Surface Tension Formulas

1. Surface tension

The free surface of a liquid behaves as stretched membrane. The force per unit length in a perpendicular direction on an imaginary line drawn in the surface of a liquid is called surface tension is also equal to the work done to increase the surface area of a liquid by unity.
T = \(\frac{F}{\ell}=\frac{\Delta W}{\Delta A}\) N/m or J/m²

2. Surface energy density

E = \(\frac{\Delta \mathrm{W}}{\Delta \mathrm{A}}\) = T

3. Shape of meniscus

• F_a = \(\frac{F_{c}}{\sqrt{2}}\) ; Plane, θ = 90° No change
• F_a < \(\frac{F_{c}}{\sqrt{2}}\) ; Convex, θ > 90° Leave fall
• F_a > \(\frac{F_{c}}{\sqrt{2}}\) ; Concave, θ < 90° Level rise

4. Rise or fall of liquid in a capillary

h = \(\frac{2 \mathrm{T} \cos \theta}{\mathrm{rdg}}\)
If capillary is inclined by an angle Φ with vertical, h is constant but length of liquid column
l = \(\frac{\mathrm{h}}{\cos \phi}\)
h ∝ \(\frac{1}{r}\)
or h₁r₁ = h₂r₂ (Jurin’s law)
If radius of capillary is r and radius of curvature of meniscus is R then-
r = R cos θ

5. Excess pressure due to surface tension

In a drop p = \(\frac{2 \mathrm{T}}{\mathrm{r}}\)
In a bubble in air p = \(\frac{4 \mathrm{T}}{\mathrm{r}}\)
In an air bubble in liquid
p = \(\frac{2 \mathrm{T}}{\mathrm{r}}\)
If surface is curved in two directions,
P = T\(\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\)

6. Radius of curvature of common surface of two bubbles in contact

r = \(\frac{r_{1} r_{2}}{r_{2}-r_{1}}\)

7. Radius of bubble formed when two bullbes coalesce

r = \(\sqrt{r_{1}^{2}+r_{2}^{2}}\)

8. Work done in blowing a bubble

W = 8πT(r₁² + r₂²)

9. Breaking of a big drop in n droplets

n = \(\frac{\mathrm{R}^{3}}{\mathrm{r}^{3}}\)
Increase in area ΔA = 4π(nr² – R²), W =4πTR³\(\left(\frac{1}{r}-\frac{1}{R}\right)\)
Fall in temperature Δθ = \(\frac{3 \mathrm{T}}{\mathrm{Jsd}}\left(\frac{1}{\mathrm{r}}-\frac{1}{\mathrm{R}}\right)\)

10. Liquid film between parallel plates

Required to separate the plates.
F = \(\frac{2 \mathrm{AT}}{\mathrm{t}}\)

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