gases temperature process molecules pressure liquid velocity bulb mixture medium
DIFFUSION. Some liquids, such. a,s mercury and w-ater, when placed in contact with each other do not mix at all, but the surface of separation remains distinct, and OK-hibits the phenomena described under CAPILLARY AcTroN. Other pairs of liquids, such as chloroform and water, mix, but only in certain proportions. The, chloroform takes up a, little water, and the water a little chloroform ; but the two mixed liquids will not mix with each other, but remain in contact separated by a surface showing capillary phenomena. The two liquids are then in a state of equilibrium with each other. The conditions of the equilibrium of heterogeneous substances have been investi-gated by Professor J. Willard Gibbs in a series of papers published in the Transactions of the Connecticut Academy of .Arts and Sciences, vol. Hi. part i. p. 10S. Other pairs of liquids, and all gases, mix in all proportions.
When two fluids are capable of being mixed, they cannot remain in equilibrium with each other ; if they are placed in contact with each other the process of mixture begins of itself, and goes on till the state of equilibrium is attained, which, in the case of fluids which mix in all proportions, is a state of uniform mixture.
This process of mixture is called diffusion. It may be easily observed by taking a glass jar half full of water and pouring a strong solution of a coloured salt, such as sulphate of copper, through a long-stemmed funnel, so as to occupy the lower part of the jar. If the jar is not disturbed we may trace the process of diffusion for weeks, months, or years, by the gradual rise of the colour into the upper part of the jar, and the weakening of the colour in the lower part.
This, however, is not a method capable of giving accurate measurements of the composition of the liquid at different depths in the vessel. For more exact determinations we may draw off a portion from a given stratum of the mixed liquid, and determine its composition either by chemical methods or by its specific gravity, or any other property from which its composition may be deduced.
But as the act of removing a portion of the fluid interferes with the process of diffusion, it is desirable to be able to ascertain the composition of any stratum of the mixture without removing it from the vessel. For this purpose Sir W. Thomson places in the jar a number of glass beads of different densities, which indicate the densities of the strata in which they are observed to float. The principal objection to this method is, that if the liquids contain air or any other gas, bubbles are apt to form on the glass beads, so as to make them float in a stratum of less density than that marked on them.
M. Voit has observed the diffusion of cane sugar in water by passing a ray of plane-polarized light horizontally through the vessel, and determining the angle through. which the plane of polarization is turned by the solution of sugar. This method is of course applicable only to those substances which cause rotation of the plane of polarized light.
Another method is to place the diffusing liquids in a hollow glass prism, with its refracting edge vertical, and to determine the deviation of a ray of light passing through the prism at different depths. The ray is bent downwards on account of the variable density of the mixture, as well as towards the thicker part of the prism ; but by making it pass as near the edge of the prism as possible, the vertical component of the refraction may be made very small; and by placing the prism within a vessel of water having parallel sides of glass, we can get rid of the constant part of the deviation, and are able to use a prism of large angle, so as to increase the part due to the diffusing substance. At the same time we can more easily control and register the temperature.
The laws of diffusion were first investigated by Graham. The diffusion of gases has recently been observed with great accuracy by Loschmidt, and that of liquids by Fick and by Voit.
Difvsion as a molecular motion. - If we observe the process of diffusion with our most powerful microscopes, we cannot follow the motion of any individual portions of the fluids. We cannot point out one place in which the lower fluid is ascending, and another in which the upper fluid is descending. There are no currents visible to us, and the motion of the material substances goes on as imperceptibly as the conduction of heat or of electricity. Hence the motion which constitutes diffusion must be distinguished from those motions of fluids which we can trace by means of floating motes. It may be described as a motion of the fluids, not in mass, but by molecules.
When we reason upon the hypothesis that a fluid is a continuous homogeneous substance, it is comparatively easy to define its density and velocity; but when we admit that it may consist of molecules of different kinds, we must revise our definitions. We therefore define ;hese quantities by considering that part of the medium which at a given instant is within a certain small region surrounding a given point. This region must be so small that the properties of the medium as a whole are sensibly the same throughout the region, and yet it must be se large as to include a large number of molecules. We then define the density of the medium at the given point as the mass of the medium within this region divided by its volume, and the velocity of the medium as the momentum of this portion of the medium divided by its mass.
If we consider the motion of the medium relative to an imaginary surface supposed to exist within the region occupied by the medium, and if we define the flow of the medium through the surface as the mass of the medium which in unit of time passes through unit of area of the surface, then it follows from the above definitions that the velocity of the medium resolved in the direction of the normal to the surface is equal to the flow divided by the density. If we suppose the surface itself to move with the same velocity as the fluid, and in the same direction, there will be no flow through it.
Having thus defined the density, velocity, and flow of the medium as a whole, or, as it is sometimes expressed, "in mass," we may now consider one of the fluids which constitute the medium, and define its density, velocity, and flow in the same way. The velocity of this fluid may be different from that of the medium in mass, and its velocity relative to that of the medium is the velocity of diffusion which we have to study.
Difusion of Gases according to the _Kinetic Theory.
So many of the phenomena of gases are found to be explained in a consistent manner by the kinetic theory of gases, that we may describe with considerable probability of correctness the kind of motion which constitutes diffusion in gases. We shall therefore consider gaseous diffusion in the light of the kinetic theory before we consider diffusion in liquids.
A. gas, according to the kinetic theory, is a collection of particles or molecules which are in rapid motion, and which, when they encounter each other, behave pretty much as elastic bodies, such as billiard balls, would do if no energy were lost in their collisions. Each molecule travels but a very small distance between one encounter and another, so that it is every now amid then altering its velocity both in direction and magnitude, and that in an exceedingly irregular manner. - The result is that the velocity of any molecule may be considered as compounded of two velocities, one of which, called the velocity of the medium, is the same for all the molecules, while the other, called the velocity of agitation, is irregular bath in magnitude and in direction, though the average magnitude of the velocity may be calculated, and any one direction is just as likely as any other.
The result of this motion is, that if in any part of the medium the molecules are more numerous than in a neighbouring region, more molecules will pass from the first region to the second than in the reverse direction, and for this reason the density of the gas will tend to become equal in all parts of the vessel containing it, except in so far as the molecules may be crowded towards one direction by the action of an external force such as gravity. Since the motion of the molecules is very swift, the process of equalization of density in a gas is a very rapid one, its velocity of propagation through the gas being that of sound.
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 mole-cules 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 of equalization of the density of each gas throughout the mixture.
The interdiffusion of two gases in a vessel is thereappears from the theory that the final result is the same, 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 th,e particular gae on oppo-site 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 movin, with different velocity.
The resistance due to encoufters 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 arc therefore of the form where the symbol of operation aet prefixed to any quantity denotes the time-variation of that quantity at a point which moves along with that medium which is distinguished by the suffix (1), or more txplicitly In the state of ultimate equilibrium ur=-2L2 = &C. = 0, and the equation is reduced to which is the ordinary form of the equations of equilibrium of a single fluid. Hence, when the process of diffusion is complete, the density of each gas at any point of the vessel is the same as if no other gas were present.
f VI is the potential of the force which acts on the gas, and if in the equation Pt =lciPi, lc, is constant, as it is when the temperature is uniform, that the equation of equilibrium becomes Hence if, as in the case of gravity, V is the same for all gases, but k is different for different gases, the composition of the mixture Nvill be different in different parts of the vessel, the proportion of the heavier gases, for which k is smaller, being greater at the bottom of the vessel than at the top. It would ba difficult, how-ever, to obtain experimental evidence of this difference of com-position except in a vessel more than 100 metres high, and it would be necessary to keep the vessel free from inequalities of tem-perature 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 experinient might, however, be made in a few minutes by placing a tube, say I 0 centimetres long, on a whirling n pparatus, 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 of hydrogen to carbonic acid would be about -/ AT 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 next consider the case of diffusion in a vertical cylinder. Let be the mass of the first gas in a column of unit area extend-ing from the bottom of the vessel to the height x,'and let v, be the volume which this mass would occupy at unit pressure, then If we add the corresponding equations togaher for all the gases, we find that the terms in C/2 destroy each other, and that if the medium is not affected with sensible currents the first term of each equation may be neglected. In ordinary experiments we may also neglect the effect of gravity, so that we get It therefore varies inversely as the total pressure of the medium, and if the coefficient of resistance, C/2 , is independent of the tem-perature, 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 dimen-sions of D are L2T-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.1 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.
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 com-ponent 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 the encounters between the naolecules 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 rnust 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 0.0000116, and Veit 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 to-gether, 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 im mense area. The effect of the stirrino. is thus to increase the arca over which the process of diff°usion 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 stir-ring 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 con-siderable 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 or-dinary 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 tak.es place, for the ratios are determined by the chemical relations between the gases and the caout-choue, 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 com-position, and not admitting of these temporary associations. When a colloid body has in different parts of its mass different proportions of water, alcohol, or solutions of crys-talloid bodies, diffusion takes place :through the colloid body, though no part of it can be shown to be in the liquid state.
On the other hand, a solution of a colloid substance is almost incapable of diffusion through a porous solid, or another colloid 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 gnm, 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 parchment-paper and other solid bodies, Graham was enabled to effect many remarkable analyses. He called this method the method of Dialysis.
The rate of evaporation of liquids is determined prin-cipally 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 i3 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 tem-perature, and p=kp, and if in be the mass evaporated in time t, and diffused into the air through a distance h,t This method is riot, of course, applicable to vapours which are rarer than the superincurnbent gas.
The solution of a salt in a liquid goes on in the same way, and so does the absorption of a 0.a.s 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 ait or other non-condensible gas is present. Hence the importance of the air-pump in the 'steam engine.
The same motion of agitation of the molecules of gases which causes two gases to diffuse through each other also 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 from those of the other when they are once mixed. If, however, the molecules of one portion have any property 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 them - or by transmission - that is by the property being com• municated from one molecule to another during their encounters. The chemical properties by whidli 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 tike place only in one of these ways.
In gases the great majority of the particles, at any instant, 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 ana 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.3 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, 0 its temperature, ps the pressure due to the aqueous vapour in it, and P the whole pressure.
If o is the specific gravity of aqueous vapour (referred to air), then the mass of water in this portion of air is 712, am .
Let this portion of air communicate with the wet bulb till its temperature sinks to 01, that of the wet bulb, and the pressure of the aqueous vapour in it rises to pl, that corresponding to the temperature 01.
The quantity of vapour which has been communicated to the air According to Apjohn's theory, this heat is supplied by the mixed air and vapour in cooling from 0 to 01.
If S is the specific heat of the air (which will not be sensibly different from that of dry air), this quantity of heat is - ms . Equating the two values we obtain Here r, is the pressure of the vapour in the atmosphere. The temperature - for which this is the maximum pressure - is the dew-point, and p, is the maximum pressure corresponding to the temperature 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, because 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 oilier portions of air.
Dr Everett has pointed out as a defect in this theory, that it does not explain how the air eau either sink in temperature or increase in humidity unless it comes into absolute contact with the wet bulb. Let us, therefore, consider what we may call the conduction and diffusion theory in calm air, taking into account the effects of radiation.
Now, if the bulb had been an electrified conductor, the conditions with respect to the potential would have been where the double integral is extended over the surface of the bulb, and dv is an element of a normal to the surface.
If II is the flow of heat iu unit of time from the bulb, where P is the ratio of the pressure of aqueous vapour to its density.
If C is the electrical capacity of the bulb, E - CV1, The heat which leaves the bulb by radiation to external objects at temperature 00 may be written h - AR(Ot where A is the surface of the bulb and R the coefficient of radiation of unit of surface.
When the temperature becomes constant tion, and radiation in a still atmosphere. It differs from the formula of the convection theory only by the factor in the last term.
K The first part of this factor D - s certainly less than unity, and probably about .77.
If the bulb is spherical and of radius r, A - 4.7rr2 and C - r, , so that the second part is - Rr .
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 and Electrolytic Conduction.
Electrolysis (see separate article) is a molecular movement of the constituents of a compound liquid in which, under the action of electromotive force, one of the components travels 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 particles having definite charges of electricity.
The force, therefore, urging an electrochemical 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, Kohlransch has determined the conductivity of the solutions of many electrolytes in water, and he finds that for very weak solutions the conductivity is proportional 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 centimetres per second of various ions in weak solutions under an electromotive 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 informsHon 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 snake 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 separated 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 iu a state of dissociation from molecules of the other kind, beimg forced away from them temporarily by the violence of 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.
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 tlse 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 quantity before the heat can be made to flow back again.
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. Fur 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 the 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 mixture 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 equilibriuna during the whole process.
If the densities of the two gases are different, WEI can effect their partial separation by a reversible process which does not involve any of the actions commonly called 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 *gases are uniformly mixed, and oecupy a volume equal to the sum af 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 be gained by allowin, first one gas and then the other to expand frorc Its original volumeqo the sum of the volumes ; and the fall of temperature and pressure is equal to that which would b,e 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 to,ether in a vessel, no external work is done, and there is no fall oef 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 equilibrimn is a quantity such that it remains constant if no he.at enters or leaves the body, and such that in general the quantity of heat which enters the body is 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 un-changed, 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 80(02 - 01) , where is the entropy at the beginning, and ,p, that at the end of the process, and 00 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 entiopy of a volume v, of a gas at pressure p, and temperature Oa exceeds its entropy where its volume is vi and its temperature 00 by the quantity Hence if volumes vi and v0 of two gases at the same temperature and pressure are mixed so as to occupy a volume vi+ v2 at the same temperature and pressure, the entropy of the system increases during the process by the quantity Since in this case the temperature does not change during the pro-cess, we play calculate the quantity of energy dissipated by multi-plying the gain of entropy by the temperature, and we thus.find for the dissipation + v, .
It is greatest when the two volumes are equal, in which case it is 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, aud 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 ulti-mately 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 different, that is to say, whether we can distinguish 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. Avail-able energy is energy which we can direct into any desired channel. Dissipated energy is energy which we cannot lay hold of and direct at pleasure, such a,s 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, whu can lay hold of some forms of energy while others elude his grasp, that energy appears to be pa,ssing inevitably from the a,vailable to the dissipated state. (J. c. m.)