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 = $\frac{F}{\ell }=\frac{\mathrm{\Delta }W}{\mathrm{\Delta }A}$ N/m or J/m²

2. Surface energy density

E = $\frac{\mathrm{\Delta }\mathrm{W}}{\mathrm{\Delta }\mathrm{A}}$ = 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 = $\frac{\mathrm{h}}{\cos \varphi }$
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$(\frac{1}{r_1}+\frac{1}{r_2})$

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$(\frac{1}{r}-\frac{1}{R})$
Fall in temperature Δθ = $\frac{3\mathrm{T}}{\mathrm{Jsd}}(\frac{1}{\mathrm{r}}-\frac{1}{\mathrm{R}})$

10. Liquid film between parallel plates

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

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