Page:EB1911 - Volume 05.djvu/274

phenomena would not take place unless the density varied rapidly near the surface. In this assertion we think he was mathematically wrong, though in his own hypothesis that the density does actually vary, he was probably right. In fact, the quantity $4\pi \rho ^{2}{\mbox{K}}$, which we may call with van der Waals the molecular pressure, is so great for most liquids (5000 atmospheres for water), that in the parts near the surface, where the molecular pressure varies rapidly, we may expect considerable variation of density, even when we take into account the smallness of the compressibility of liquids.

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

$p={\mbox{P}}+{\mbox{A}}\rho ^{2},$

where ${\mbox{A}}$ is a constant [equal to Laplace’s intrinsic pressure ${\mbox{K}}$. But this equation is applicable only at points in the interior, where $\rho$ is not varying.]

[The intrinsic pressure and the surface-tension of a uniform mass are perhaps more easily found by the following process. 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. If the density be $\sigma$, the attraction between the whole of one side and a layer upon the other distant ${\mbox{z}}$ from the plane and of thickness $dz$ is $2\pi \sigma ^{2}\psi ({\mbox{z}})dz$, reckoned per unit of area. The expression for the intrinsic pressure is thus simply

${\mbox{K}}=2\pi \sigma ^{2}\int _{0}^{\infty }\psi (z)dz$

(28)

In Laplace’s investigation $\sigma$ is supposed to be unity. We may call the value which (28) then assumes K₀, so that as above

${\mbox{K}}_{0}=2\pi \int _{0}^{\infty }\psi (z)dz$

(29)

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. From this consideration we may derive Laplace’s expression, as has been done by Dupre (Théorie mécanique de la chaleur, Paris, 1869), and Kelvin ("Capillary Attraction," Proc. Roy. Inst., January 1886. Reprinted, Popular Lectures and Addresses, 1889). 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.

If $\sigma _{1}$, $\sigma _{2}$ represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area

$2\pi \sigma _{1}\sigma _{2}\int _{z}^{\infty }\psi (z)dz,$

(30)

or $2\pi \sigma _{1}\sigma _{2}\theta (z)$, if we write

$\int _{z}^{\infty }\psi (z)dz=\theta (z)$

(31)

The work required to produce the separation in question is thus

$2\pi \sigma _{1}\sigma _{2}\int _{0}^{\infty }\theta (z)dz$

(32)

and for the tension of a liquid of density $\sigma$ we have

${\mbox{T}}=\pi \sigma ^{2}\int _{0}^{\infty }\theta (z)dz$

(33)

The form of this expression may be modified by integration by parts. For

$\int \theta (z)dz=\theta (z).z-\int z{\frac {d\theta (z)}{dz}}dz=\theta (z).z+\int z\psi (z)dz$

Since $\theta (0)$ is finite, proportional to ${\mbox{K}}$, the integrated term vanishes at both limits, and we have simply

$\int _{0}^{\infty }\theta (z)dz=\int _{0}^{\infty }z\psi (z)dz$

(34)

and

${\mbox{T}}=\pi \sigma ^{2}\int _{0}^{\infty }z\psi (z)dz$

(35)

In Laplace's notation the second member of (34), multiplied by $2\pi$, is represented by ${\mbox{H}}$.

As Laplace has shown, the values for ${\mbox{K}}$ and ${\mbox{T}}$ may also be expressed in terms of the function $\phi$, with which we started. Integrating by parts, we get

$\int \psi (z)dz=z\psi (z)+{\tfrac {1}{3}}z^{3}\Pi (z)+{\tfrac {1}{3}}\int z^{3}\phi (z)dz,$

$\int z\psi (z)dz={\tfrac {1}{2}}z^{2}\psi (z)+{\tfrac {1}{8}}z^{4}\Pi (z)+{\tfrac {1}{8}}\int z^{3}\phi (z)dz,$

In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write

$\int _{0}^{\infty }\psi (z)dz={\tfrac {1}{3}}\int _{2}^{\infty }z^{3}\phi (z)dz,\quad \int _{0}^{\infty }z\psi (z)dz={\tfrac {1}{8}}\int _{0}^{\infty }z^{4}\phi dz;$

(36)

so that

${\mbox{K}}_{0}={\frac {2\pi }{3}}\int _{0}^{\infty }z^{3}\phi (z)dz,\quad {\mbox{T}}_{0}={\frac {\pi }{8}}\int _{0}^{\infty }z^{4}\phi (z)dz$

(37)

A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that

$\phi (f)=e^{-\beta }f$

(38)

From this we obtain

$\Pi (z)=\beta ^{-1}e^{-\beta z},\quad \psi (z)=\beta ^{-3}(\beta z+1)e^{-\beta z}$

(39)

${\mbox{K}}_{0}=4\pi \beta ^{-4},\quad {\mbox{T}}_{0}=3\pi \beta ^{-5}$

(40)

The range of the attractive force is mathematically infinite, but practically of the order $\beta ^{-1}$, and we see that ${\mbox{T}}$ is of higher order in this small quantity than ${\mbox{K}}$. That ${\mbox{K}}$ is in all cases of the fourth order and ${\mbox{T}}$ of the fifth order in the range of the forces is obvious from (37) without integration.

An apparently simple example would be to suppose $\phi (z)=z^{n}$. We get

${\begin{matrix}\Pi (z)&=&-{\frac {z^{n+1}}{n+1}},\quad \psi (z)={\frac {z^{n+3}}{n+3.n+1}},\\K_{0}&=&{\frac {2\pi z}{n+4.n+3.n+1}}{\big |}_{0}^{\infty }\end{matrix}}$

(41)

The intrinsic pressure will thus be infinite whatever n may be. If $n+4$ be positive, the attraction of infinitely distant parts contributes to the result; while if $n+4$ be negative, the parts in immediate contiguity act with infinite power. For the transition case, discussed by William Sutherland (Phil. Mag. xxiv. p. 113, 1887), of $n+4=0$, $K_{0}$ is also infinite. It seems therefore that nothing satisfactory can be arrived at under this head.

As a third example, we will take the law proposed by Young, viz.

$\left.{\begin{matrix}\phi (z)=1&{\mbox{ from }}&z=0&{\mbox{ to }}&z=a,\\\phi (z)=0&{\mbox{ from }}&z=a&{\mbox{ to }}&z=\infty ;\end{matrix}}\right\}$

(42)

and corresponding therewith,

$\left.{\begin{matrix}\Pi (z)=&a-z&{\mbox{ from }}&z=0&{\mbox{ to }}&z=a,\\\Pi (z)=&0&{\mbox{ from }}&z=a&{\mbox{ to }}&z=\infty ,\end{matrix}}\right\}$

(43)

$\left.{\begin{matrix}\psi (z)&=&{\tfrac {1}{2}}a(a^{2}-z^{2})-{\tfrac {1}{3}}(a^{3}-z^{3})\\{}&&{\mbox{ from }}z=0{\mbox{ to }}z=a,\\\psi (z)&=&0{\mbox{ from }}z=a{\mbox{ to }}z=\infty ,\end{matrix}}\right\}$

(44)

Equations (37) now give

${\mbox{K}}_{0}={\frac {2\pi }{3}}\int _{0}^{\infty }z^{3}dz={\frac {\pi a^{3}}{6}},$

(45)

${\mbox{T}}_{0}={\frac {\pi }{8}}\int _{0}^{a}z^{4}dz={\frac {\pi a^{5}}{40}}$

(46)

The numerical results differ from those of Young, who finds that "the contractile force is one-third of the whole cohesive force of a stratum of particles, equal in thickness to the interval to which the primitive equable cohesion extends," viz. ${\mbox{T}}={\tfrac {1}{3}}a{\mbox{K}}$; whereas according to the above calculation ${\mbox{T}}={\tfrac {3}{20}}a{\mbox{K}}$. The discrepancy seems to depend upon Young having treated the attractive force as operative in one direction only. For further calculations on Laplace’s principles, see Rayleigh, Phil. Mag., Oct. Dec. 1890, or Scientific Papers, vol. iii. p. 397.]

On Surface-Tension

Definition.—The tension of a liquid surface across any line drawn on the surface is normal to the line, and is the same for all directions of the line, and is measured by the force across an element of the line divided by the length of that element.

Experimental Laws of Surface-Tension.—1. 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. 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.

2. 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