There can be no doubt that if the amount of matter compressed into unit volume (i.e., the density p) be increased the value of the intrinsic pressure would be increased. Moreover, a factor r (akin to the gravitation constant) is also needed if K is to be expressed in ordinary dynamical units. Hence finally The quantity, K, is called the Laplace pressure or intrinsic pres sure. More generally we could assume that the force between elementary volumes is f (r). The successive integrals then be come where Oa, VI are written as the values of successive integrals. Finally K is obtained in ordinary units by multiplying by rp2 For example if f(r)= where k is a constant we have Cs) = k3 E –k s • The difficulty in testing these formulae arises from the occur rence of three variables (k, s, F). The only one of these about which we know the value independently is s which for liquids must be of the same order as the molecular diameter. Caution is necessary in considering forces which vary so fast with the distance as cohesive forces undoubtedly do. A small error in the assumed value for s would lead to very considerable error in k andT. The intrinsic pressure at the breaking point of a solid in tension should be the breaking stress itself. The latter as actually measured is, however, only a lower limit of K since flaws in the material and irregularities in the application of the tension will lead to fracture occurring at a value lower than the ideal stress. The effect of cohesion on the characteristic equa tion of a gas was allowed for by van der Waals who, following Laplace, took the intrinsic pressure as varying with the density square alone. The equation becomes b) = RT where at the same time b is introduced to represent the fact that the volume of the gas cannot be reduced to zero. Here K= which for water at ordinary densities is between 10-20 Atmos. The fact that the equation turns out to be approximate only makes caution necessary. Moreover a is found to vary nearly inversely as the absolute temperature. We are obliged to as sume, therefore, either that r is a function of the temperature or, what is more likely, that s is such a function. The latter assumption is indeed a very plausible one, for the least average distance of approach of the molecules must be affected by the motion of agitation amongst them and this increases as the square root of the absolute temperature.
Besides these considerations it must be pointed out that the modification worked out by the Dutch school only partly allows for the effects arising from the molecular character of matter. Knowing the forces between two molecules at a given distance apart, the actual value of the intrinsic pressure could only be correctly obtained by summing up the components (normal to the interface at which the pressure is being calculated) of all the forces between every pair of molecules each being calculated for the actual positions in which the molecules instantaneously are. It is true, that with forces that vary so fast with distance it is only the molecules which are fairly near to a point whose individual contributions are important : the contributions of the remainder may be estimated by an approximate method such as that followed by Laplace. Something has been done in this direction (Lennard-Jones, Faraday Society, "Discussion on co hesion," 1927) but the fact remains that though the law of force is probably greater than the inverse fifth it may vary up to the inverse seventh so far as present knowledge goes. If we
take the higher of these two estimates the force between mole cules which are neighbours and those that are next door to neighbours will be as to I, i.e., in the ratio 128 tor or that between any layer and the next and next but one contiguous layer as to I, i.e., in the ratio of 64 to 1. Hence all but a small percentage of the intrinsic pressure is due to the molecules in a small volume surrounding the point. This conclusion is important since it signifies that in any material (gas, liquid, solid) the intrinsic pressure will be uniform at all points except for a thin layer of usually negligible volume near its boundary.
Theory of Surface Forces.—This surface layer has theoreti cally no inner boundary but it may be taken, for most purposes, as being only a few molecules thick (mol. diameter is of the order of r cm.). The most prominent peculiarity is that it possesses more potential energy than the rest of the body per unit volume.
Now increase of free energy at constant temperature is equal to the external work done on a system. If we write dW =a dA is called the surface tension. To determine its value theoretically we estimate dW for the formation of a surface of unit area. Now if a body is split by a plane section and the two halves separated, a fresh surface is formed. The attraction per unit area between the halves at any distance x is co The total work of separation isf K (x) dx and in the process a fresh area of 2 units is formed. Hence expressing the .work done in ordinary units (In previous editions many theorems were given [some of which date from Laplace] which depend, however, upon the lower limit being zero. It is sufficient to say that these theorems are not applicable to the formulae given here.) If a power law is taken as the law of force between molecules it follows that Young took the functions F as being identical being all the same function of zero instead of the molecular diameters; whence PO' F(o) or 1.1 cri2= const. X PO. Hence for three liquids taken pair by pair At an+ -?./ .723+I Q31=0.
These simplified relations are known not to be true. On the other hand if the power law is valid and is the same law for different substances the equations given here lead to and the square root law should not be expected to be true ex cepting in special cases in which the molecular diameters of the three substances were the same.
All surface-tensions with which we have to deal are in reality interfacial tensions since a liquid is always in contact with its own vapour or the gas in which it is immersed. The effect due to the gas is known, however, to be very small.
The normal pressure to be expected from the existence of a superficial tension is best obtained by considering in the first place a cylinder.