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Mechanics
Surface Energy and Surface Tension
Surface energy is the extra potential energy possessed by molecules at the surface of a liquid due to unbalanced intermolecular attractive forces. Surface tension is numerically equal to the work done in increasing the surface area by unity under isothermal conditions.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of surface tension, molecular theory of surface tension, relation between surface energy and surface tension, excess pressure inside liquid drops and soap bubbles, angle of contact, and capillary rise.
PGT extension
Work done in blowing bubbles, splitting and coalescence of liquid drops, thermodynamics of surface films, temperature dependence of surface tension, and rigorous derivation of capillary height accounting for meniscus.
Prerequisites
Intermolecular forces and range of molecular attraction
Work, energy, and conservation principles in mechanics
Hydrostatics and pressure distribution in fluids
Molecular Theory of Surface Tension
Molecules deep inside a liquid experience equal attractive forces from surrounding molecules in all directions, resulting in a net zero force.
Molecules located within a surface layer of thickness equal to the molecular range (sphere of influence) experience a net downward attractive force due to the absence of upper liquid molecules.
To bring a molecule from the interior of the liquid to the surface, work must be done against this inward resultant force, which gets stored as potential energy at the surface.
Surface Energy vs Surface Tension
Surface tension (T or σ) is defined as the force acting per unit length imaginarilly drawn on the liquid surface, perpendicular to the line, with dimensions [ML&sup0;T&sup-2;].
Surface energy (E) is the potential energy per unit surface area of the liquid film, with dimensions [MT&sup-2;].
Numerically and dimensionally, surface tension and surface energy per unit area are identical, expressed in J m&sup-2; or N m&sup-1;.
Excess Pressure across Curved Surfaces
Due to surface tension, molecules on a curved liquid surface pull inward, creating a pressure difference between the concave and convex sides.
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 (inner and outer), the excess pressure is given by $P = 4T / R$.
Capillarity and Angle of Contact
The angle of contact (θ) is the angle subtended by the tangent to the liquid surface at the point of contact with the solid surface, measured inside the liquid.
Liquids that wet glass (e.g., water) form a concave meniscus and have an acute angle of contact (θ < 90°), showing capillary rise.
Liquids that do not wet glass (e.g., mercury) form a convex meniscus and have an obtuse angle of contact (θ > 90°), showing capillary depression.
Derivations
Work Done in Increasing Surface Area and Surface Energy Relation
Conditions: Isothermal expansion of a rectangular liquid film on a wire frame with one movable wire of length l.
Let a wire frame support a liquid film of length l. The film has two free surfaces (front and back), so the total effective length of the surface acting against the pulling force is 2l.
If the movable wire is pulled outward by a small distance dx, the increase in total surface area of the film is d$A = 2 * l * dx$.
The magnitude of the inward pulling force due to surface tension is $F = T * (2l)$.
The work done (dW) in pulling the wire by distance dx is d$W = F * dx = (T * 2l) * dx = T * (2 * l * dx) = T * dA$.
This work done is stored in the film as potential surface energy (dU). Thus, d$U = T * dA$, which implies $T = dU / dA$.
Result: Surface tension is equal to the surface energy per unit surface area (T = E).
Jurin's Law: Rise of Liquid in a Capillary Tube
Conditions: A capillary tube of radius r dipped in a liquid of surface tension T and density ρ with contact angle θ.
Let the liquid rise to a height h in the capillary tube. The upper meniscus is hemispherical of radius r, such that $r = R \cos(\theta)$, where R is the radius of curvature of the meniscus.
The upward force supporting the liquid column is due to the vertical component of surface tension acting along the circumference of the meniscus: $F_up = (2 \pi r) * T * \cos(\theta)$.
The downward force is the weight of the liquid column of height h (including the meniscus correction): $F_down = m * g = (\pi r^2 * h) * \rho * g$.
Equating the upward and downward forces for equilibrium: (2 π r) * T * cos(θ) = π r^2 * h * ρ * g.
Solving for capillary rise h gives $h = (2 T \cos(\theta)) / (r \rho g)$.
Result: h = \frac{2T \cos\theta}{r \rho g}
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Surface Tension Definition
$$T = \frac{F}{l}$$
T = Surface tension, F = Force, l = Length N m&sup-1;
Isothermal conditions
Surface Energy
$$W = T \cdot \Delta A$$
W = Work done, T = Surface tension, \Delta A = Change in surface area Joule (J)
Constant temperature
Excess Pressure in Liquid Drop
$$P = \frac{2T}{R}$$
P = Excess pressure, T = Surface tension, R = Radius of drop Pa (N m&sup-2;)
Spherical drop with single free surface
Excess Pressure in Soap Bubble
$$P = \frac{4T}{R}$$
P = Excess pressure, T = Surface tension, R = Radius of bubble Pa (N m&sup-2;)
Spherical bubble with two free surfaces
Capillary Rise (Jurin's Law)
$$h = \frac{2T \cos\theta}{r \rho g}$$
h = Height, T = Surface tension, \theta = Angle of contact, r = Tube radius, \rho = Density, g = Acceleration due to gravity m
Narrow capillary tube, clean glass
Work Done in Blowing a Soap Bubble from Radius R1 to R2
$$W = 8 \pi T (R_2^2 - R_1^2)$$
W = Work done, T = Surface tension, R_1, $R_2 = Initial$ and final radii Joule (J)
Soap bubble (two free surfaces)
Work Done in Merging n Identical Drops into One Big Drop
$$W = 4 \pi R^2 T (n^{1/3} - 1)$$
R = Radius of big drop, r = Radius of small drop, n = Number of drops, T = Surface tension Joule (J)
Volume conservation ($R = n^{1/3}r)$
Temperature Dependence of Surface Tension
$$T_t = T_0 (1 - \alpha t)$$
$T_t = Surface tension at t°C$, $T_0 = Surface tension at 0°C$, \alpha = Temperature coefficient N m&sup-1;
Valid for temperatures well below the boiling point
Graphs and experiments
Variation of Surface Tension with Temperature
Draw: A linear downward sloping graph of Surface Tension (T) on the y-axis versus Temperature (T_c in Celsius) on the x-axis, meeting zero at the critical temperature.
Interpret: Surface tension decreases linearly with an increase in temperature because thermal agitation increases average molecular separation, reducing cohesive forces.
Exam use: Direct conceptual questions and numerical calculations involving temperature coefficients.
Jaeger's Method for Determining Surface Tension
Draw: Experimental setup showing an air bubble forming at the tip of a capillary immersed in liquid, connected to a manometer measuring maximum pressure head.
Interpret: The maximum excess pressure required to detach an air bubble is related to surface tension by $P_max = (2T / r) + h\rhog$.
Exam use: Practical physics questions and experimental error analysis in PGT exams.
Capillary Rise vs Radius Graph
Draw: A rectangular hyperbola representing h vs r, showing that height decreases rapidly as tube radius increases (h \propto 1/r).
Exam use: Interpreting proportionalities in objective multiple-choice questions.
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⁻¹.
Identify given values: Radius R = 2 cm = 0.02 m, Surface tension $T = 0.03 N m^{-1}$.
Recall that a soap bubble has two free surfaces, so the excess pressure formula is $P = 4T / R$.
Substitute the values: $P = (4 * 0.03) / 0.02$.
Calculate the final value: $P = 0.12 / 0.02 = 6 Pa$.
Answer: 6 Pa
PGT extension
8 identical water drops, each of radius r = 1 mm, merge to form a single large drop. If the surface tension of water is 0.072 N m⁻¹, calculate the amount of energy liberated in this process.
Equating volumes of 8 small drops to one large drop: (4/3) * π * $R^3 = 8 * ((4/3) * \pi * r^3)$.
Simplify to find the radius of the big drop: $R^3 = 8r^3 => R = 2r = 2 * 1 mm = 2 mm = 2 * 10^{-3} m$.
Initial surface area of 8 drops: $A_i = 8 * (4 \pi r^2) = 32 \pi r^2$.
Final surface area of the big drop: $A_f = 4 \pi R^2 = 4 \pi (2r)^2 = 16 \pi r^2$.
A capillary tube of internal radius 0.5 mm is dipped vertically into water. To what height does the water rise in the tube? (Surface tension of water = 0.072 N m⁻¹, angle of contact = 0°, density of water = 1000 kg m⁻³, g = 9.8 m s⁻²)
List given parameters: $r = 0.5 mm = 5 * 10^{-4} m$, $T = 0.072 N m^{-1}$, $\theta = 0° (\cos\theta = 1)$, $\rho = 1000 kg m^{-3}$, $g = 9.8 m s^{-2}$.
Write Jurin's formula for capillary rise: $h = (2 T \cos(\theta)) / (r \rho g)$.
The liquid temperature is assumed constant during isothermal surface area changes; localized heating occurs during rapid adiabatic drops.
Capillary rise formulas assume a perfectly cylindrical tube and a spherical meniscus, which breaks down for large radius tubes where the meniscus flattens.
Surface tension is assumed uniform, ignoring contamination or surfactant impurities that locally alter surface tension.
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 due to surface tension act along the plane of the liquid surface perpendicular to any imaginary line drawn on it.
Common error: A soap bubble has only one surface area to consider for energy calculations.
Correct: A soap bubble has two free surfaces (inner and outer).
A very thin liquid film in air exposes both sides to air, doubling the active surface area compared to a solid liquid drop.
Common error: Water rises in a capillary tube because of atmospheric pressure pushing it up.
Correct: Water rises due to the upward component of adhesive forces and surface tension pulling the contact line upward.
Atmospheric pressure acts equally on all liquid surfaces and does not drive the net upward capillary ascent against gravity.
Quick revision
Surface tension $T = F/l = W/\DeltaA with units N m⁻¹ or J m⁻²$.
Excess pressure is 2T/R for drops and 4T/R for soap bubbles.
Capillary rise is inversely proportional to tube radius (h ∝ 1/r).
Surface tension decreases linearly with an increase in temperature.
Surface tension ($T$ or $\sigma$) is defined as force per unit length with dimensions $[ML^0T^{-2}]$ and units $\text{N m}^{-1}$.
Surface energy ($E$) is the potential energy per unit surface area, numerically and dimensionally equal to surface tension.
Excess pressure inside a spherical liquid drop is $P = \frac{2T}{R}$ (one free surface).
Excess pressure inside a spherical soap bubble is $P = \frac{4T}{R}$ (two free surfaces).
Liquid wetting glass (e.g., water) forms a concave meniscus with an acute angle of contact ($\theta < 90^\circ$) and shows capillary rise.
Liquid not wetting glass (e.g., mercury) forms a convex meniscus with an obtuse angle of contact ($\theta > 90^\circ$) and shows capillary depression.
Capillary rise height is given by Jurin's Law: $h = \frac{2T \cos\theta}{r \rho g}$.
Work done in blowing a soap bubble from radius $R_1$ to $R_2$ is $W = 8\pi T(R_2^2 - R_1^2)$ due to two free surfaces.
Work done in merging $n$ identical small drops into one big drop is $W = 4\pi R^2 T(n^{1/3} - 1)$.
Surface tension decreases with temperature according to the relation $T_t = T_0(1 - \alpha t)$.
Glossary
Surface Tension ($T$)The tensile force acting per unit length imaginarily drawn on a liquid surface, perpendicular to the line.
Surface Energy ($E$)The extra potential energy stored in the molecules present in the surface layer per unit surface area.
Sphere of InfluenceA sphere centered on a liquid molecule with a radius equal to the molecular range, within which intermolecular attractive forces are effective.
Angle of Contact ($\theta$)The angle subtended by the tangent to the liquid surface at the point of contact with a solid surface, measured inside the liquid.
Excess Pressure ($P$)The pressure difference between the concave and convex sides of a curved liquid surface caused by surface tension.
CapillarityThe phenomenon of rise or fall of a liquid inside a narrow tube (capillary tube) immersed in it.
Jurin's LawThe physical law stating that the height of capillary rise is inversely proportional to the radius of the capillary tube and density of the liquid.
Liquid Drop vs. Soap Bubble
Parameter
Liquid Drop
Soap Bubble
Free Surfaces
One free surface
Two free surfaces (inner and outer)
Excess Pressure ($P$)
$P = \frac{2T}{R}$
$P = \frac{4T}{R}$
Work Done (Radius $0$ to $R$)
$W = 4\pi R^2 T$
$W = 8\pi R^2 T$
Water (Wetting) vs. Mercury (Non-Wetting)
Property
Water in Glass Tube
Mercury in Glass Tube
Meniscus Shape
Concave
Convex
Angle of Contact ($\theta$)
Acute ($\theta < 90^\circ$)
Obtuse ($\theta > 90^\circ$)
Capillary Action
Capillary Rise ($h > 0$)
Capillary Depression ($h < 0$)
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 ($T = \frac{F}{l}$). Dimensional formula of force is $[MLT^{-2}]$ and length is $[L]$. Thus, $[MT^{-2}]$. \text{MT}^{-2}$.
2. Molecules deep inside a liquid experience a net force of zero because:
Explanation: Molecules in the interior of a liquid are surrounded symmetrically by other molecules within their sphere of influence, leading to cancellation of attractive forces in all directions.
3. What is the excess pressure inside a spherical liquid drop of radius $R$ and surface tension $T$?
Explanation: A liquid drop has only one free surface. The excess pressure across its curved spherical surface is given by $P = \frac{2T}{R}$.
4. 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) exposed to air, which doubles the inward force due to surface tension, making the excess pressure $P = \frac{4T}{R}$.
5. The angle of contact for liquids that wet glass, such as water, is:
Explanation: Liquids that wet glass form a concave meniscus, corresponding to an acute angle of contact where adhesive forces exceed cohesive forces.
6. Which of the following phenomena is responsible for capillary rise?
Explanation: Capillary rise is governed by Jurin's Law, which depends on surface tension ($T$), angle of contact ($\theta$), tube radius ($r$), and liquid density ($\rho$).
7. If $n$ identical small liquid drops each of radius $r$ merge to form a single big drop of radius $R$, what is the relation between $R$ and $r$?
Explanation: By conservation of volume, the total volume of $n$ small drops equals the volume of the big drop: $n \cdot \frac{4}{3}\pi r^3 = \frac{4}{3}\pi R^3$, which gives $R = n^{1/3}r$.
8. What is the work done in blowing a soap bubble from radius $R_1$ to radius $R_2$?
Explanation: The change in surface area for a soap bubble with two free surfaces is $\Delta A = 2 \times 4\pi (R_2^2 - R_1^2) = 8\pi(R_2^2 - R_1^2)$. Work done is $W = T \cdot \Delta A = 8\pi T(R_2^2 - R_1^2)$.
9. How does surface tension generally vary with an increase in temperature?
Explanation: With an increase in temperature, thermal agitation increases, weakening intermolecular cohesive forces, which causes surface tension to decrease according to $T_t = T_0(1 - \alpha t)$.
10. Why does mercury form a convex meniscus inside a glass tube?
Explanation: Mercury exhibits strong cohesion and weak adhesion with glass, resulting in an obtuse angle of contact ($\theta > 90^\circ$) and a convex meniscus.
11. What is the unit of surface energy?
Explanation: Surface energy is potential energy per unit area, expressed in Joules per square meter ($\text{J m}^{-2}$) or equivalently $\text{N m}^{-1}$.
12. If the radius of a capillary tube is halved, the height to which liquid rises in the tube will:
Explanation: According to Jurin's Law, $h = \frac{2T\cos\theta}{r\rho g}$, which means $h$ is inversely proportional to radius $r$. If $r$ is halved, $h$ is doubled.
13. What happens to the temperature of a liquid when two or more small drops coalesce into a single large drop?
Explanation: When drops coalesce, total surface area decreases, releasing surface energy as thermal energy, which slightly raises the temperature of the drop.
14. Which incorrect misconception about surface tension is frequently encountered?
Explanation: Surface tension is a tensile force acting tangentially along the surface of the liquid, not a downward bulk force.
15. At the critical temperature of a liquid, its surface tension:
Explanation: At the critical temperature, the distinction between liquid and vapor phases disappears, molecular range forces vanish, and surface tension drops to zero.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If $8$ identical drops of water of radius $r$ merge to form a big drop of radius $R$, then the ratio of surface energy of the big drop to the total initial surface energy of the small drops is:
Explanation: For $n = 8$, $R = 8^{1/3}r = 2r$. Initial surface area = $8 \times 4\pi r^2 = 32\pi r^2$. Final surface area = $4\pi R^2 = 4\pi(2r)^2 = 16\pi r^2$. Ratio of final surface energy to initial surface energy = $16\pi r^2 : 32\pi r^2 = 1 : 2$.
2. The excess pressure inside a soap bubble of radius $3\text{ cm}$ is balanced by a liquid column of height $2\text{ cm}$. If the density of the liquid is $10^3 \text{ kg m}^{-3}$, what is the surface tension of the soap solution? (Take $g = 10 \text{ m s}^{-2}$)
3. A capillary tube of radius $r$ is immersed in water vertically. The height of water rise is $h$. The mass of water in the capillary tube is $m$. If the radius of the tube is doubled, the mass of water that will rise in the capillary tube is:
Explanation: Capillary height $h \propto \frac{1}{r}$. When radius becomes $2r$, height becomes $h/2$. Mass of water $m = \pi r^2 h \rho$. New mass $m' = \pi (2r)^2 \left(\frac{h}{2}\right) \rho = \pi (4r^2) \left(\frac{h}{2}\right) \rho = 2 \pi r^2 h \rho = 2m$.
4. Work done in blowing a soap bubble of surface tension $T$ from radius $R$ to $2R$ is:
Explanation: Work done $W = 8\pi T (R_2^2 - R_1^2) = 8\pi T ((2R)^2 - R^2) = 8\pi T (4R^2 - R^2) = 8\pi T (3R^2) = 24\pi R^2 T$.
5. The angle of contact between a pure liquid and glass is $0^\circ$. A capillary tube is dipped in the liquid. The shape of the meniscus of the liquid in the tube is:
Explanation: When the angle of contact is $0^\circ$, the tangent to the liquid surface at the wall is parallel to the tube wall, forming a perfectly concave hemispherical meniscus.
6. Two soap bubbles of radii $R_1$ and $R_2$ coalesce under isothermal conditions to form a single soap bubble of radius $R$. The correct expression for $R$ is:
Explanation: In isothermal coalescence of soap bubbles, the total surface area remains conserved because volume of air inside is conserved: $8\pi R_1^2 + 8\pi R_2^2 = 8\pi R^2 \Rightarrow R = \sqrt{R_1^2 + R_2^2}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The sphere of influence is defined as a sphere with a radius equal to:
Explanation: The sphere of influence has a radius equal to the molecular range within which intermolecular attractive forces are active.
2. What is the dimensional formula of surface energy?
Explanation: Surface energy is energy per unit area: $\frac{[ML^2T^{-2}]}{[L^2]} = [MT^{-2}]$, which is identical to the dimensions of surface tension.
3. The excess pressure inside a spherical drop of radius $R$ is $P_1$ and that inside a soap bubble of the same radius and same liquid is $P_2$. The ratio $P_1 : P_2$ is:
4. A drop of water of radius $r$ is broken into $1000$ identical droplets. If $T$ is the surface tension, the work done in this process is:
Explanation: Number of droplets $n = 1000$. Radius of each small droplet $r' = r / (1000)^{1/3} = r / 10 = 0.1r$. Increase in surface area $\Delta A = 1000 \times 4\pi (0.1r)^2 - 4\pi r^2 = 1000 \times 0.01 \times 4\pi r^2 - 4\pi r^2 = 40\pi r^2 - 4\pi r^2 = 36\pi r^2$. Work done $W = T \cdot \Delta A = 36\pi r^2 T$.
5. The height of liquid column in a capillary tube is $h$. If the tube is inclined at an angle $\alpha$ to the vertical, the length of the liquid column $l$ inside the tube will be:
Explanation: When a capillary tube is inclined at angle $\alpha$ to the vertical, the vertical height $h$ of the liquid column remains constant. The length $l$ of the liquid column along the inclined tube is given by $l \cos\alpha = h \Rightarrow l = \frac{h}{\cos\alpha}$.
6. Which of the following statements about surface tension and temperature is correct?
Explanation: Surface tension decreases approximately linearly with an increase in temperature as molecular thermal motion disrupts cohesive bonding.
7. Two capillary tubes of radii $r_1$ and $r_2$ are dipped in water. The heights of water rise are $h_1$ and $h_2$ respectively. The ratio $h_1 / h_2$ is equal to:
Explanation: By Jurin's Law, $h \propto \frac{1}{r}$, so $\frac{h_1}{h_2} = \frac{r_2}{r_1}$.
8. If a capillary tube is immersed in water and water rises to a height $h$, what happens if the length of the capillary tube is cut to a value less than $h$?
Explanation: When a capillary tube is cut shorter than the equilibrium rise height $h$, water rises to the top of the tube and the radius of curvature of the meniscus increases (radius becomes larger) so that $R \cdot h' = \text{constant}$, avoiding overflow.
9. What is the excess pressure inside a cylindrical liquid jet or column of radius $R$ and surface tension $T$?
Explanation: For a cylindrical liquid jet or bubble, curvature exists only in one principal cross-sectional plane, yielding an excess pressure of $P = \frac{T}{R}$.
10. The work done in blowing a soap bubble of radius $R$ with surface tension $T$ is $W_1$. The work done in further blowing it from radius $R$ to $2R$ is $W_2$. The ratio $W_1 / W_2$ is:
Explanation: $W_1 = 8\pi T R^2$. $W_2$ for radius $R$ to $2R$ is $8\pi T ((2R)^2 - R^2) = 8\pi T (3R^2) = 3W_1$. Therefore, $W_1 / W_2 = 1 / 3$.