" is applied to an ideal membrane or partition through which one gas can pass, and which offers an insuperable barrier to any diffusion whatever of a second gas. By means of two semi-permeable partitions acting oppositely with respect to two different gases A and B these gases could be mixed or separated by reversible methods. The annexed figure shows a diagrammatic representation of the process.

We suppose the gases contained in a cylindrical tube; P, Q, R, S are
four pistons, of which P and R are joined to one connecting rod, Q and
S to another. P, S are impermeable to both gases; Q is semi-permeable,
allowing the gas A to pass through but not B, similarly R allows the
gas B to pass through but not A. The distance PR is equal to the
distance QS, so that if the rods are pushed towards each other as far
as they will go, P and Q will be in contact, as also R and S. Imagine
the space RQ filled with a mixture of the two gases under these
conditions. Then by slowly drawing the connecting rods apart until R,
Q touch, the gas A will pass into the space PQ, and B will pass into
the space RS, and the gases will finally be completely separated;
similarly, by pushing the connecting rods together, the two gases will
be remixed in the space RQ. By performing the operations slowly enough
we may make the processes as nearly reversible as we please, so that
no available energy is lost in either change. The gas A being at every
instant in equilibrium on the two sides of the piston Q, its density,
and therefore its partial pressure, is the same on both sides, and the
same is true regarding the gas B on the two sides of R. Also _no work
is done in moving the pistons_, for the partial pressures of B on the
two sides of R balance each other, consequently, the resultant thrust
on R is due to the gas A alone, and is equal and opposite to its
resultant thrust on P, so that the connecting rods are at every
instant in a state of mechanical equilibrium so far as the pressures
of the gases A and B are concerned. We conclude that in the reversible
separation of the gases by this method at constant temperature without
the production or absorption of mechanical work, the densities and the
partial pressures of the two separated gases are the same as they were
in the mixture. These conclusions are in entire agreement with those
of the preceding section. If this agreement did not exist it would be
possible, theoretically, to obtain perpetual motion from the gases in
a way that would be inconsistent with the second law of
thermodynamics.

Most physicists admit, as Planck does, that it is impossible to obtain an ideal semi-permeable substance; indeed such a substance would necessarily have to possess an infinitely great resistance to diffusion for such gases as could not penetrate it. But in an experiment performed under actual conditions the losses of available energy arising from this cause would be attributable to the imperfect efficiency of the partitions and not to the gases themselves; moreover, these losses are, in every case, found to be completely in accordance with the laws of irreversible thermodynamics. The reasoning in this article being somewhat condensed the reader must necessarily be referred to treatises on thermodynamics for further information on points of detail connected with the argument. Even when he consults these treatises he may find some points omitted which have been examined in full detail at some time or other, but are not sufficiently often raised to require mention in print.

II. _Kinetic Models of Diffusion._--Imagine in the first instance that a very large number of red balls are distributed over one half of a billiard table, and an equal number of white balls over the other half. If the balls are set in motion with different velocities in various directions, diffusion will take place, the red balls finding their way among the white ones, and vice versa; and the process will be retarded by collisions between the balls. The simplest model of a perfect gas studied in the kinetic theory of gases (see MOLECULE) differs from the above illustration in that the bodies representing the molecules move in space instead of in a plane, and, unlike billiard balls, their motion is unresisted, and they are perfectly elastic, so that no kinetic energy is lost either during their free motions, or at a collision.

The mathematical analysis connected with the application of the
kinetic theory to diffusion is very long and cumbersome. We shall
therefore confine our attention to regarding a medium formed of
elastic spheres as a mechanical model, by which the most important
features of diffusion can be illustrated. We shall assume the results
of the kinetic theory, according to which:--(1) In a dynamical model
of a perfect gas the mean kinetic energy of translation of the
molecules represents the absolute temperature of the gas. (2) The
pressure at any point is proportional to the product of the number of
molecules in unit volume about that point into the mean square of the
velocity. (The mean square of the velocity is different from but
proportional to the square of the mean velocity, and in the subsequent
arguments either of these two quantities can generally be taken.) (3)
In a gas mixture represented by a mixture of molecules of unequal
masses, the mean kinetic energies of the different kinds are equal.

Consider now the problem of diffusion in a region containing two kinds
of molecules A and B of unequal mass. The molecules of A in the
neighbourhood of any point will, by their motion, spread out in every
direction until they come into collision with other molecules of
either kind, and this spreading out from every point of the medium
will give rise to diffusion. If we imagine the velocities of the A
molecules to be equally distributed in all directions, as they would
be in a homogeneous mixture, it is obvious that the process of
diffusion will be greater, _ceteris paribus_, the greater the velocity
of the molecules, and the greater the length of the free path before a
collision takes place. If we assume consistently with this, that the
coefficient of diffusion of the gas A is proportional to the mean
value of Wala, where wa is the velocity and la is the length of the
path of a molecule of A, this expression for the coefficient of
diffusion is of the right dimensions in length and time. If, moreover,
we observe that when diffusion takes place in a fixed direction, say
that of the axis of x, it depends only on the resolved part of the
velocity and length of path in that direction: this hypothesis readily
leads to our taking the mean value of 1/3w_a l_a as the coefficient of
diffusion for the gas A. This value was obtained by O. E. Meyer and
others.

Unfortunately, however, it makes the coefficients of diffusion unequal
for the two gases, a result inconsistent with that obtained above from
considerations of the coefficient of resistance, and leading to the
consequence that differences of pressure would be set up in different
parts of the gas. To equalize these differences of pressure, Meyer
assumed that a counter current is set up, this current being, of
course, very slow in practice; and J. Stefan assumed that the
diffusion of one gas was not affected by collisions between molecules
of the _same gas_. When the molecules are mixed in equal proportions
both hypotheses lead to the value 1/6([w_a l_a] + [w_b l_b]), (square
brackets denoting mean values). When one gas preponderates largely
over the other, the phenomena of diffusion are too difficult of
observation to allow of accurate experimental tests being made.
Moreover, in this case no difference exists unless the molecules are
different in size or mass.

Instead of supposing a velocity of translation added after the
mathematical calculations have been performed, a better plan is to
assume from the outset that the molecules of the two gases have small
velocities of translation in opposite directions, superposed on the
distribution of velocity, which would occur in a medium representing a
gas at rest. When a collision occurs between molecules of different
gases a transference of momentum takes place between them, and the
quantity of momentum so transferred in one second in a unit of volume
gives a dynamical measure of the resistance to diffusion. It is to be
observed that, however small the relative velocity of the gases A and
B, it plays an all-important part in determining the coefficient of
resistance; for without such relative motion, and with the velocities
evenly distributed in all directions, no transference of momentum
could take place. The coefficient of resistance being found, the
motion of each of the two gases may be discussed separately.

One of the most important consequences of the kinetic theory is that if the volume be kept constant the coefficient of diffusion varies as the square root of the absolute temperature. To prove this, we merely have to imagine the velocity of each molecule to be suddenly increased n fold; the subsequent processes, including diffusion, will then go on n times as fast; and the temperature T, being proportional to the kinetic energy, and therefore to the square of the velocity, will be increased n^2 fold. Thus K, the coefficient of diffusion, varies as [sqrt]T.

The relation of K to the density when the temperature remains constant is more difficult to discuss, but it may be sufficient to notice that if the number of molecules is increased n fold, the chances of a collision are n times as great, and the distance traversed between collisions is (not _therefore_ but as the result of more detailed reasoning) on the average 1/n of what it was before. Thus the free path, and therefore the coefficient of diffusion, varies inversely as the density, or directly as the volume. If the pressure p and temperature T be taken as variables, K varies inversely as p and directly as [sqrt]T^3.

Now according to the experiments first made by J. C. Maxwell and J. Loschmidt, it appeared that with constant density K was proportional to T more nearly than to [sqrt]T. The inference is that in this respect a medium formed of colliding spheres fails to give a correct mechanical model of gases. It has been found by L. Boltzmann, Maxwell and others that a system of particles whose mutual actions vary according to the inverse fifth power of the distance between them represents more correctly the relation between the coefficient of diffusion and temperature in actual gases. Other recent theories of diffusion have been advanced by M. Thiesen, P. Langevin and W. Sutherland. On the other hand, J. Thovert finds experimental evidence that the coefficient of diffusion is proportional to molecular velocity in the cases examined of non-electrolytes dissolved in water at 18 deg. at 2.5 grams per litre.

BIBLIOGRAPHY.--The best introduction to the study of theories of
diffusion is afforded by O. E. Meyer's Kinetic _Theory of Gases_,
translated by Robert E. Baynes (London, 1899). The mathematical
portion, though sufficient for ordinary purposes, is mostly of the
simplest possible character. Another useful treatise is R. Ruhlmann's
_Handbuch der mechanischen Warmetheorie_ (Brunswick, 1885). For a
shorter sketch the reader may refer to J. C. Maxwell's _Theory of
Heat_, chaps, xix. and xxii., or numerous other treatises on physics.
The theory of the semi-permeable membrane is discussed by M. Planck
in his _Treatise on Thermodynamics_, English translation by A. Ogg
(1903), also in treatises on thermodynamics by W. Voigt and other
writers. For a more detailed study of diffusion in general the
following papers may be consulted:--L. Boltzmann, "Zur Integration der
Diffusionsgleichung," _Sitzung. der k. bayer. Akad math.-phys. Klasse_
(May 1894); T. des Coudres, "Diffusionsvorgange in einem Zylinder,"
_Wied. Ann._ lv. (1895), p. 213; J. Loschmidt,
"Experimentaluntersuchungen uber Diffusion," _Wien. Sitz._ lxi., lxii.
(1870); J. Stefan, "Gleichgewicht und ... Diffusion von Gasmengen,"
_Wien. Sitz._ lxiii., "Dynamische Theorie der Diffusion," _Wien.
Sitz._ lxv. (April 1872); M. Toepler, "Gas-diffusion," _Wied. Ann._
lviii. (1896), p. 599; A. Wretschko, "Experimentaluntersuchungen uber
die Diffusion von Gasmengen," _Wien. Sitz._ lxii. The mathematical
theory of diffusion, according to the kinetic theory of gases, has
been treated by a number of different methods, and for the study of
these the reader may consult L. Boltzmann, _Vorlesungen uber
Gastheorie_ (Leipzig, 1896-1898); S. H. Burbury, _Kinetic Theory of
Gases_ (Cambridge, 1899), and papers by L. Boltzmann in _Wien. Sitz._
lxxxvi. (1882), lxxxvii. (1883); P. G. Tait, "Foundations of the
Kinetic Theory of Gases," _Trans. R.S.E._ xxxiii., xxxv., xxvi., or
_Scientific Papers_, ii. (Cambridge, 1900). For recent work reference
should be made to the current issues of _Science Abstracts_ (London),
and entries under the heading "Diffusion" will be found in the general
index at the end of each volume. (G. H. BR.)

DIGBY, SIR EVERARD (1578-1606), English conspirator, son of Everard Digby of Stoke Dry, Rutland, was born on the 16th of May 1578. He inherited a large estate at his father's death in 1592, and acquired a considerable increase by his marriage in 1596 to Mary, daughter and heir of William Mulsho of Gothurst (now Gayhurst), in Buckinghamshire. He obtained a place in Queen Elizabeth's household and as a ward of the crown was brought up a Protestant; but about 1599 he came under the influence of the Jesuit, John Gerard, and soon afterwards joined the Roman Catholics. He supported James's accession and was knighted by the latter on the 23rd of April 1603. In a letter to Salisbury, the date of which has been ascribed to May 1605, Digby offered to go on a mission to the pope to obtain from the latter a promise to prevent Romanist attempts against the government in return for concessions to the Roman Catholics; adding that if severe measures were again taken against them "within brief there will be massacres, rebellions and desperate attempts against the king and state." Digby had suffered no personal injury or persecution on account of his religion, but he sympathized with his co-religionists; and when at Michaelmas, 1605, the government had fully decided to return to the policy of repression, the authors of the Gunpowder Plot (q.v.) sought his financial support, and he joined eagerly in the conspiracy. His particular share in the plan was the organization of a rising in the Midlands; and on the pretence of a hunting party he assembled a body of gentlemen together at Danchurch in Warwickshire on the 5th of November, who were to take action immediately the news arrived from London of the successful destruction of the king and the House of Lords, and to seize the person of the princess Elizabeth, who was residing in the neighbourhood. The conspirators arrived late on the evening of the 6th to tell their story of failure and disaster, and Digby, who possibly might have escaped the more serious charge of high treason, was persuaded by Catesby, with a false tale that the king and Salisbury were dead, to further implicate himself in the plot and join the small band of conspirators in their hopeless endeavour to raise the country. He accompanied them, the same day, to Huddington in Worcestershire and on the 7th to Holbeche in Staffordshire. The following morning, however, he abandoned his companions, dismissed his servants except two, who declared "they would never leave him but against their will," and attempted with these to conceal himself in a pit. He was, however, soon discovered and surrounded. He made a last effort to break through his captors on horseback, but was taken and conveyed a prisoner to the Tower. His trial took place in Westminster Hall, on the 27th of January 1606, and alone among the conspirators he pleaded guilty, declaring that the motives of his crime had been his friendship for Catesby and his devotion to his religion. He was condemned to death, and his execution, which took place on the 31st, in St Paul's Churchyard, was accompanied by all the brutalities exacted by the law.

Digby was a handsome man, of fine presence. Father Gerard extols his skill in sport, his "riding of great horses," as well as his skill in music, his gifts of mind and his religious devotion, and concludes "he was as complete a man in all things, that deserved estimation or might win affection as one should see in a kingdom." Some of Digby's letters and papers, which include a poem before his execution, a last letter to his infant sons and correspondence with his wife from the Tower, were published in _The Gunpowder Treason_ by Thomas Barlow, bishop of Lincoln, in 1679. He left two sons, of whom the elder, Sir Kenelm Digby, was the well-known author and diplomatist.

See works on the Gunpowder Plot; Narrative of Father Gerard, in
_Condition of the Catholics under James I._ by J. Morris (1872), &c. A
life of Digby under the title of _A Life of a Conspirator_, by a
Romish Recusant (Thomas Longueville), was published in 1895.
(P. C. Y.)

DIGBY, SIR KENELM (1603-1665), English author, diplomatist and naval commander, son of Sir Everard Digby (q.v.), was born on the 11th of July 1603, and after his father's execution in 1606 resided with his mother at Gayhurst, being brought up apparently as a Roman Catholic. In 1617 he accompanied his cousin, Sir John Digby, afterwards 1st earl of Bristol, and then ambassador in Spain, to Madrid. On his return in April 1618 he entered Gloucester Hall (now Worcester College), Oxford, and studied under Thomas Allen (1542-1632), the celebrated mathematician, who was much impressed with his abilities and called him the _Mirandula_, i.e. the infant prodigy, of his age.[1] He left the university without taking a degree in 1620, and travelled in France, where, according to his own account, he inspired an uncontrollable passion in the queen-mother, Marie de' Medici, now a lady of more than mature age and charms; he visited Florence, and in March 1623 joined Sir John Digby again at Madrid, at the time when Prince Charles and Buckingham arrived on their adventurous expedition. He joined the prince's household and returned with him to England on the 5th of October 1623, being knighted by James I. on the 23rd of October and receiving the appointment of gentleman of the privy chamber to Prince Charles. In 1625 he married secretly Venetia, daughter of Sir Edward Hanley of Tonge Castle, Shropshire, a lady of extraordinary beauty and intellectual attainments, but of doubtful virtue. Digby was a man of great stature and bodily strength. Edward Hyde, afterwards earl of Clarendon, who with Ben Jonson was included among his most intimate friends, describes him as "a man of very extraordinary person and presence which drew the eyes of all men upon him, a wonderful graceful behaviour, a flowing courtesy and civility, and such a volubility of language as surprised and delighted."[2] Digby for some time was excluded from public employment by Buckingham's jealousy of his cousin, Lord Bristol. At length in 1627, on the latter's advice, Digby determined to attempt "some generous action," and on the 22nd of December, with the approval of the king, embarked as a privateer with two ships, with the object of attacking the French ships in the Venetian harbour of Scanderoon. On the 18th of January he arrived off Gibraltar and captured several Spanish and Flemish vessels. From the 15th of February to the 27th of March he remained at anchor off Algiers on account of the sickness of his men, and extracted a promise from the authorities of better treatment of the English ships. He seized a rich Dutch vessel near Majorca, and after other adventures gained a complete victory over the French and Venetian ships in the harbour of Scanderoon on the 11th of June. His successes, however, brought upon the English merchants the risk of reprisals, and he was urged to depart. He returned home in triumph in February 1629, and was well received by the king, and was made a commissioner of the navy in October 1630, but his proceedings were disavowed on account of the complaints of the Venetian ambassador. In 1633 Lady Digby died, and her memory was celebrated by Ben Jonson in a series of poems entitled _Eupheme_, and by other poets of the day. Digby retired to Gresham College, and exhibited extravagant grief, maintaining a seclusion for two years. About this time Digby professed himself a Protestant, but by October 1635, while in France, he had already returned to the Roman Catholic faith.[3] In a letter dated the 27th of March 1636 Laud remonstrates with him, but assures him of the continuance of his friendship.[4] In 1638 he published _A Conference with a Lady about choice of a Religion_, in which he argues that the Roman Church, possessing alone the qualifications of universality, unity of doctrine and uninterrupted apostolic succession, is the only true church, and that the intrusion of error into it is impossible. The same subject is treated in letters to George Digby, afterwards 2nd earl of Bristol, dated the 2nd of November 1638 and the 29th of November 1639, which were published in 1651, as well as in a further _Discourse concerning Infallibility in Religion_ in 1652. Returning to England he associated himself with the queen and her Roman Catholic friends, and joined in the appeal to the English Romanists for money to support the king's Scottish expedition.[5] In consequence he was summoned to the bar of the House of Commons on the 27th of January 1641, and the king was petitioned to remove him with other recusants from his councils. He left England, and while at Paris killed in a duel a French lord who had insulted Charles I. in his presence. Louis XIII. took his part, and furnished him with a military escort into Flanders. Returning home he was imprisoned, by order of the House of Commons, early in 1642, successively in the "Three Tobacco Pipes nigh Charing Cross," where his delightful conversation is said to have transformed the prison into "a place of delight,"[6] and at Winchester House. He was finally released and allowed to go to France on the 30th of July 1643, through the intervention of the queen of France, Anne of Austria, on condition that he would neither promote nor conceal any plots abroad against the English government.

Before leaving England an attempt was made to draw from him an admission that Laud, with whom he had been intimate, had desired to be made a cardinal, but Digby denied that the archbishop had any leanings towards Rome. On the 1st of November 1643 it was resolved by the Commons to confiscate his property. He published in London the same year _Observations on the 22nd stanza in the 9th canto of the 2nd book of Spenser's "Faerie Queene,"_ the MS. of which is in the Egerton collection (British Museum, No. 2725 f. 117 b), and _Observations_ on a surreptitious and unauthorized edition of the _Religio Medici_, by Sir Thomas Browne, from the Roman Catholic point of view, which drew a severe rebuke from the author. After his arrival in Paris he published his chief philosophical works, _Of Bodies_ and _Of the Immortality of Man's Soul_ (1644), autograph MSS. of which are in the Bibliotheque Ste Genevieve at Paris, and made the acquaintance of Descartes. He was appointed by Queen Henrietta Maria her chancellor, and in the summer of 1645 he was despatched by her to Rome to obtain assistance. Digby promised the conversion of Charles and of his chief supporters. At first his eloquence made a great impression. Pope Innocent X. declared that he spoke not merely as a Catholic but as an ecclesiastic. But the absence of any warrant from Charles himself roused suspicions as to the solidity of his assurances, and he obtained nothing but a grant of 20,000 crowns. A violent quarrel with the pope followed, and he returned in 1646, having consented in the queen's name to complete religious freedom for the Roman Catholics, both in England and Ireland, to an independent parliament in Ireland, and to the surrender of Dublin and all the Irish fortresses into the hands of the Roman Catholics, the king's troops to be employed in enforcing the articles and the pope granting about L36,000 with a promise of further payments in obtaining direct assistance. In February 1649 Digby was invited to come to England to arrange a proposed toleration of the Roman Catholics, but on his arrival in May the scheme had already been abandoned. He was again banished on the 31st of August, and it was not till 1654 that he was allowed by the council of state to return. He now entered into close relations with Cromwell, from whom he hoped to obtain toleration for the Roman Catholics, and whose alliance he desired to secure for France rather than for Spain, and was engaged by Cromwell, much to the scandal of both Royalists and Roundheads, in negotiations abroad, of which the aim was probably to prevent a union between those two foreign powers. He visited Germany, in 1660 was in Paris, and at the Restoration returned to England. He was well received in spite of his former relations with Cromwell, and was confirmed in his post as Queen Henrietta Maria's chancellor. In January 1661 he delivered a lecture, which was published the same month, at Gresham College, on the vegetation of plants, and became an original member of the Royal Society in 1663. In January 1664 he was forbidden to appear at court, the cause assigned being that he had interposed too far in favour of the 2nd earl of Bristol, disgraced by the king on account of the charge of high treason brought by him against Clarendon into the House of Lords. The rest of his life was spent in the enjoyment of literary and scientific society at his house in Covent Garden. He died on the 11th of June 1665. He had five children, of whom two, a son and one daughter, survived him.

Digby, though he possessed for the time a considerable knowledge of natural science, and is said to have been the first to explain the necessity of oxygen to the existence of plants, bears no high place in the history of science. He was a firm believer in astrology and alchemy, and the extraordinary fables which he circulated on the subject of his discoveries are evidence of anything rather than of the scientific spirit. In 1656 he made public a marvellous account of a city in Tripoli, petrified in a few hours, which he printed in the _Mercurius Politicus_. Malicious reports had been current that his wife had been poisoned by one of his prescriptions, viper wine, taken to preserve her beauty. Evelyn, who visited him in Paris in 1651, describes him as an "errant mountebank." Henry Stubbes characterizes him as "the very Pliny of our age for lying," and Lady Fanshawe refers to the same "infirmity."[7] His famous "powder of sympathy," which seems to have been only powder of "vitriol," healed without any contact, by being merely applied to a rag or bandage taken from the wound, and Digby records a miraculous cure by this means in a lecture given by him at Montpellier on this subject in 1658, published in French and English the same year, in German in 1660 and in Dutch in 1663; but Digby's claim to its original discovery is doubtful, Nathaniel Highmore in his _History of Generation_ (1651, p. 113) calling the powder "Talbot's powder," and ascribing its invention to Sir Gilbert Talbot. Some of Digby's pills and preparations, however, described in _The Closet of the Eminently Learned Sir Kenelm Digby Knt. Opened_ (publ. 1677), are said to make less demand upon the faith of patients, and his injunction on the subject of the making of tea, to let the water "remain upon it no longer than you can say the Miserere Psalm very leisurely," is one by no means to be ridiculed. As a philosopher and an Aristotelian Digby shows little originality and followed the methods of the schoolmen. His Roman Catholic orthodoxy mixed with rationalism, and his political opinions, according to which any existing authority should receive support, were evidently derived from Thomas White (1582-1676), the Roman Catholic philosopher, who lived with him in France. White published in 1651 _Institutionum Peripateticorum libri quinque_, purporting to expound Digby's "peripatetic philosophy," but going far beyond Digby's published treatises. Digby's _Memoirs_ are composed in the high-flown fantastic manner then usual when recounting incidents of love and adventure, but the style of his more sober works is excellent. In 1632 he presented to the Bodleian library a collection of 236 MSS., bequeathed to him by his former tutor Thomas Allen, and described in _Catalogi codicum manuscriptorum bibliothecae Bodleianae_, by W. D. Macray, part ix. Besides the works already mentioned Digby translated _A Treatise of adhering to God written by Albert the Great, Bishop of Ratisbon_ (1653); and he was the author of _Private Memoirs_, published by Sir N. H. Nicholas from _Harleian MS. 6758_ with introduction (1827); _Journal of the Scanderoon Voyage in 1628_, printed by J. Bruce with preface (Camden Society, 1868); _Poems from Sir Kenelm Digby's Papers_... with preface and notes (Roxburghe Club, 1877); in the _Add. MSS._ 34,362 f. 66 is a poem _Of the Miserys of Man_, probably by Digby; _Choice of Experimental Receipts in Physick and Chirurgery_ ... _collected by Sir K. Digby_ (1668), and _Chymical Secrets and Rare Experiments_ (1683), were published by G. Hartman, who describes himself as Digby's steward and laboratory assistant.

See the _Life of Sir Kenelm Digby by one of his Descendants_ (T.
Longueville), 1896. (P. C. Y.)

FOOTNOTES:

[1] _Letters by Eminent Persons_ (Aubrey's Lives), ii. 324.

[2] _Life and Continuation._

[3] Strafford's _Letters_, i. 474.

[4] Laud's _Works_, vi. 447.

[5] _Thomason Tracts_, Brit. Mus. E 164 (15).

[6] _Archaeologia Cantiana_, ii. 190.

[7] _Dict. of Nat. Biog._ sub "Digby." See also Robert Boyle's _Works_
(1744), v. 302.

DIGBY, KENELM HENRY (1800-1880), English writer, youngest son of William Digby, dean of Clonfert, was born at Clonfert, Ireland, in 1800. He was educated at Trinity College, Cambridge, and soon after taking his B.A. degree there in 1819 became a Roman Catholic. He spent most of his life, which was mainly devoted to literary pursuits, in London, where he died on the 22nd of March 1880. Digby's reputation rests chiefly on his earliest publication, _The Broadstone of Honour, or Rules for the Gentlemen of England_ (1822), which contains an exhaustive survey of medieval customs, full of quotations from varied sources. The work was subsequently enlarged and issued (1826-1827) in four volumes entitled: _Godefridus_, _Tancredus_, _Morus_ and _Orlandus_ (numerous re-impressions, the best of which is the edition brought out by B. Quaritch in five volumes, 1876-1877).

Among Digby's other works are: _Mores Catholici, or Ages of Faith_ (11
vols., London, 1831-1840); _Compitum; or the Meeting of the Ways at
the Catholic Church_ (7 vols., London, 1848-1854); _The Lovers' Seat,
Kathemerina; or Common Things in relation to Beauty, Virtue and Faith_
(2 vols., London, 1856). A complete list is given in J. Gillow's
_Bibliographical Dictionary of English Catholics_, ii. 81-83.

DIGENES ACRITAS, BASILIUS, Byzantine national hero, probably lived in the 10th century. He is named Digenes (of double birth) as the son of a Moslem father and a Christian mother; Acritas ([Greek: akra], frontier, boundary), as one of the frontier guards of the empire, corresponding to the Roman _milites limitanei_. The chief duty of these _acritae_ consisted in repelling Moslem inroads and the raids of the _apelatae_ (cattle-lifters), brigands who may be compared with the more modern Klephts. The original Digenes epic is lost, but four poems are extant, in which the different incidents of the legend have been worked up by different hands. The first of these consists of about 4000 lines, written in the so-called "political" metre, and was discovered in the latter part of the 19th century, in a 16th-century MS., at Trebizond; the other three MSS. were found at Grotta Ferrata, Andros and Oxford. The poem, which has been compared with the _Chanson de Roland_ and the _Romance of the Cid_, undoubtedly contains a kernel of fact, although it cannot be regarded as in any sense an historical record. The scene of action is laid in Cappadocia and the district of the Euphrates.