- Gioacchino Murat Stock Photos & Gioacchino Murat Stock Images - Alamy
- Napoleon I, Emperor of the French, 1769-1821
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- Joachim Murat

He married Napoleon Bonaparte's sister Caroline. American politician. Stipple engraving, American, Artist: Unknown. He contributed to victories at Marengo, Austerlitz, Jena and Eylau.

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After Napoleon's final defeat, he was court-martialled and shot. Lithograph c Portrait of Joachim Murat , King of Naples MuratMiniature Portrait of Joachim Murat , c. Congress of Vienna, - American lawyer and legislator. Jackson at the Battle of New Orleans during the War of Steel engraving, 19th century.

## Gioacchino Murat Stock Photos & Gioacchino Murat Stock Images - Alamy

Jackson, Andrew J. Marshal of France and brother-in-law of Napoleon Bonaparte. Reimagined by Gibon. View on a Lake Rev. Engraving Portrait of Joachim Murat 19th century. The Victoria history of the county of Surrey. Natural history. Representative and U. Ebenezer Storer II. If its tension is T the work required to effect this increase of surface will be T dS, and the energy of the film will be increased by this amount. Hence in this case. Hence the tension of a thick film is equal to the sum of the tensions of its two surfaces as already calculated equation 7.

The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics.

The actual force between the particles arises in part from their mutual gravitation, which is inversely as the square of the distance. It is easy to show that a force subject to this law would not account for capillary action. Let us examine the case in which the particle m is placed at a distance z from a curved stratum of the substance, whose principal radii of curvature are R 1 and R 2. Let P fig. This expression, when integrated, gives 1 the work done on a particle m while it is brought from an infinite distance to the point P, or 2 the attraction on a long slender column normal to the surface and terminating at P, the mass of unit of length of the column being m.

In the form of the theory given by Laplace, the density of the liquid was supposed to be uniform. For thinner films. Now, the force between the particle and the liquid is certainly, on the whole, attractive; but if between any two small values of c it should be repulsive, then for films whose thickness lies between these values the tension will increase as the thickness diminishes, but for all other cases the tension will diminish as the thickness diminishes. We have given several examples in which the density is assumed to be uniform, because Poisson has asserted that capillary phenomena would not take place unless the density varied rapidly near the surface.

### Napoleon I, Emperor of the French, 1769-1821

In this assertion we think he was mathematically wrong, though in his own hypothesis that the density does actually vary, he was probably right. The pressure at any point of the liquid arises from two causes, the external pressure P to which the liquid is subjected, and the pressure arising from the mutual attraction of its molecules. If we suppose that the number of molecules within the range of the attraction of a given molecule is very large, the part of the pressure arising from attraction will be proportional to the square of the number of molecules in unit of volume, that is, to the square of the density.

Hence we may write. The former can be found at once by calculating the mutual attraction of the parts of a large mass which lie on opposite sides of an imaginary plane interface. The expression for the intrinsic pressure is thus simply.

We may call the value which 28 then assumes K 0 , so that as above. The expression for the superficial tension is most readily found with the aid of the idea of superficial energy, introduced into the subject by Gauss. Since the tension is constant, the work that must be done to extend the surface by one unit of area measures the tension, and the work required for the generation of any surface is the product of the tension and the area. Reprinted, Popular Lectures and Addresses , For imagine a small cavity to be formed in the interior of the mass and to be gradually expanded in such a shape that the walls consist almost entirely of two parallel planes.

The distance between the planes is supposed to be very small compared with their ultimate diameters, but at the same time large enough to exceed the range of the attractive forces. The work required to produce this crevasse is twice the product of the tension and the area of one of the faces. If we now suppose the crevasse produced by direct separation of its walls, the work necessary must be the same as before, the initial and final configurations being identical; and we recognize that the tension may be measured by half the work that must be done per unit of area against the mutual attraction in order to separate the two portions which lie upon opposite sides of an ideal plane to a distance from one another which is outside the range of the forces.

It only remains to calculate this work. Integrating by parts, we get. In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write. A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that. That K is in all cases of the fourth order and T of the fifth order in the range of the forces is obvious from 37 without integration. The intrinsic pressure will thus be infinite whatever n may be. For the transition case, discussed by William Sutherland Phil.

It seems therefore that nothing satisfactory can be arrived at under this head. The discrepancy seems to depend upon Young having treated the attractive force as operative in one direction only. Experimental Laws of Surface-Tension. For any given liquid surface, as the surface which separates water from air, or oil from water, the surface-tension is the same at every point of the surface and in every direction. It is also practically independent of the curvature of the surface, although it appears from the mathematical theory that there is a slight increase of tension where the mean curvature of the surface is concave, and a slight diminution where it is convex.

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The amount of this increase and diminution is too small to be directly measured, though it has a certain theoretical importance in the explanation of the equilibrium of the superficial layer of the liquid where it is inclined to the horizon. The surface-tension diminishes as the temperature rises, and when the temperature reaches that of the critical point at which the distinction between the liquid and its vapour ceases, it has been observed by Andrews that the capillary action also vanishes.

The early writers on capillary action supposed that the diminution of capillary action was due simply to the change of density corresponding to the rise of temperature, and, therefore, assuming the surface-tension to vary as the square of the density, they deduced its variations from the observed dilatation of the liquid by heat. This assumption, however, does not appear to be verified by the experiments of Brunner and Wolff on the rise of water in tubes at different temperatures.

## Joachim Murat

The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air. If three fluids which do not mix are in contact with each other, the three surfaces of separation meet in a line, straight or curved. Let O fig. The three angles between the tangent planes to the three surfaces of separation at the point O are completely determined by the tensions of the three surfaces.

For if in the triangle abc the side ab is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides bc and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at O for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab.

Thus the angles at which the surfaces of separation meet are the same at all parts of the line of concourse of the three fluids. If four fluids, a, b, c, d, meet in a point O, and if a tetrahedron ABCD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at O to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at O to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at O to the other three lines of concourse of the liquids, and the other five edges of the tetrahedron will be normal to the tangent planes at O to the other five surfaces of contact.

We must not, however, raise the upper net too much, or the system of films will become unstable. When a drop of one liquid, B, is placed on the surface of another, A, the phenomena which take place depend on the relative magnitude of the three surface-tensions corresponding to the surface between A and air, between B and air, and between A and B. If no one of these tensions is greater than the sum of the other two, the drop will assume the form of a lens, the angles which the upper and lower surfaces of the lens make with the free surface of A and with each other being equal to the external angles of the triangle of forces.

Such lenses are often seen formed by drops of fat floating on the surface of hot water, soup or gravy. But when the surface-tension of A exceeds the sum of the tensions of the surfaces of contact of B with air and with A, it is impossible to construct the triangle of forces; so that equilibrium becomes impossible. The edge of the drop is drawn out by the surface-tension of A with a force greater than the sum of the tensions of the two surfaces of the drop. The drop, therefore, spreads itself out, with great velocity, over the surface of A till it covers an enormous area, and is reduced to such extreme tenuity that it is not probable that it retains the same properties of surface-tension which it has in a large mass.

Thus a drop of train oil will spread itself over the surface of the sea till it shows the colours of thin plates. But even when thus attenuated, the film may be proved to be present, since the surface-tension of the liquid is considerably less than that of pure water. This may be shown by placing another drop of oil on the surface. This drop will not spread out like the first drop, but will take the form of a flat lens with a distinct circular edge, showing that the surface-tension of what is still apparently pure water is now less than the sum of the tensions of the surfaces separating oil from air and water.

The spreading of drops on the surface of a liquid has formed the subject of a very extensive series of experiments by Charles Tomlinson; van der Mensbrugghe has also written a very complete memoir on this subject Sur la tension superficielle des liquides , Bruxelles, When a solid body is in contact with two fluids, the surface of the solid cannot alter its form, but the angle at which the surface of contact of the two fluids meets the surface of the solid depends on the values of the three surface-tensions.

If a and b are the two fluids and c the solid then the equilibrium of the tensions at the point O depends only on that of thin components parallel to the surface, because the surface-tensions normal to the surface are balanced by the resistance of the solid. Hence if the angle ROQ fig. As an experiment on the angle of contact only gives us the difference of the surface-tensions at the solid surface, we cannot determine their actual value.

It is theoretically probable that they are often negative, and may be called surface-pressures. If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest. If the quantity of the first fluid is small it will stand in a drop on the surface of the solid without wetting it.

If the quantity of the second fluid is small it will spread itself over the surface and wet the solid. If a drop of alcohol be made to touch one side of a drop of oil on a glass plate, the alcohol will appear to chase the oil over the plate, and if a drop of water and a drop of bisulphide of carbon be placed in contact in a horizontal capillary tube, the bisulphide of carbon will chase the water along the tube. In both cases the liquids move in the direction in which the surface-pressure at the solid is least.

If T 12 denote the interfacial tension, the energy corresponding to unit of area of the interface is also T 12 , as we see by considering the introduction through a fine tube of one body into the interior of the other. A comparison with another method of generating the interface, similar to that previously employed when but one body was in question, will now allow us to evaluate T The work required to cleave asunder the parts of the first fluid which lie on the two sides of an ideal plane passing through the interior, is per unit of area 2T 1 , and the free surface produced is two units in area.

So for the second fluid the corresponding work is 2T 2. This having been effected, let us now suppose that each of the units of area of free surface of fluid 1 is allowed to approach normally a unit area of 2 until contact is established.