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rapid one, its velocity of propagation through the gas being

that of sound.

of hydrogen to carbonic acid would be about greater at the end of the tube nearest the axis. The experimental verification of the result is important, as it establishes a method of effecting the partial separation of gases without the selective action of chemical agents.

Let us now consider two gases in the same vessel, the proportion of the gases being different in different parts of the vessel, but the pressure being everywhere the same. The agitation of the molcules will still cause more molecules of the first gas to pass from places where that gas is dense to places where it is rare than in the opposite direction, but since the second gas is dense where the first one is rare, its molecules will be for the most part travelling in the opposite direction. Hence the molecules of the two gases will encounter each other, and every encounter will act as a check to the process and the equation of motion becomes of equalization of the density of each gas throughout the mixture.

Let us next consider the case of diffusion in a vertical cylinder. Let m, be the mass of the first gas in a column of unit area extending from the bottom of the vessel to the height x, and let r, be the volume which this mass would occupy at unit pressure, then k1m1 = v1,

The interdiffusion of two gases in a vessel is therefore a much slower process than that by which the density of a single gas becomes equalized, though it appears from the theory that the final result is the same, and that each gas is distributed through the vessel in precisely the same way as if no other gas had been present, and this even when we take into account the effect of gravity.

If we apply the ordinary language about fluids to a single gas of the mixture, we may distinguish the forces which act on an element of volume as follows:

1st. Any external force, such as gravity or electricity. 2d. The difference of the pressure of the particular gas on opposite sides of the element of volume. [The pressure due to other gases is to be considered of no account].

3d. The resistance arising from the percolation of the gas through the other gases which are moving with different velocity.

The resistance due to encounters with the molecules of any other gas is proportional to the velocity of the first gas relative to the second to the product of their densities, and to a coefficient which depends on the nature of the gases and on the temperature. The equations of motion of one gas of a mixture are therefore of

the form

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dm P1 = d'

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dm dt

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d3v1 dv ̧3 _ d2v, dr,

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dt

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dx2 D is called the coefficient of diffusion.

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It is equal to

It therefore varies inversely as the total pressure of the medium, and if the coefficient of resistance, C12, is independent of the temperature, it varies directly as the product kik,, i.e., as the square of the absolute temperature. It is probable, however, that the effect of temperature is not so great as this would make it.

In liquids D probably depends on the proportion of the ingredients of the mixed medium as well as on the temperature. The dimensions of D are L'T-1, where L is the unit of length and T the unit of time.

The values of the coefficients of diffusion of several pairs of gases have been determined by Loschmidt. They are referred in the following table to the centimetre and the second as units, for the temperature 0°C and the pressure of 76 centimetres of mercury.

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Hence if, as in the case of gravity, is the same for all gases, but k is different for different gases, the composition of the mixture will be different in different parts of the vessel, the proportion of the heavier gases, for which is smaller, being greater at the bottom of the vessel than at the top. It would be difficult, how. ever, to obtain experimental evidence of this difference of composition except in a vessel more than 100 metres high, and it would be necessary to keep the vessel free from inequalities of temperature for more than a year, in order to allow the process of diffusion to advance to a state even half-way towards that of ultimate equilibrium. The experiment might, however, be made in a few minutes by placing a tube, say 10 centimetres long, on a whirling apparatus, so that one end shall be close to the axis, while the other is moving at the rate, say, of 50 metres per second. Thus if equal volumes of hydrogen and carbonic acid were used, the proportion

Oxygen and carbonic oxide,
Carbonic oxide and hydrogen,

Diffusion in Liquids.

The nature of the motion of the molecules in liquids is less understood than in gases, but it is easy to see that if there is any irregular displacement among the molecules in a mixed liquid, it must, on the whole, tend to cause each component to pass from places where it forms a large proportion of the mixture to places where it is less abundant. It is also manifest that any relative motion of two constituents of the mixture will be opposed by a resistance arising from Imperial Academy of Vienna, 10th March 1870.

1

the encounters between the molecules of these components. The value of this resistance, however, depends, in liquids, on more complicated conditions than in gases, and for the present we must regard it as a function of all the physical properties of the mixture at the given place, that is to say, its temperature and pressure, and the proportions of the different components of the mixture.

The coefficient of interdiffusion of two liquids must therefore be considered as depending on all the physical properties of the mixture according to laws which can be ascertained only by experiment.

Thus Fick has determined the coefficient of diffusion for common salt in water to be 0.00000116, and Voit has found that of cane-sugar to be 0·00000365.

It appears from these numbers that in a vessel of the same size the process of diffusion of liquids requires a greater number of days to reach a given stage than the process of diffusion of gases in the same vessel requires seconds.

When we wish to mix two liquids, it is not sufficient to place them in the same vessel, for if the vessel is, say, a metre in depth, the lighter liquid will lie above the denser, and it will be many years before the mixture becomes even sensibly uniform. We therefore stir the two liquids together, that is to say, we move a solid body through the | vessel, first one way, then another, so as to make the liquid contents eddy about in as complicated a manner as possible. The effect of this is that the two liquids, which originally formed two thick horizontal layers, one above the other, are now disposed in thin and excessively convoluted strata, which, if they could be spread out, would cover an iminense area. The effect of the stirring is thus to increase the area over which the process of diffusion can go on, and to diminish the distance between the diffusing liquids; and since the time required for diffusion varies as the square of the thickness of the layers, it is evident that by a moderate amount of stirring the process of mixture which would otherwise require years may be completed in a few seconds. That the process is not instantaneous is easily ascertained by observing that for some time after the stirring the mixture appears full of streaks, which cause it to lose its transparency. This arises from the different indices of refraction of different portions of the mixture which have been brought near each other by stirring. The surfaces of separation are so drawn out and convoluted, that the whole mass has a woolly appearance, for no ray of light can pass through it without being turned many times out of its path.

Graham observed that the diffusion both of liquids and gases takes place through porous solid bodies, such as plugs of plaster of Paris or plates of pressed plumbago, at a rate not very much less than when no such body is interposed, and this even when the solid partition is amply sufficient to check all ordinary currents, and even to sustain a considerable difference of pressure on its opposite sides.

But there is another class of cases in which a liquid or a gas can pass through a diaphragm, which is not, in the ordinary sense, porous. For instance, when carbonic acid gas is confined in a soap bubble it rapidly escapes. The gas is absorbed at the inner surface of the bubble, and forms a solution of carbonic acid in water. This solution diffuses from the inner surface of the bubble, where it is strongest, to the outer surface, where it is in contact with air, and the carbonic acid evaporates and diffuses out into the atmosphere. It is also found that hydrogen and other gases can pass through a layer of caoutchouc.

Graham

showed that it is not through pores, in the ordinary sense, that the motion takes place, for the ratios are determined by the chemical relations between the gases and the caoutshone, or the liquid film.

According to Graham's theory, the caoutchouc is a colloid substance,—that is, one which is capable of combining, in a temporary and very loose manner, with indeterminate proportions of certain other substances, just as glue will form a jelly with various proportions of water. Another class of substances, which Graham called crystalloid, are distinguished from these by being always of definite composition, and not admitting of these temporary associations. When a colloïd body has in different parts of its mass different proportions of water, alcohol, or solutions of crystalloïd bodies, diffusion takes place through the colloïd body, though no part of it can be shown to be in the liquid state.

On the other hand, a solution of a colloïd substance is almost incapable of diffusion through a porous solid, or another colloïd body. Thus, if a solution of gum and salt in water is placed in contact with a solid jelly of gelatine and alcohol, alcohol will be diffused into the gum, and salt and water will be diffused into the gelatine, but the gum and the gelatine will not diffuse into each other.

There are certain metals whose relations to certain gases Graham explained by this theory. For instance, hydrogen can be made to pass through iron and palladium at a high temperature, and carbonic oxide can be made to pass through iron. The gases form colloidal unions with the metals, and are diffused through them as water is diffused through a jelly. Root has lately found that hydrogen can pass through platinum, even at ordinary temperatures.

By taking advantage of the different velocities with which different liquids and gases pass through parchmentpaper and other solid bodies, Graham was enabled to effect many remarkable analyses. He called this method the method of Dialysis.

Diffusion and Evaporation, Condensation, Solution, and Absorption.

The rate of evaporation of liquids is determined principally by the rate of diffusion of the vapour through the air or other gas which lies above the liquid. Indeed, the coefficient of diffusion of the vapour of a liquid through air can be determined in a rough but easy manner by placing a little of the liquid in a test tube, and observing the rate at which its weight diminishes by evaporation day by day. For at the surface of the liquid the density of the vapour is that corresponding to the temperature, whereas at the mouth of the test tube the air is nearly pure. Hence, if p be the pressure of the vapour corresponding to the temperature, and p= kp, and if m be the mass evaporated in time t, and diffused into the air through a distance,1 then khon D = pt

This method is not, of course, applicable to vapours which are rarer than the superincumbent gas.

The solution of a salt in a liquid goes on in the same way, and so does the absorption of a gas by a liquid. These processes are all accelerated by currents, for the reason already explained.

The processes of evaporation and condensation go on much more rapidly when no air or other non-condensible is present. Hence the importance of the air-pump in the steam engine.

gas

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om

(P1-Po)

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causes two portions of the same gas to diffuse through each other, although we cannot observe this kind of diffusion, because we cannot distinguish the molecules of one portion and if L is the latent heat of vapour at the temperature & the from those of the other when they are once mixed. If, quantity of heat required to produce this vapour is however, the molecules of one portion have any property (P-Pc) L. whereby they can be distinguished from those of the other. then that property will be communicated from one part of the medium to an adjoining part, and that either by convection-that is by the molecules themselves passing out of one part into the other. carrying the property with themor by transmission-that is by the property being com. municated from one molecule to another during their en

counters.

The chemical properties by which different substances are recognized are inseparable from their molecules, so that the diffusion of such properties can take place only by the transference of the molecules themselves, but the momentum of a molecule in any given direction and its energy are also properties which may be different in different molecules, but which may be communicated from one molecule to another. Hence the diffusion of momentum and that of energy through the medium can take place in two different ways, whereas the diffusion of matter can take place only in one of these ways.

In gases the great majority of the particles, at any instaut, are describing free paths, and it is therefore possible to show that there is a simple numerical relation between the coefficients of the three kinds of diffusion,-the diffusion of matter, the lateral diffusion of velocity (which is the phenomenon known as the internal friction or viscosity of fluids), and the diffusion of energy (which is called the conduction of heat). But in liquids the majority of the molecules are engaged at close quarters with one or more other molecules, so that the transmission of momentum and of energy takes place in a far greater degree by communication from one molecule to another, than by convection by the molecules themselves. Hence the ratios of the coefficient of diffusion to those of viscosity and thermal conductivity are much smaller in liquids than in gases.

Theory of the Wet Bulb Thermometer.

The temperature indicated by the wet bulb thermometer is determined in great part by the relation between the coefficients of diffusion and thermal conductivity. As the water evaporates from the wet bulb heat must be supplied to it by convection, conduction, or radiation. This supply of heat will not be sufficient to maintain the temperature constant till the temperature of the wet bulb has sunk so far below that of the surrounding air and other bodies that the flow of heat due to the difference of temperature is equal to the latent heat of the vapour which leaves the bulb.

The use of the wet bulb thermometer as a means of estimating the humidity of the atmosphere was employed by Hutton and Leslie, but the formula by which the dew-point is commonly deduced from the readings of the wet and dry thermometers was first given by Dr Apjohn.

Dr Apjohn assumes that, when the temperature of the wet bulb is stationary, the heat required to convert the water into vapour is given out by portions of the surrounding air in cooling from the temperature of the atmosphere to that of the wet bulb, and that the air thus cooled becomes saturated with the vapour which it receives from the bulb.

Let m be the mass of a portion of air at a distance from the wet bulb, its temperature, p, the pressure due to the aqueous vapour in it, and P the whole pressure.

If is the specific gravity of aqueous vapour (referred to air), then the mass of water in this portion of air is om.

Let this portion of air communicate with the wet bulb till its temperature sinks to 6,, that of the wet bulb, and the pressure of the aqueous vapour in it rises to p1, that corresponding to the temperature 01.

is

The quantity of vapour which has been communicated to the air

Playfair's "Life of Hutton," Edinburgh Transactions, vol. v. p. 67, note.

Encyc. Brit., 8th ed. vol. i.. "Dissertation Fifth," p. 764. 'Trans. Royal Irish Academy, 1834.

According to Apjobn's theory, this heat is supplied by the mixed air and vapour in cooling from 0 to 01.

ferent from that of dry air), this quantity of heat is
If S is the specific heat of the air (which will not be sensibly dif

(00-01) ms.

Equating the two values we obtain

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The

Here ro is the pressure of the vapour in the atmosphere. temperature-for which this is the maximum pressure is the dew. point, and p, is the maxinium pressure corresponding to the tempera. ture 0, of the wet bulb. Hence this formula, combined with tables of the pressure of aqueous vapour, enables us to find the dew-point from observations of the wet and dry bulb thermometers.

We may call this the convection theory of the wet bulb, becaus we consider the temperature and humidity of a portion of air brought from a distance to be affected directly by the wet bulb without communication either of heat or of vapour with other portions of air.

Dr Everett has pointed out as a defect in this theory, that it does in humidity unless it comes into absolute contact with the wet not explain how the air can either sink in temperature or increase bulb. Let us, therefore, consider what we may call the conduction and diffusion theory in calm air, taking into account the effects of radiation.

The steady conduction of heat is determined by the conditions-
0-0, at a great distance from the bulb,

0-0 at the surface of the bulb,
20-0 at any point of the medium.

The steady diffusion of vapour is determined by the conditions-
pPo at a great distance from the bulb,

p-p at the surface of the bulb,

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at any point of the medium.

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tion, and radiation in a still atmosphere. It differs from the formula of the convection theory only by the factor in the last is certainly less than unity, and

The first part of this factor probably about 77.

.

K D If the bulb is spherical and of radius r, A-4 and C-r, so that the second part Rr is PSD Hence, the larger the wet bulb, the greater will be the ratio of the effect of radiation to that of conduction. If, on the other hand, the air is in motion, this will increase both conduction and diffusion, so as to increase the ratio of the first part to the second. By comparing actual observations of the dew-point with Apjohn's formula, it has been found that the factor should be somewhat greater than unity. According to our theory it ought to be greater if the bulb is larger, and smaller if there is much wind. Relation between Diffusion und Electrolytic Conduction. Electrolysis (see separate article) is a molecular movement of the constituents of a componnd liquid in which, under the action of electromotive force, one of the components trave in the positive and the other in the negative direction, the flow of each component, when reckoned in electrochemical equivalents, being in all cases numerically equal to the flow of electricity.

Electrolysis resembles diffusion in being a molecular movement of two currents in opposite directions through the same liquid; but since the liquid is of the same composition throughout, we cannot ascribe the currents to the molecular agitation of a medium whose composition varies from one part to another as in ordinary diffusion, but we must ascribe it to the action of the electromotive force on articles having definite charges of electricity.

The force, therefore, urging an electro-chemical equivalent of either component, or ion, as it is called, in a given direction is numerically equal to the electromotive force at a given point of the electrolyte, and is therefore comparable with any ordinary force. The resistance which prevents the current from rising above a certain value is that arising from the encounters of the molecules of the ion with other molecules as they struggle forward through the liquid, and this depends on their relative velocity, and also on the nature of the ion, and of the liquid through which it has to flow.

The average velocity of the ions will therefore increase, till the resistance they meet with is equal to the force which urges them forward, and they will thus acquire a definite velocity proportional to the electric force at the point, but depending also on the nature of the liquid.

If the resistance of the liquid to the passage of the ion is the same for different strengths of solution, the velocity of the ion will be the same for different strengths, but the quantity of it, and therefore the quantity of electricity which passes in a given time, will be proportional to the strength of the solution.

Now, Kohlrausch has determined the conductivity of the solutions of many electrolytes in water, and he finds that for very weak solutions the conductivity is propor tional to the strength. When the solution is strong the liquid through which the ions struggle can no longer be considered sensibly the same as pure water, and consequently this proportionality does not hold good for strong solutions.

Kohlrausch has determined the actual velocity in cenlimetres per second of various ions in weak solutions under an electro-motive force of unit value. From these velocities he has calculated the conductivities of weak solutions of electrolytes different from those of which he made use in calculating the velocity of the ions, and he finds the results consistent with direct experiments on those electrolytes.

It is manifest that we have here important informa

tion as to the resistance which the ion meets with in travelling through the liquid. It is not easy, however, to make a numerical comparison between this resistance and any results of ordinary diffusion, for, in the first place, we cannot make experiments on the diffusion of ions. Many electrolytes, indeed, are decomposed by the current into components, one or both of which are capable of diffusion, but these components, when once sepa rated out of the electrolyte, are no longer ions they are no longer acted on by electric force, or charged with definite quantities of electricity. Some of them, as the metals, are insoluble, and therefore incapable of diffusion; others, like the gases, though soluble in the liquid electrolyte, are not, when in solution, acted on by the current.

Besides this, if we accept the theory of electrolysis proposed by Clausius, the molecules acted on by the electromotive force are not the whole of the molecules which form the constituents of the electrolyte, but only those which at a given instant are in a state of dissociation from molecules of the other kind, being forced away from them temporarily by the violence of the molecular agitation. If these dissociated molecules form a small proportion of the whole, the velocity of their passage through the medium must be much greater than the mean velocity of the whole, which is the quantity calculated by Kohlrausch.

On Processes by which the Mixture and Separation of Fluids can be effected in a Reversible Manner.

A physical process is said to be reversible when the material system can be made to return from the final state to the original state under conditions which at every stage of the reverse process differ only infinitesimally from the conditions at the corresponding stage of the direct process. All other processes are called irreversible.

Thus the passage of heat from one body to another is a reversible process if the temperature of the first body exceeds that of the second only by an infinitesimal quantity, because by changing the temperature of either of the the bodies by an infinitesimal quantity, the heat may be made to flow back again from the second body to first.

But if the temperature of the first body is higher than that of the second by a finite quantity, the passage of heat from the first body to the second is not a reversible process, for the temperature of one or both of the bodies must be altered by a finite cuantity before the heat can be made to flow back again.

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In like manner the interdiffusion of two gases is in general an irreversible process, for in order to separate the two gases the conditions must be very considerably changed. For instance, if carbonic acid is one of the gases, we can separate it from the other by means of quicklime; but the absorption of carbonic acid by quicklime at ordinary temperatures and pressures is an irreversible process, for in order to separate the carbonic acid from thə lime it must be raised to a high temperature.

In all reversible processes the substances which are in contact must be in complete equilibrium throughout the process; and Professor Gibbs has shown the condition of equilibrium to be that not only the temperature and the pressure of the two substances must be the same, but also that the potential of each of the component substances must be the same in both compounds, and that there is an additional condition which we need not here specify.

Now, we may obtain complete equilibrium between quicklime and the mixture containing carbonic acid if we raise the whole to a temperature at which the pressure. of dissociation of the carbonic acid in carbonate of lime is equal to the pressure of the carbonic acid in the mixed gases. By altering the temperature or the pressure very slowly we may cause carbonic acid to pass from the mix

ture to the lime, or from the lime to the mixture, in such a manner that the conditions of the system differ only by infinitesimal quantities at the corresponding stages of the direct and the inverse processes. The same thing may be done at lower temperatures by means of potash or soda.

If one of the gases can be condensed into a liquid, and if during the condensation the pressure is increased or the temperature diminished so slowly that the liquid and the mixed gases are always very nearly in equilibrium, the separation, and mixture of the gases can be effected in a reversible manner.

The same thing can be done by means of a liquid which absorbs the gases in different proportions, provided that we can maintain such conditions as to temperature and pressure as shall keep the system in equilibrium during the whole process.

If the densities of the two gases are different, we can effect their partial separation by a reversible process which does not involve any of the actions commonly called chemical. We place the mixed gases in a very long horizontal tube, and we raise one end of the tube till the tube is vertical. If this is done so slowly that at every stage of the process the distribution of the two gases is sensibly the same as it would be at the same stage of the reverse process, the process will be reversible, and if the tube is long enough the separation of the gases may be carried to any extent.

In the Philosophical Magazine for 1876, Lord Rayleigh has in. vestigated the thermodynamics of diffusion, and has shown that if two portions of different gases are given at the same pressure and temperature, it is possible, by mixing them by a reversible process to obtain a certain quantity of work. At the end of the process the two cases are uniformly mixed, and occupy a volume equal to the sum of the volumes they occupied when separate, but the temperature and pressure of the mixture is lower than before.

The work which can be gained during the mixture is equal to that which would he gained by allowing first one gas and then the other to expand from its original volume to the sum of the volumes; and the fall of temperature and pressure is equal to that which would be produced in the mixture by taking away a quantity of heat equivalent to this work.

If the diffusion takes place by an irreversible process, such as goes on when the gases are placed together in a vessel, no external work is done, and there is no fall of temperature or of pressure during the process.

We may arrive at this result by a method which, if not so instructive as that of Lord Rayleigh, is more general, by the use of the physical quantity called by Clausius the Entropy of the system. The entropy of a body in equilibrium is a quantity such that it remains constant if no heat enters or leaves the body, and such that in general the quantity of heat which enters the body is

Sedo,

where is the entropy, and @ the absolute temperature. The entropy of a material system is the sum of the entropy of its parts.

In reversible processes the entropy of the system remains unchanged, but in all irreversible processes the entropy of the system increases.

The increase of entropy involves a diminution of the available energy of the system, that is to say, the total quantity of work which can be obtained from the system. This is expressed by Sir W. Thomson by saying that a certain amount of energy is dissipated. The quantity of energy which is dissipated in a given process is

equal to

00(P2-1),

where is the entropy at the beginning, and 4, that at the end of the process, and e, is the temperature of the system in its ultimate state, when no more work can be got out of it.

When we can determine the ultimate temperature we can calculate the amount of energy dissipated by any process; but it is sometimes difficult to do this, whereas the increase of entropy is determined by the known states of the system at the beginning and end of the process.

The entropy of a volume V1 of a gas at pressure p, and temperature 91 exceeds its entropy where its volume is v, and its temperature by the quantity

Pit, log + log

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or the sum of the work which would be done by the two portions of gas if each expanded under constant temperature to the volume D1 + D2.

It is greatest when the two volumes are equal, in which case it is 1.386 pv,

where p is the pressure and v the volume of one of the portions.

Let us now suppose that we have in a vessel two separate portions of gas of equal volume, and at the same pressure and temperature, with a movable partition between them. If we remove the partition the agitation of the molecules will carry them from one side of the partition to the other in an irregular manner, till ultimately the two portions of gas will be thoroughly and uniformly mixed together. This motion of the molecules will take place whether the two gases are the same or can distinguish different, that is to say, whether, we between the properties of the two gases or not.

If the two gases are such that we can separate them by a reversible process, then, as we have just shown, we might gain a definite amount of work by allowing them to mix under certain conditions; and if we allow them to mix by ordinary diffusion, this amount of work is no longer available, but is dissipated for ever. If, on the other hand, the two portions of gas are the same, then no work can be gained by mixing them, and no work is dissipated by allowing them to diffuse into each other.

It appears, therefore, that the process of diffusion does not involve dissipation of energy if the two gases are the same, but that it does if they can be separated from each other by a reversible process.

Now, when we say that two gases are the same, we mean that we cannot distinguish the one from the other by any known reaction. It is not probable, but it is pos sible, that two gases derived from different sources, but hitherto supposed to be the same, may hereafter be found to be different, and that a method may be discovered of separating them by a reversible process. If this should happen, the process of interdiffusion which we had formerly supposed not to be an instance of dissipation of energy would now be recognized as such an instance.

It follows from this that the idea of dissipation of energy depends on the extent of our knowledge. Available energy is energy which we can direct into any desired channel. Dissipated energy is energy which we cannot loy hold of and direct at pleasure, such as the energy of the confused agitation of molecules which we call heat. Now, confusion, like the correlative term order, is not a property of material things in themselves, but only in relation to the mind which perceives them. A memorandum-book does not, provided it is neatly written, appear confused to an illiterate person, or to the owner who understands it thoroughly, but to any other person able to read it appears to be inextricably confused. Similarly the notion of dis sipated energy could not occur to a being who could not turn any of the energies of nature to his own account, or to one who could trace the motion of every molecule and seize it at the right moment. It is only to a being in the intermediate stage, who can lay hold of some forms of

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