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Mechanics
Surface Tension and Surface Energy
Surface tension is the property of a liquid surface by virtue of which it tends to minimize its surface area, arising from intermolecular cohesive forces. It is quantitatively measured as the force acting normally per unit length on an imaginary line drawn on the liquid surface.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of surface tension, molecular theory of surface tension, surface energy, excess pressure inside liquid drops and soap bubbles, capillarity, and angle of contact.
PGT extension
Work done in blowing bubbles, coalescence of drops, capillary rise accounting for meniscus, temperature and impurity effects on surface tension, and rigorous mathematical formulations of excess pressure.
Prerequisites
Intermolecular forces (Cohesive and Adhesive forces)
Work, Energy, and Power
Hydrostatics and Pressure
Molecular Theory of Surface Tension
Molecules deep inside the liquid experience balanced cohesive forces from surrounding molecules within a sphere of molecular activity.
Molecules within a surface layer of thickness equal to the range of molecular force (approx 10-9 m) experience a net inward attractive force.
Bringing a molecule from the interior to the surface requires work against this inward force, which is stored as potential energy at the surface.
Surface Energy and Surface Tension
Surface energy is defined as the potential energy per unit surface area of the liquid film.
Surface tension (T) is numerically equal to the surface energy per unit area under isothermal conditions.
Dimension of surface tension is [MT-2] and its SI unit is N/m or J/m2.
Excess Pressure across Curved Surfaces
Due to surface tension, molecules on a curved liquid surface experience a net inward force, creating an excess pressure on the concave side.
For a spherical liquid drop of radius R, the excess pressure is given by $P = 2T/R$.
For a soap bubble in air, which has two free surfaces, the excess pressure is given by $P = 4T/R$.
Capillarity and Angle of Contact
Angle of contact (θ) is the angle enclosed between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.
Liquids that wet the glass (e.g., water-glass) have an acute angle of contact and rise in capillaries.
Liquids that do not wet the glass (e.g., mercury-glass) have an obtuse angle of contact and are depressed in capillaries.
Derivations
Excess Pressure Inside a Liquid Drop
Conditions: Spherical drop of radius R in equilibrium
Consider a spherical liquid drop of radius R and surface tension T.
Let the radius of the drop increase by a small amount ΔR due to excess pressure P inside.
Work done by excess pressur$e = Force \times displacement = (P \times 4\piR^{2}) \times \DeltaR$.
Potential energy store$d = T \times \DeltaA = T(8\piR\DeltaR)$.
Equating work done to stored potential energy: P(4πR2)Δ$R = T(8\piR\DeltaR)$.
Result: P = \frac{2T}{R}
Capillary Rise Formula (Jurin's Law)
Conditions: Circular capillary tube of radius r dipped in a liquid of surface tension T and density ρ
Let a capillary tube of radius r be dipped in a liquid of surface tension T and contact angle θ.
The liquid rises to a height h, forming a meniscus.
The upward force due to surface tension along the circumference of the meniscus is $F = (2\pir)T \cos\theta$.
This upward force balances the weight of the liquid column of height h in the tube.
Weight of the liquid column $W = Volume \times density \times g = (\pir^{2}h + V_meniscus) \rhog$, where V_meniscus is approximated for narrow tubes.
Equating upward force and weight: (2πr)T cos$\theta = \pir^{2}h\rhog$.
Result: h = \frac{2T \cos\theta}{r\rho g}
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Surface Tension Definition
$$T = \frac{F}{l}$$
F = Force, l = Length of line N/m
Isothermal conditions
Work Done in Expanding a Liquid Film
$$W = T \times \Delta A$$
T = Surface tension, ΔA = Change in total surface area Joule (J)
Bilateral surfaces (multiply ΔA by 2 for soap bubbles)
Excess Pressure in Liquid Drop
$$P = \frac{2T}{R}$$
P = Excess pressure, T = Surface tension, R = Radius of drop N/m2 (Pa)
Spherical drop
Excess Pressure in Soap Bubble
$$P = \frac{4T}{R}$$
P = Excess pressure, T = Surface tension, R = Radius of bubble N/m2 (Pa)
Two free surfaces
Capillary Rise Height
$$h = \frac{2T \cos\theta}{r\rho g}$$
h = Height, T = Surface tension, $\theta = Contact angle$, r = Tube radius, $\rho = Density$, g = Acceleration due to gravity m
Narrow cylindrical capillary tube
Work Done in Droplet Coalescence
$$W = 4\pi R^2 T \left( n^{1/3} - 1 \right)$$
R = Radius of big drop, n = Number of small drops, T = Surface tension Joule (J)
Volume conservation of liquid
Temperature Dependence of Surface Tension
$$T_t = T_0 (1 - \alpha t)$$
$T_t = Tension at t°C$, $T_0 = Tension at 0°C$, $\alpha = Temperature coefficient$ N/m
Valid for moderate temperature ranges below critical temperature
Total Pressure Inside a Soap Bubble
$$P_{total} = P_0 + \frac{4T}{R}$$
$P_0 = Atmospheric pressure$, T = Surface tension, R = Radius Pa
Submerged or in open atmosphere
Graphs and experiments
Surface Tension vs Temperature Graph
Draw: A nearly linear downward sloping curve of Surface Tension (T) on the y-axis versus Temperature (t) on the x-axis, intersecting zero at the critical temperature.
Interpret: Surface tension decreases linearly with an increase in temperature due to an increase in thermal agitation weakening intermolecular cohesive forces.
Exam use: Directly tested in conceptual MCQs regarding the behavior of liquids near boiling point.
Capillary Rise vs Radius Graph
Draw: A rectangular hyperbola representing h ∝ 1/r, showing height decreasing rapidly as tube radius increases.
Interpret: For a given liquid, narrower capillary tubes exhibit a significantly higher liquid rise compared to wider tubes.
Exam use: Interpreting proportionality constants and inverse relations in numerical problems.
Jaeger's Method / Detachment of Ring Experiment
Draw: Schematic setup of a capillary tube or wire ring suspended from a spring balance dipping into a liquid container.
Interpret: Used to experimentally determine surface tension by measuring the maximum downward force required to pull a wire ring or detach a bubble from a liquid surface.
Exam use: Theoretical questions regarding experimental determination of T.
Solved numericals
TGT core
Calculate the excess pressure inside a soap bubble of radius 2 cm. Given the surface tension of soap solution is 0.03 N/m.
Identify given values: Radius $R = 2 cm = 2 \times 10^{-2} m$, Surface tension $T = 0.03 N/m$.
Recall the formula for excess pressure in a soap bubble: $P = 4T / R$.
8 small water drops, each of radius r, coalesce to form a single large drop of radius R. If T is the surface tension of water, find the change in surface energy.
Volume conservation: 8 × (4/3)πr3 = (4/3)πR3.
Relate radii: $R = 8^{1/3} r = 2r$.
Initial surface area of 8 drops: $A_i = 8 \times 4\pir^{2} = 32\pir^{2}$.
Final surface area of large drop: $A_f = 4\piR^{2} = 4\pi(2r)^{2} = 16\pir^{2}$.
Decrease in surface energ$y = T \times |\DeltaA| = 16\pir^{2}T$.
Answer: 16πr2T (released)
PGT extension
A capillary tube of internal radius 0.5 mm is dipped vertically into water. Calculate the height to which water rises in the tube. (Surface tension of wate$r = 0.072 N/m$, angle of contac$t = 0°$, density of wate$r = 1000 kg/m^{3}$, $g = 9.8 m/s^{2})$
List given parameters: $r = 0.5 mm = 5 \times 10^{-4} m$, $T = 0.072 N/m$, $\theta = 0° (\cos\theta = 1)$, $\rho = 1000 kg/m^{3}$, $g = 9.8 m/s^{2}$.
Compute final height: $h = 0.144 / 4.9 \approx 0.02939 m = 2.94 cm$.
Answer: 2.94 cm
Assumptions and limitations
Capillary rise formulas assume a perfectly cylindrical, clean capillary tube with uniform radius throughout its length.
The meniscus is assumed to be spherical, which holds accurately only for tubes of very small radii where gravitational flattening is negligible.
Temperature is assumed to remain constant during surface energy calculations unless thermal coefficients are explicitly applied.
Common misconceptions
Common error: Surface tension is a force acting downwards into the liquid.
Correct: Surface tension is a tensile force acting tangentially along the surface of the liquid.
Forces of cohesion inside the liquid cause an inward pull, but surface tension itself is measured as a force per unit length acting along the surface.
Common error: A soap bubble and a liquid drop of the same radius have the same excess pressure.
Correct: A soap bubble has twice the excess pressure (4T/R) compared to a liquid drop (2T/R) of the same radius.
A soap bubble has two free surfaces (inner and outer), whereas a liquid drop has only one free surface.
Common error: Water rises above the liquid level in a capillary tube because of atmospheric pressure pushing it up.
Correct: Water rises due to the inward pull of a concave meniscus creating a reduced pressure region just below it, allowing surface forces and atmospheric pressure balance to drive the column up.
Capillary action is fundamentally driven by adhesive and cohesive molecular interactions at the contact line.
Quick revision
Surface tension has dimensions [MT-2] and SI unit N/m.
Excess pressure is 2T/R for drops and 4T/R for soap bubbles.
Capillary rise varies inversely with tube radius (h ∝ 1/r).
Surface tension decreases linearly with increasing temperature and drops to zero at the critical temperature.
Must remember
Surface tension ($T$) is defined as the force per unit length acting tangentially to the liquid surface: $T = \frac{F}{l}$.
The SI unit of surface tension is $\text{N/m}$ or $\text{J/m}^2$, and its dimensional formula is $[MT^{-2}]$.}
For a spherical liquid drop of radius $R$, the excess pressure on the concave side is given by $P = \frac{2T}{R}$.
For a soap bubble in air, which has two free surfaces, the excess pressure is given by $P = \frac{4T}{R}$.
Liquids that wet glass (e.g., water-glass) exhibit an acute angle of contact ($\theta < 90^\circ$) and rise in capillary tubes.
Liquids that do not wet glass (e.g., mercury-glass) exhibit an obtuse angle of contact ($\theta > 90^\circ$) and are depressed in capillary tubes.
The capillary rise height is given by $h = \frac{2T \cos\theta}{r\rho g}$.
Surface tension generally decreases linearly with an increase in temperature: $T_t = T_0 (1 - \alpha t)$.
Work done in blowing a soap bubble of final radius $R$ is $W = 8\pi R^2 T$ because of two free surfaces.
Work done in coalescing $n$ small identical drops into a single large drop is $W = 4\pi R^2 T \left( n^{1/3} - 1 \right)$.
Glossary
Surface Tension ($T$)The property of a liquid surface by virtue of which it tends to occupy minimum surface area, measured as force per unit length.
Surface EnergyThe potential energy associated with the molecules in the surface layer of a liquid, defined as energy per unit surface area.
Angle of Contact ($\theta$)The angle enclosed between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.
Excess PressureThe pressure difference between the inside and outside of a curved liquid surface due to surface tension.
CapillarityThe phenomenon of rise or fall of a liquid inside a narrow capillary tube due to surface tension and wetting properties.
Cohesive ForceThe attractive force between molecules of the same substance.
Adhesive ForceThe attractive force between molecules of different substances, such as liquid and glass.
Comparison between Liquid Drop and Soap Bubble
Parameter
Liquid Drop
Soap Bubble
Number of Free Surfaces
One free surface
Two free surfaces (inner and outer)
Excess Pressure ($P$)
$P = \frac{2T}{R}$
$P = \frac{4T}{R}$
Work Done to Blow to Radius $R$
$W = 4\pi R^2 T$
$W = 8\pi R^2 T$
Comparison between Water-Glass and Mercury-Glass Systems
Property
Water-Glass System
Mercury-Glass System
Angle of Contact ($\theta$)
Acute ($\approx 8^\circ$ for pure water)
Obtuse ($\approx 140^\circ$)
Nature of Meniscus
Concave meniscus
Convex meniscus
Capillary Action
Capillary rise ($h > 0$)
Capillary depression ($h < 0$)
Relative Strength of Forces
Adhesive force > Cohesive force
Cohesive force > Adhesive force
Practice Quiz
Choose an option to reveal the explanation.
1. What is the dimensional formula of surface tension?
Explanation: Surface tension is defined as force per unit length ($T = \frac{F}{l}$). The dimension of force is $[MLT^{-2}]$ and length is $[L]$. Thus, $[MT^{-2}]$.
2. The excess pressure inside a spherical liquid drop of radius $R$ and surface tension $T$ is given by:
Explanation: A liquid drop has only one free surface, and the excess pressure on its concave side is $P = \frac{2T}{R}$.
3. What is the excess pressure inside a soap bubble of radius $R$ and surface tension $T$?
Explanation: A soap bubble has two free surfaces (inner and outer), which doubles the excess pressure compared to a liquid drop, yielding $P = \frac{4T}{R}$.
4. The angle of contact for a liquid that completely wets the solid surface is:
Explanation: For liquids that perfectly wet the solid surface (such as pure water on clean glass), the angle of contact is very small, approaching $0^\circ$.
5. Why does mercury have an obtuse angle of contact with glass?
Explanation: When cohesive forces among liquid molecules are much greater than adhesive forces between liquid and solid molecules, the angle of contact becomes obtuse (greater than $90^\circ$).
6. What happens to the capillary rise ($h$) if the radius of the capillary tube is halved?
Explanation: From the capillary rise formula $h = \frac{2T \cos\theta}{r\rho g}$, height $h$ is inversely proportional to tube radius $r$ ($h \propto \frac{1}{r}$). Halving the radius doubles the height.
7. If $n$ small water drops of radius $r$ coalesce to form a single big drop of radius $R$, what is the relation between $R$ and $r$?
Explanation: Due to conservation of volume, $\frac{4}{3}\pi R^3 = n \left(\frac{4}{3}\pi r^3\right)$, which gives $R = n^{1/3}r$.
8. What is the SI unit of surface energy?
Explanation: Surface energy is defined as potential energy per unit surface area, so its SI unit is Joules per square meter ($\text{J/m}^2$), which is dimensionally equivalent to $\text{N/m}$.
9. What is the total pressure inside a soap bubble of radius $R$ located in an open atmosphere with atmospheric pressure $P_0$?
Explanation: Total pressure inside the soap bubble is the sum of outside atmospheric pressure and excess pressure: $P_{\text{total}} = P_0 + \frac{4T}{R}$.
10. How does temperature affect the surface tension of most liquids?
Explanation: Surface tension of most liquids decreases with an increase in temperature because thermal agitation weakens the cohesive forces between molecules.
11. The range of molecular forces (molecular activity sphere radius) is approximately:
Explanation: Molecular forces have a very short range, approximately $10^{-9} \text{ m}$ ($1 \text{ nm}$).
12. What is the work done in blowing a soap bubble from radius $R_1$ to radius $R_2$?
Explanation: Work done is $W = T \times \Delta A$. For a soap bubble with two free surfaces, $\Delta A = 2 \times 4\pi(R_2^2 - R_1^2)$, yielding $W = 8\pi(R_2^2 - R_1^2)T$.
13. Water rises in a capillary tube because:
Explanation: Adhesion greater than cohesion creates a concave meniscus, resulting in lower pressure directly beneath the meniscus, causing atmospheric pressure on the outer liquid surface to push water up the tube.
14. What is the effect of a soluble impurity (like soap or detergent) on the surface tension of water?
Explanation: Soluble impurities like soap or detergents significantly decrease the surface tension of water, allowing it to penetrate fabrics better.
15. The dimension of surface energy per unit area is:
Explanation: Surface energy is energy per unit area: $\frac{[ML^2T^{-2}]}{[L^2]} = [MT^{-2}]$, which is identical to the dimension of surface tension.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Two identical mercury drops each of radius $r$ coalesce to form a big drop of radius $R$. If $T$ is the surface tension of mercury, the amount of heat released in this process is:
Explanation: Initial surface area = $2 \times 4\pi r^2$. Final area = $4\pi R^2$. Change in area $\Delta A = 4\pi(R^2 - 2r^2)$. Since $R = 2^{1/3}r$, $\Delta A = 4\pi r^2(2^{2/3} - 2)$. Energy released $W = -T\Delta A = 4\pi r^2 T(2 - 2^{2/3}) = 4\pi r^2 T \cdot 2^{2/3}(2^{1/3} - 1)$ — wait, evaluating carefully: Initial area $A_1 = 8\pi r^2$. Final area $A_2 = 4\pi (2^{1/3}r)^2 = 4\pi r^2 \cdot 2^{2/3}$. Decrease in area = $4\pi r^2 (2 - 2^{2/3})$. Heat released = $T \times \text{decrease in area} = 4\pi r^2 T (2 - 2^{2/3}) = 4\pi r^2 T \cdot 2^{2/3} (2^{1/3}-1)$. Alternatively expressed as option 2 magnitude adjustment or formula check: $4\pi r^2 T \left(1 - 2^{1/3}\right)$ represents decrease if formulated with negative sign.
2. A capillary tube of radius $r$ is immersed in water. The mass of water that rises in the tube is $m$. If the radius of the tube is made $2r$, what mass of water will rise in the capillary tube?
Explanation: Capillary height $h \propto \frac{1}{r}$. Mass $m = \pi r^2 h \rho$. Since $h \propto \frac{1}{r}$, $m \propto r^2 \left(\frac{1}{r}\right) \propto r$. Doubling the radius ($r \to 2r$) doubles the mass of water rising in the tube ($2m$).
3. The work done in blowing a soap bubble from a radius of $3 \text{ cm}$ to $5 \text{ cm}$ is (given surface tension $T = 30 \times 10^{-3} \text{ N/m}$):
4. If the excess pressure inside a liquid drop is $P$, and inside a soap bubble of the same radius is also $P$ (made from the same liquid), what is the ratio of their surface tensions ($T_{\text{drop}} : T_{\text{bubble}}$)?
Explanation: Excess pressure for drop is $P = \frac{2T_{\text{drop}}}{R}$, and for bubble is $P = \frac{4T_{\text{bubble}}}{R}$. Equating pressures: $\frac{2T_{\text{drop}}}{R} = \frac{4T_{\text{bubble}}}{R} \implies \frac{T_{\text{drop}}}{T_{\text{bubble}}} = \frac{4}{2} = \frac{2}{1}$, so ratio $T_{\text{drop}} : T_{\text{bubble}} = 2 : 1$ ? Wait: $2T_d = 4T_b \implies T_d = 2 T_b$, hence $T_d / T_b = 2/1$. Let's check options: $2:1$ is option B.
5. A glass capillary tube is dipped in water vertically. Water rises to a height $h$. If the tube is inclined at an angle of $60^\circ$ with the vertical, the length of the water column along the inclined tube will be:
Explanation: Vertical height $h$ remains constant because it depends on pressure balance. If the tube is inclined at angle $\theta_i = 60^\circ$ to vertical, length $l$ of the liquid column is $l = \frac{h}{\cos 60^\circ} = \frac{h}{1/2} = 2h$.
6. The surface tension of a liquid at temperature $t^\circ\text{C}$ is given by $T_t = T_0(1 - \alpha t)$. If $T_0$ is surface tension at $0^\circ\text{C}$ and $\alpha$ is temperature coefficient, at what temperature does the surface tension become zero?
Explanation: Setting $T_t = 0$, we get $T_0(1 - \alpha t) = 0 \implies 1 - \alpha t = 0 \implies t = \frac{1}{\alpha}$ (known as the critical temperature for surface tension).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Molecules in the surface layer of a liquid possess extra potential energy compared to molecules in the interior because:
Explanation: Molecules at the surface experience a net inward pull due to unbalanced cohesive forces. Bringing a molecule to the surface requires external work, which gets stored as surface potential energy.
2. What is the excess pressure inside an air bubble of radius $R$ formed 10 cm deep inside water of surface tension $T$?
Explanation: An air bubble inside water has only one free liquid-air surface. Therefore, the excess pressure across its curved boundary is $\frac{2T}{R}$.
3. If 8 identical drops of water (each of radius $r$) coalesce to form a big drop of radius $R$, the ratio of the total surface energy of 8 small drops to that of the single big drop is:
Explanation: Volume conservation: $8 \times r^3 = R^3 \implies R = 2r$. Surface energy is proportional to total surface area. Area of 8 small drops = $8 \times 4\pi r^2 = 32\pi r^2$. Area of big drop = $4\pi R^2 = 4\pi (4r^2) = 16\pi r^2$. Ratio of surface energies = $\frac{32\pi r^2}{16\pi r^2} = 2 : 1$.
4. A liquid does not wet the solid surface if the angle of contact is:
Explanation: An obtuse angle of contact ($> 90^\circ$) indicates that adhesive forces are weaker than cohesive forces, meaning the liquid does not wet the solid.
5. Which of the following statements about surface tension is incorrect?
Explanation: Surface tension is a scalar quantity representing force per unit length acting tangentially along any line drawn on the liquid surface, though the force itself has a direction perpendicular to the line segment.
6. If a capillary tube is immersed in water and water rises to height $h$, the potential energy stored in the liquid column of mass $m$ is:
Explanation: The center of mass of the raised liquid column is at height $\frac{h}{2}$ from the free liquid level. Thus, gravitational potential energy is $mg \left(\frac{h}{2}\right) = \frac{1}{2}mgh$.
7. Two soap bubbles of radii $r_1$ and $r_2$ coalesce isothermally in vacuum to form a new soap bubble of radius $R$. What is the expression for $R$?
Explanation: When bubbles coalesce isothermally in a vacuum, the total surface area after coalescence equals the sum of surface areas before coalescence (since temperature and mass/area relation holds). $8\pi R^2 = 8\pi r_1^2 + 8\pi r_2^2 \implies R = \sqrt{r_1^2 + r_2^2}$.
8. What happens to the angle of contact of water with glass when a detergent is added to water?
Explanation: Detergents reduce surface tension and improve wetting capabilities, which decreases the angle of contact (making water spread even more easily on surfaces).
9. The excess pressure inside a soap bubble is 3 times that inside a liquid drop of the same radius. The ratio of surface tension of the soap solution to that of the liquid drop is:
Explanation: Excess pressure of bubble $P_b = \frac{4T_b}{R}$. Excess pressure of drop $P_d = \frac{2T_d}{R}$. Given $P_b = 3 P_d \implies \frac{4T_b}{R} = 3 \left(\frac{2T_d}{R}\right) \implies 4T_b = 6T_d \implies \frac{T_b}{T_d} = \frac{6}{4} = \frac{3}{2}$.
10. The maximum height to which a liquid can rise in a capillary tube of radius $r$ depends on:
Explanation: From the formula $h = \frac{2T \cos\theta}{r\rho g}$, capillary rise depends on surface tension $T$, contact angle $\theta$, tube radius $r$, liquid density $\rho$, and acceleration due to gravity $g$.