13Surface Tension

📚
SURFACE TENSION
▢ Introduction:
❖ Definition: Property of liquid surface by virtue of which it behaves like stretched membrane and tends to occupy minimum surface area

Table 1: Surface Tension Basics

Quantity
Formula / Point
Surface tension
\(T=\frac{F}{L}\)
Surface energy relation
\(T=\frac{Surface\ energy}{Area}=\frac{W}{\Delta A}\)
SI unit
\(N\ m^{-1}\)
Dimension
\([MT^{-2}]\)
Nature
Scalar quantity
Cause
Intermolecular cohesive force between liquid molecules
❖ Special Force Formulae:

Table 1: Force Due to Surface Tension

Case
Formula
Ring of radius \(R\) floating on liquid surface
\(F=T\times 2(2\pi R)=4\pi RT\)
Straight thread of length \(l\) on liquid surface
\(F=T\times 2l=2Tl\)
Disc radius \(R\) with concentric hole radius \(r\)
\(F=T\cdot2\pi(R+r)\)
▢ Factors Affecting Surface Tension:

Table 1: Factors Affecting Surface Tension

Factor
Effect
Temperature increases
Surface tension generally decreases
Critical temperature
Surface tension becomes zero
Molten cadmium and copper
Surface tension increases with temperature
Highly soluble impurities e.g. NaCl
Surface tension increases
Slightly soluble impurities e.g. soap, camphor, phenol
Surface tension decreases
▢ Shape of Meniscus and Angle of Contact:

Table 1: Meniscus, Wetting and Capillary Action

Condition
\(F_c<\sqrt2F_a\)
\(F_c=\sqrt2F_a\)
\(F_c>\sqrt2F_a\)
Angle of contact
< 90°
= 90°
> 90°
Wetting
Liquid wets solid
Liquid does not wet solid properly
Liquid does not wet solid
Meniscus
Concave upward
Plane surface
Convex upward
Capillary action
Rises up
No capillary rise/fall
Falls down
Example
Water/kerosene in glass vessel
Water in silver vessel
Mercury in glass vessel
❖ Symbols:
  • •
    \(F_c\) = cohesive force
  • •
    \(F_a\) = adhesive force
▢ Angle of Contact:
❖ Definition: Angle between tangent drawn on liquid surface and solid surface inside liquid at point of contact

Table 1: Angle of Contact

Point
Value / Effect
Range
0° to 180°
Glass-water
< 10° ≈ 8° ≈ 0°
Glass-mercury
≈ 135°
Inclination of solid
No effect
Temperature increases
Angle of contact decreases
Highly soluble impurity added to water
Angle of contact increases
Sparingly soluble impurity added
Angle of contact decreases
▢ Excess Pressure:
❖ Definition: Pressure difference across a curved liquid surface
❖ Key Point: Pressure on concave side is greater than pressure on convex side

Table 1: Excess Pressure Formulae

Case
Formula
Different medium / single surface
\(\Delta P=\frac{2T}{R}\)
Same medium / two surfaces
\(\Delta P=\frac{4T}{R}\)
Soap bubble within solution
\(\Delta P=\frac{2T}{R}\)
Soap bubble in air
\(\Delta P=\frac{4T}{R}\)
Liquid drop in air
\(\Delta P=\frac{2T}{R}\)
Liquid film of thickness \(d\) between two plates
\(\Delta P=\frac{2T}{d}\)
Cylindrical surface
\(\Delta P=\frac{T}{R}\)
Air bubble radius \(r\) at depth \(h\) in water
\(P'=P+h\rho g+\frac{2T}{r}\)
▢ Capillarity:
❖ Definition: Rise or fall of liquid column inside a capillary tube
❖ Capillary Rise / Fall:

Table 1: Capillary Formulae

Quantity
Formula
Height of liquid column
\(h=\frac{2T\cos\theta}{r\rho g}=\frac{2T}{R\rho g}\)
Radius of meniscus
\(R=\frac{r}{\cos\theta}\)
For pure water
\(\theta=0^\circ,\ \cos\theta=1\)
Pure water capillary rise
\(h=\frac{2T}{r\rho g}\)
Volume of liquid raised
\(V=\frac{2\pi rT\cos\theta}{\rho g}\)
Mass of liquid raised
\(m=\frac{2\pi rT\cos\theta}{g}\)
Excess potential energy
\(U=mg\frac{h}{2}=\pi rTh\cos\theta\)
❖ Symbols:
  • •
    \(T\) = surface tension
  • •
    \(\theta\) = angle of contact
  • •
    \(r\) = radius of tube
  • •
    \(\rho\) = density of liquid
  • •
    \(g\) = acceleration due to gravity
  • •
    \(R\) = radius of meniscus
❖ Jurin's Law: \(h\propto \frac{1}{r}\)
❖ Inclined Capillary Tube:

Table 1: Tilted Capillary Tube

Condition
Result
Capillary tilted by angle \(\alpha\) with vertical
Vertical height remains \(h\)
Length of liquid column
\(l=\frac{h}{\cos\alpha}\)
Angle of contact
Remains unchanged
❖ Depends On:
  • •
    Liquid
  • •
    Solid
  • •
    Angle of contact
  • •
    Surface tension
  • •
    Radius of capillary tube
  • •
    Density of liquid
  • •
    Acceleration due to gravity
▢ Glass Plates and Liquid Film:

Table 1: Liquid Film Between Glass Plates

Case
Formula / Point
Minimum force to separate two plates of area \(A\), film thickness \(d\)
\(F=\Delta P\cdot A=\frac{2T}{d}A\)
Water layer squeezed between parallel glass plates
Pressure inside water layer is less than pressure on plates
▢ Bubbles and Drops:

Table 1: Bubbles and Drops Formulae

Case
Formula / Result
Air passes between connected bubbles
From smaller bubble to larger bubble
Reason
\(\Delta P\) is larger for smaller radius
Two bubbles of radii \(r_1>r_2\) in contact
\(R=\frac{r_1r_2}{r_1-r_2}\)
Common interface
Concave towards smaller bubble
Two soap bubbles coalesce in vacuum isothermally
\(R=\sqrt{r_1^2+r_2^2}\)
Several soap bubbles coalesce in vacuum isothermally
\(R^2=r_1^2+r_2^2+...+r_n^2\)
\(n\) identical soap bubbles radius \(r\) coalesce
\(R=\sqrt n\ r\)
Several liquid droplets coalesce
Volume conserved: \(R^3=r_1^3+r_2^3+...+r_n^3\)
\(n\) identical droplets radius \(r\) coalesce
\(R=n^{1/3}r\)
Two bubbles coalesce
Energy is released
Soap bubble charged positive/negative
Radius increases
▢ Work Done and Energy:

Table 1: Surface Energy Formulae

Case
Work / Energy
Forming liquid drop of radius \(R\)
\(W=4\pi R^2T\)
Increasing liquid drop radius \(r_1\to r_2\)
\(W=4\pi T(r_2^2-r_1^2)\)
Forming soap bubble radius \(R\) in air
\(W=8\pi R^2T\)
Increasing soap bubble radius \(r_1\to r_2\)
\(W=8\pi T(r_2^2-r_1^2)\)
Liquid drop radius \(R\) breaks into \(n\) equal drops
\(W=4\pi R^2T(n^{1/3}-1)\)
\(n\) identical drops radius \(r\) combine to form big drop radius \(R\)
\(W=4\pi T(nr^2-R^2)\)
Coalescence of small drops
Energy is released
Breaking drop into smaller drops
Energy is absorbed
❖ Temperature Rise on Coalescence:
◉ Condition: Small drops of density \(\rho\), specific heat \(c\), surface tension \(T\), radius \(r\), coalesce into big drop radius \(R\)
◉ Formula: \(\Delta\theta=\frac{3T}{J\rho c}\left(\frac{1}{r}-\frac{1}{R}\right)\)
◉ J: Mechanical equivalent of heat
▢ Insufficient Capillary Length:
❖ Condition: Available capillary height \(h_1\) is less than actual capillary rise \(h\)

Table 1: Insufficient Capillary Tube

Point
Result
Overflow
Liquid does not overflow
Liquid level
Rises to entire length of tube
Radius of curvature
Increases
Nature of meniscus
Unchanged
Relation
\(h_1R=hr\)
▢ Read and Digest:

Table 1: Important Surface Tension Points

Fact
Answer
Surface tension at boiling point
Zero
Liquid drops are spherical
To have minimum surface area
Small droplets more spherical than large drops
Surface tension dominates gravity
Waterproofing agent
Changes angle of contact from acute to obtuse
Waterproofing materials
Increase surface tension and angle of contact
Pressure just below water meniscus in glass tube
Less than pressure above it
Surface tension, elasticity, viscosity
Arise due to intermolecular cohesive force
Wider capillary tube
More volume and mass raised
Potential energy of raised capillary liquid
Independent of tube radius
Capillary rise on Moon
Six times that on Earth
Surface molecule P.E.
Greater than molecule inside liquid
Rise of oil in lamp wick
Due to capillarity
Weightlessness capillary tube
Water rises to other end but does not overflow
Spiders/insects on water
Surface tension acts like elastic membrane
▢ High-Yield Recall:

Table 1: Surface Tension One-Liners

Fact
Answer
Surface tension
\(T=\frac{F}{L}=\frac{W}{\Delta A}\)
Dimension
\([MT^{-2}]\)
Nature
Scalar
Cause
Cohesive intermolecular force
Temperature increases
Surface tension decreases
At critical temperature
Surface tension = 0
NaCl added
Surface tension increases
Soap/camphor/phenol added
Surface tension decreases
Angle of contact range
0° to 180°
Glass-water angle
≈ 8° ≈ 0°
Glass-mercury angle
≈ 135°
Concave meniscus
Liquid wets solid; angle < 90°
Convex meniscus
Liquid does not wet solid; angle > 90°
Soap bubble in air excess pressure
\(\Delta P=\frac{4T}{R}\)
Liquid drop excess pressure
\(\Delta P=\frac{2T}{R}\)
Cylindrical surface excess pressure
\(\Delta P=\frac{T}{R}\)
Capillary rise
\(h=\frac{2T\cos\theta}{r\rho g}\)
Jurin's law
\(h\propto\frac{1}{r}\)
Mass raised in capillary
\(m=\frac{2\pi rT\cos\theta}{g}\)
Volume raised in capillary
\(V=\frac{2\pi rT\cos\theta}{\rho g}\)
Inclined capillary
Vertical height unchanged; column length increases
Air between connected bubbles
Moves from smaller to larger bubble
Two soap bubbles coalesce
\(R=\sqrt{r_1^2+r_2^2}\)
\(n\) liquid drops coalesce
\(R=n^{1/3}r\)
\(n\) soap bubbles coalesce
\(R=\sqrt n\ r\)
Liquid drop formation work
\(4\pi R^2T\)
Soap bubble formation work
\(8\pi R^2T\)
Capillary rise on Moon
6 times Earth
Oil rises in wick
Capillarity
Q1.
A soap bubble (surface tension = 30×10⁻³ N/m) has radius 2 cm. The work done in doubling the radius is:
📅BP 2014
Q2.
Two capillary tubes made of same material but different radius were dipped into water:
Q3.
The surface energy of a soap bubble is proportional to its radius as:
📅BP 2011
Q4.
Oil kept in frying pan spreads more easily when hot due to:
Q5.
When two drops combine to form a big drop, ratio of surface energies (2 drops:big drop) is:
📅BP 2010
Q6.
Excess pressure inside 1 cm diameter soap bubble (T=25×10⁻³ N/m) is:
📅MOE 2014
Q7.
When liquid is cooled, its surface tension:
📅MOE 2012
Q8.
With rise in temperature, surface tension:
📅IOM 2013•KU 2010
Q9.
Water rises 3 cm in vertical capillary. If inclined at 30°, rise will be:
📅IOM 2011
Q10.
Waterproofing agent changes angle of contact:
📅IOM 2011
Q11.
Two unequal soap bubbles connected:
📅KU 2014
Q12.
Work to expand soap film from 10×6 cm to 10×11 cm (T=3×10⁻² N/m):
📅IE 2011
Q13.
Reason for water droplet being spherical:
📅BP 2013
Q14.
Capillary tube (5 cm long, 0.1 mm radius) in water (T=25 dyne/cm):
📅Bangladesh 09
Q15.
Water rise in 0.044 mm diameter capillary (T=73 dyne/cm):
📅IOM 08
Q16.
Two radius r bubbles coalesce into one bubble of radius R:
📅MOE 2062
Q17.
Capillary rise when cross-section reduced to 1/4th original:
📅MOE 2010
Q18.
Tension in string when stone falls freely:
📅IE-01
Q19.
Work to blow soap bubble of radius r (surface tension T):
📅IE-04
Q20.
Capillary rise at 60° inclination vs. vertical 2 cm rise:
📅BPKIHS-07
Q21.
Detergents remove grease by:
📅BPKIHS 05
Q22.
Work to double soap bubble radius R:
📅BPKIHS-06
Q23.
Contact angle when liquid doesn't wet surface:
📅MOE/BPKIHS-97
Q24.
Depth for 0.4mm air bubble equilibrium (T=72×10⁻³ N/m):
Q25.
Length of water column in vertical 2mm radius capillary (T=73.5×10⁻³ N/m):
Q26.
Work to expand soap film from 10×6 cm to 10×10 cm (T=0.030 N/m):
Q27.
Radius of common interface when 3mm and 4mm soap bubbles coalesce:
Q28.
Ratio of liquid heights in capillaries (SG ratio 0.4:0.8, T ratio 6:5):
Q29.
Work to double radius of 2cm soap bubble (T=3.0×10⁻² N/m):
Q30.
Volume ratio of bubbles with internal pressures 1.01:1.02 atm:
Q31.
Force to pull 5cm radius plate from water (T=75×10⁻³ N/m):
Q32.
Excess pressure inside soap bubble of radius r:
Q33.
Work to blow bubble of volume 2V vs. volume V:
📅KU 2015
Q34.
Ratio of final to initial surface energy when 1000 drops combine:
Q35.
Work to increase bubble radius from R to 3R (initial work W):
Q36.
Work to break 1 cm mercury drop into 10⁶ droplets (T=35×10⁻³ N/m):
Q37.
Mass of water in capillary when radius changes from r to 2r:
Q38.
Excess pressure ratio for bubbles with radii 2:1:
Q39.
Length of liquid column when 3 cm vertical capillary tilted 60°:
Q40.
Work to expand soap bubble diameter from D to 3D (T=surface tension):
Q41.
Apparent contact angle when capillary tip is 1 cm above liquid (original rise 2 cm):
Q42.
Radius of capillary supporting 6.28×10⁻⁴ N liquid weight (T=5×10⁻² N/m):
Q43.
Capillary rise is maximum when water temperature is:
Q44.
Capillary rise in satellite compared to 0.1 m on Earth:
Q45.
Surface tension force on disc with hole (outer R, inner r):
Q46.
Height of water column in 1 mm radius vertical capillary (T=73.5×10⁻³ N/m):
Q47.
Force to pull 5 cm radius plate from water (T=75 dyne/cm):
📅KU 2015
Q48.
Liquid height when vessel length halved (original height h' < h):
📅KU 2017