Surface tension | PhysicsRank

A soap film pulls on its boundary, while a small curved drop can have greater pressure inside than outside. These are connected effects of an interface whose energy changes with area. The number of surfaces matters as much as the formula.

This chapter derives the familiar factors of two from actual interfaces and geometry. We will distinguish a liquid drop, a soap film and a gas bubble in liquid before combining capillary and hydrostatic effects.

What you will learn to do

Interpret surface tension as force per length and reversible surface work per area.
Count one or two interfaces in a film, drop or bubble.
Derive spherical pressure jumps and use local external pressure.
Use a through-liquid contact angle for capillary forces.
Derive static capillary rise and state its geometric limits.
Conserve liquid volume before comparing drop surface energies.

Useful before this chapterPressure, floating and fluid support →Work, stored energy and power →Units: keep the quantity, change the number →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

If making extra interface requires energy, why can a stretched film pull its boundary inward?

Compare your reasoning

Moving the boundary inward can reduce area and interfacial energy. The force reflects that energy change. A soap film has two liquid–air surfaces that must both be counted.

Revisit the explanation →

01 / Surface tension, curved interfaces and capillary rise

Creating an interface requires energy

A liquid surface is a boundary with a different molecular environment from the liquid interior. At fixed temperature and composition, increasing the interface area generally requires work. Surface tension S can be described as the reversible work required per increase in area for a simple liquid interface under those conditions.

Surface tension also acts mechanically as a force per unit length along an interface, perpendicular to a chosen boundary line within the surface. The units N/m and J/m² are equivalent. It is not a pressure, which has units N/m², and it is not a force that must always point vertically upward.

For constant S, an area increase ΔA requires surface work SΔA. Count every interface whose area changes. A liquid drop in air has one liquid–gas interface. A thin soap film separating air on both sides has two interfaces, and both contribute to the work. Temperature, surfactant concentration and contamination can alter S, so treat it as constant only when that is part of the model.

\Delta E_{\mathrm{surface}}=S\Delta A,\qquad[S]=\mathrm{N/m}=\mathrm{J/m^2}

Here ΔA is the total changing interfacial area. For two faces of a thin film, include both areas.

STOP & THINK

Which unit is appropriate for surface tension?

N/m
N/m²
N·m

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: Creating an interface requires energy
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
\Delta E_{\mathrm{surface}}=S\Delta A,\qquad[S]=\mathrm{N/m}=\mathrm{J/m^2}
Meaning: Here ΔA is the total changing interfacial area. For two faces of a thin film, include both areas.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Surface tension, curved interfaces and capillary rise

A movable wire exposes the factor of two

Imagine a thin liquid film held in a rectangular frame with one movable straight wire of length l. Moving the wire outward by dx increases the area of each face by l dx. With two faces, the total area increase is 2l dx, so quasistatic work balance gives F dx = 2Sl dx and F = 2Sl.

The factor two counts interfaces, not two ends of the wire. If a different arrangement contains only one changing interface, its contribution is Sl. If the pull direction is not aligned with the film’s local force direction, resolve the appropriate components rather than using the full magnitude automatically.

The force follows from a reversible area change in an ideal film of constant surface tension. A wire’s weight, friction in guides or viscous motion can require additional applied force. Keep those contributions separate from the film’s surface-tension force.

WORKED EXAMPLE

Check the boundary force with surface work

A thin soap film has a movable wire 0.080 m long and surface tension 0.030 N/m on each face. The wire is drawn quasistatically by 0.020 m, neglecting its weight and friction. Find the required force and work.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: A movable wire exposes the factor of two
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Surface tension, curved interfaces and capillary rise

A curved interface needs a pressure difference to balance its pull

For a small spherical liquid drop of radius R, imagine cutting it into two hemispheres. The pressure difference across the interface acts over the circular cut area πR². Surface tension acts around the circular boundary of length 2πR. Balancing them gives Δp = 2S/R.

A thin spherical soap bubble has two nearly equal-radius interfaces, so both boundary contributions act and the excess internal pressure is approximately 4S/R. A gas bubble inside a liquid has one gas–liquid interface, giving 2S/R when gravity-induced pressure variation over the small bubble is negligible.

The excess pressure is a difference across the interface. To obtain absolute internal pressure, add the local external pressure. For a bubble submerged in liquid, that external pressure includes hydrostatic depth. Do not add atmospheric pressure alone when the bubble is well below the free surface.

\Delta p_{\mathrm{drop}}=\frac{2S}{R},\qquad\Delta p_{\mathrm{thin\ soap\ bubble}}\approx\frac{4S}{R}

These spherical expressions use radius, not diameter. The thin-film approximation treats the two soap-bubble radii as equal.

WORKED EXAMPLE

Compare a drop and a soap bubble of equal radius

A spherical liquid drop and a thin soap bubble each have radius 3.0 mm. For comparison, take the relevant surface tension as 0.030 N/m at each interface and neglect gravity-induced distortion. Find their excess internal pressures.

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STOP & THINK

A small gas bubble completely inside a liquid has how many gas–liquid interfaces in the elementary pressure-jump model?

One
Two because it has an inside and an outside
Four

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: A curved interface needs a pressure difference to balance its pull
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
\Delta p_{\mathrm{drop}}=\frac{2S}{R},\qquad\Delta p_{\mathrm{thin\ soap\ bubble}}\approx\frac{4S}{R}
Meaning: These spherical expressions use radius, not diameter. The thin-film approximation treats the two soap-bubble radii as equal.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Surface tension, curved interfaces and capillary rise

Use the pressure just outside the curved surface

A bubble’s excess pressure is measured relative to the surrounding liquid at the bubble’s location. For a small bubble at depth h in a static incompressible liquid, that local pressure is approximately patm + ρgh. Adding 2S/R gives the gas pressure inside the bubble.

The assumption of a spherical bubble requires surface-tension effects to dominate appreciable shape distortion. A useful comparison is the dimensionless ratio ρgR²/S: when it is small, the gravitational pressure variation across the radius is small relative to the capillary pressure scale. A large bubble can deform and require a more detailed curvature balance.

If surface tension is the same, a smaller spherical bubble has a larger excess pressure. Connecting bubbles or allowing gas transfer can therefore create flow from higher to lower pressure, but changes of radius, gas amount and temperature must be considered as the process continues. An initial pressure comparison is not a complete final-state model.

WORKED EXAMPLE

Add hydrostatic and capillary contributions separately

A small spherical gas bubble of radius 0.20 mm lies 0.20 m below the surface of a liquid with density 1000 kg/m³ and surface tension 0.050 N/m. Atmospheric pressure is 100 kPa and g = 10 m/s². Neglect shape distortion. Find its internal absolute pressure.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: Use the pressure just outside the curved surface
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Surface tension, curved interfaces and capillary rise

The contact angle determines the vertical component at a wall

Where a liquid interface meets a solid, its contact angle is conventionally measured through the liquid between the solid surface and the tangent to the liquid–gas interface. It reflects the balance of interfacial properties for the liquid, solid and gas under the stated conditions.

A wetting liquid with an acute contact angle can rise near a wall and form a concave meniscus. A nonwetting liquid with an obtuse contact angle can form a depressed convex meniscus. Surface preparation and contamination can change the contact behaviour; the liquid name alone does not determine one immutable angle on every solid.

In a narrow circular vertical capillary, the contact-line circumference is 2πr. The vertical component of the surface-tension support is 2πrS cos θ. The cosine supplies both magnitude and sign. It is positive for a rise when θ < 90°, zero at 90° in the elementary model and negative for a depression when θ > 90°.

F_{\mathrm{vertical}}=2\pi rS\cos\theta

The radius r is the capillary’s internal radius. The angle θ is measured through the liquid at the wall.

Explain this equation

Read it in words

The vertical surface-tension force around a circular contact line is the circumference times the tension, multiplied by its vertical projection factor.

Symbols and units

$F_{\mathrm{vertical}}$: Signed vertical contact-line force on the liquid, upward taken positive.Newtons (N).
$r$: Internal radius of the circular tube at the contact line.Metres (m).
$S$: Surface tension of the liquid–gas interface.Newtons per metre (N/m).
$\theta$: Contact angle measured through the liquid against the vertical tube wall.Angle; cosθ is dimensionless.

Signs and operations

2πr is a circumference, not the cross-sectional area πr². cosθ selects the vertical part of the interface tension: positive for a wetting angle below 90°, negative above 90°.

When it applies

Use a circular vertical tube, a uniform contact angle and surface tension, and a static contact line. A capillary-height formula also needs hydrostatic balance and a small-meniscus-volume approximation.

Make it concrete

For r = 0.50 mm, S = 0.072 N/m and θ = 60°, the upward force is about 1.13 × 10⁻⁴ N. At 90° its vertical component vanishes, although surface tension itself has not vanished.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): F_{\mathrm{vertical}}=2\pi rS\cos\theta
Read it as: The vertical surface-tension force around a circular contact line is the circumference times the tension, multiplied by its vertical projection factor.
Symbols:
F_{\mathrm{vertical}}: Signed vertical contact-line force on the liquid, upward taken positive.; Newtons (N).
r: Internal radius of the circular tube at the contact line.; Metres (m).
S: Surface tension of the liquid–gas interface.; Newtons per metre (N/m).
\theta: Contact angle measured through the liquid against the vertical tube wall.; Angle; cosθ is dimensionless.
Signs and operations: 2πr is a circumference, not the cross-sectional area πr². cosθ selects the vertical part of the interface tension: positive for a wetting angle below 90°, negative above 90°.
Conditions: Use a circular vertical tube, a uniform contact angle and surface tension, and a static contact line. A capillary-height formula also needs hydrostatic balance and a small-meniscus-volume approximation.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

STOP & THINK

In the elementary capillary model, what sign does cos θ have for a contact angle greater than 90° but less than 180°?

Positive, predicting a rise
Negative, predicting a depression
Always zero

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: The contact angle determines the vertical component at a wall
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
F_{\mathrm{vertical}}=2\pi rS\cos\theta
Meaning: The radius r is the capillary’s internal radius. The angle θ is measured through the liquid at the wall.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Surface tension, curved interfaces and capillary rise

Balance a contact-line force against a column’s weight

For a narrow vertical capillary in a large reservoir, balance the vertical surface-tension contribution with the weight of the elevated liquid column. Neglect the small meniscus-volume correction and the gas density. Then 2πrS cos θ = ρgπr²h, giving h = 2S cos θ/(ρgr).

The height is measured vertically relative to the reservoir surface. The capillary radius is not the spherical radius of curvature of the meniscus; the two are related through the contact angle in the ideal geometry. Mixing them can drop a cosine or a factor of two.

The rise grows as the tube radius becomes smaller because the supporting contact length scales as r while the column area scales as r². This explains why narrow tubes show large capillary effects. The result is a static equilibrium height: it does not predict how quickly the liquid rises, since viscosity and transient motion control that process.

WORKED EXAMPLE

Predict the rise and test the radius dependence

A liquid with surface tension 0.050 N/m and density 1000 kg/m³ completely wets a narrow vertical tube, so θ = 0°. The internal radius is 0.50 mm. Take g = 10 m/s² and neglect the meniscus-volume correction. Find the equilibrium rise and the rise if the radius is halved.

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STOP & THINK

Two otherwise identical narrow capillaries have internal radii r and 3r and the same acute contact angle. What is h(r)/h(3r)?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: Balance a contact-line force against a column’s weight
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Surface tension, curved interfaces and capillary rise

Conserve liquid volume before comparing surface energies

When incompressible liquid drops merge without evaporation, their total liquid volume is conserved. Surface area is not. For N equal spherical drops of radius r combining into one sphere, the final radius is N^(1/3)r. The final area relative to the initial total area is N^(−1/3).

At constant surface tension, a decrease of interface area releases surface energy. That released energy can become motion, internal energy or other transfers; the surface-energy difference alone does not specify how it is partitioned. Reversing the process to create many small drops requires at least the corresponding ideal surface work, with additional real losses possible.

Do not apply liquid-volume conservation directly to gas volumes inside merging soap bubbles at arbitrary pressure. Gas can compress and expand, and the internal pressure itself depends on radius. A bubble-merging calculation needs gas amount, temperature and mechanical conditions as well as the interfacial energies.

WORKED EXAMPLE

Compare area without inserting a numerical surface tension

Eight equal spherical liquid drops merge into one spherical drop. Assume incompressible liquid, no evaporation and constant surface tension. Find the final radius in units of the original radius r and the fraction of initial surface energy released.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: Conserve liquid volume before comparing surface energies
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Surface tension, curved interfaces and capillary rise

Use surface tension when the geometry and scale require it

Surface effects matter most when characteristic lengths are small. The ratio of gravitational and capillary effects scales with length squared, so a tiny drop can be nearly spherical while a much larger puddle spreads under gravity. A force balance should reflect that scale rather than use a sphere formula for every shape.

A light object supported at a liquid surface can receive both hydrostatic buoyancy and forces along its contact line. This is why surface-supported needles or particles are not fully described by displaced-volume buoyancy alone. The contact geometry, wetting and vertical components must be included.

A useful calculation sequence is to identify the interfaces, draw their geometry, decide whether force or surface work is needed, and check the pressure reference and radius definitions. Count two faces only when two physical interfaces participate. These checks prevent most factor-of-two and radius-versus-diameter errors.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Surface tension, curved interfaces and capillary rise
Focus: Use surface tension when the geometry and scale require it
Learning goals:
- Interpret surface tension as force per length and reversible surface work per area.
- Count one or two interfaces in a film, drop or bubble.
- Derive spherical pressure jumps and use local external pressure.
- Use a through-liquid contact angle for capillary forces.
- Derive static capillary rise and state its geometric limits.
- Conserve liquid volume before comparing drop surface energies.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Surface tension has units N/m, equivalent to J/m².
A thin soap film has two interfaces; a simple liquid drop or gas cavity has one.
Spherical excess pressure scales as inverse radius.
Absolute internal pressure includes the local external pressure.
Capillary rise follows a vertical force balance and depends on cos θ and internal tube radius.
A change in surface energy does not by itself determine its eventual partition into other energy forms.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A thin two-faced soap film has surface tension 0.025 N/m and a straight movable boundary of length 0.10 m. What surface-tension force acts on it, in millinewtons?

Answer in mN.

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SELF CHECK 2

A spherical liquid drop has surface tension 0.040 N/m and radius 2.0 mm. What is its excess internal pressure?

Answer in Pa.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

At constant surface tension, halving a spherical drop’s radius does what to its excess pressure?

Halves it
Doubles it
Leaves it unchanged

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SELF CHECK 4

Does the static capillary-rise formula alone determine how quickly liquid reaches its final height?

Yes
No; transient flow and viscosity need additional modelling
Only for a larger tube

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SELF CHECK 5

Why is conserving the sum of initial gas-bubble volumes not generally sufficient to determine a final merged bubble radius?

Gas volume can change with pressure and temperature
Gas amount is never conserved
Surface tension has no effect on bubbles

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

How high does the liquid climb?Intermediate · Free sample · Hints and guided reasoning Pay for newly created surfaceAdvanced · Hints and guided reasoning Two surfaces support the bubbleIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Why do the simple pressure-jump formulas differ for a liquid drop and a thin soap bubble?

Compare your reasoning

A liquid drop has one curved liquid–gas interface; a thin soap bubble has two. For the same radius and surface tension in the ideal models, the bubble’s pressure jump is twice the drop’s.

Revisit the explanation →

A few days later: change the situation

Within the simple capillary model, what happens to the rise if tube radius is halved while liquid and contact angle stay unchanged?

Compare your reasoning

The rise doubles. Upward surface-tension force scales with circumference, while the supported liquid weight scales with cross-sectional area times height. Their balance gives height proportional to 1/r.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑