JEE Physics Mechanical Properties Of Fluids Notes Set 05

Surface Tension: Origin and Definition

A molecule inside a liquid is pulled equally in all directions by its neighbours, so the net cohesive force on it is zero. A molecule at the surface has neighbours only below it, so it is pulled inwards. As a result the surface behaves like a stretched elastic membrane that tries to shrink to the smallest possible area.
Surface molecules also have higher energy than interior molecules, so the liquid lowers its energy by reducing the number of surface molecules. This is why small drops and bubbles are spherical.
The force exerted by a surface on a line of contact is proportional to the length of the line: \( F=T\,\ell \). The constant T is the surface tension, the force per unit length acting along the surface, perpendicular to the line. Its SI unit is N/m, and its dimensions are \( [MT^{-2}] \).
A film such as a soap film has two surfaces, so the contact length is counted twice: for a slider of length x the pull is \( F=2Tx \). A soap bubble also has two surfaces.
Surface tension falls with a rise in temperature (it vanishes at the critical temperature), falls when sparingly soluble substances like soap or phenol are added, and rises when highly soluble salts like NaCl are added.

Surface Energy

Extending a surface requires work against surface tension. The surface energy is \( U=T\,A \), so surface tension is also the surface energy per unit area. Increasing the area by \( \Delta A \) needs work \( W=T\,\Delta A \) (use \( \Delta A \) of both faces for a film or bubble).
Splitting a drop of radius R into n equal drops of radius \( r=R\,n^{-1/3} \) needs work \( W=4\pi R^2T\left(n^{1/3}-1\right) \). When n drops of radius r merge into one of radius R, the energy released is \( 4\pi T\,R^3\left(\frac1r-\frac1R\right) \); if all of it becomes heat, the temperature rise is \( \Delta\theta=\frac{3T}{J\,s}\left(\frac1r-\frac1R\right) \) for water in CGS units.

Cohesion, Adhesion and Angle of Contact

Cohesive force acts between molecules of the same substance and adhesive force acts between different substances. The shape of the surface where a liquid meets a solid (the meniscus) depends on which is stronger.
The angle of contact is the angle between the tangent to the liquid surface and the solid surface, measured inside the liquid. If adhesion is strong the angle is acute and the liquid wets the solid (water on glass, concave meniscus). If the angle is obtuse, as for mercury on glass, the meniscus is convex and the liquid does not wet the solid. A 90 degree angle gives a flat surface. The angle depends on the pair of materials, not on the shape or mass of the solid.

Excess Pressure

A curved liquid surface has higher pressure on its concave side. The excess pressure is \( \frac{2T}{R} \) for one surface, where R is the radius of curvature.
Liquid drop (one surface): \( P_{in}-P_{out}=\frac{2T}{R} \).
Air bubble inside a liquid (one surface): \( \frac{2T}{R} \); at depth h, the pressure inside is \( P_0+h\rho g+\frac{2T}{R} \).
Soap bubble in air (two surfaces): \( P_{in}-P_{out}=\frac{4T}{R} \).
Two bubbles in contact share a film of radius \( R=\frac{r_1r_2}{r_2-r_1} \). When two soap bubbles merge isothermally in vacuum, \( R^2=r_1^2+r_2^2 \). If two bubbles are joined by a tube, air flows from the smaller (higher pressure) to the larger one.

Capillarity

The rise or fall of a liquid in a narrow tube is called capillarity. For a tube of radius r with contact angle \( \theta \), the meniscus has radius of curvature \( r/\cos\theta \), and balancing pressures gives Jurin's law \( h=\frac{2T\cos\theta}{r\rho g} \).
For an acute angle the liquid rises, for an obtuse angle (mercury) it falls. The rise is inversely proportional to the radius, and inversely proportional to the effective gravity: it is larger in a lift accelerating downwards and infinite in a freely falling frame. In an inclined tube the vertical height stays h and the length along the tube is \( h/\cos\alpha \).
Between two parallel plates a distance d apart: \( h=\frac{2T\cos\theta}{d\rho g} \). The weight of the raised liquid column equals the vertical pull of the surface, \( 2\pi rT\cos\theta \).
If a column of liquid has two free surfaces (as in a drop trapped in a tube open at both ends), both menisci contribute and the column length is \( \frac{4T\cos\theta}{r\rho g} \).
Energy in capillary rise: the work done by surface tension is \( 2\pi rTh \), of which half is stored as potential energy \( \pi rTh \) and the other half is dissipated as heat.

Worked Examples

1. A soap bubble of radius 2 cm has surface tension 0.03 N/m. Excess pressure: \( \frac{4\times0.03}{0.02}=6 \) N/m².
2. 1000 droplets of radius r form a drop of radius 10r. Energy lost: \( T\,4\pi r^2(1000-100)=900\times4\pi r^2T \).
3. A needle floats on water because the vertical component of the surface tension along its length supports its weight. A thin disc of radius r floats if \( mg=W_{displaced}+2\pi rT\cos\theta \).