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Young-Laplace Equation Calculator

Ready to calculate
ΔP = 2γ cos θ / a = 2γ / R.
13 fluid presets.
6 pressure units + cap rise.
100% Free.
No Data Stored.

How it Works

01Pick a Fluid (or Custom γ)

13 preset fluids with auto-filled surface tension, or enter γ manually. Multi-unit: N/m, mN/m, dyn/cm, J/m².

02Enter Tube Geometry

Inner radius (a) of capillary tube + contact angle (θ) — OR meniscus radius (R) directly.

03Apply ΔP = 2γ/R = 2γ cos θ / a

Young-Laplace formula. Auto-converts units; handles both circular-tube and direct-meniscus inputs.

04Get ΔP + Capillary Rise

Output in 6 pressure units (Pa, kPa, bar, atm, mmHg, psi) plus capillary rise via Jurin's law.

What is a Young-Laplace Equation Calculator?

The Young-Laplace equation is the foundational result of capillarity and surface chemistry — it describes the pressure jump ΔP across a curved liquid-vapor interface as ΔP = 2γ/R (for a spherical meniscus of radius R) or equivalently ΔP = 2γ cos θ / a (for a circular tube of radius a with contact angle θ). Independently derived by Thomas Young in 1805 and Pierre-Simon Laplace in 1806, the equation explains capillary rise (water climbing in a thin tube), capillary depression (mercury sinking in a tube), water transport in tree xylem, ink-jet droplet formation, microfluidic chip operation, soil-water retention, and dozens of other natural and engineered phenomena involving curved liquid surfaces.

Our Young-Laplace Equation Calculator implements both forms with full unit support. Inputs: surface tension γ (4 unit options: N/m, mN/m, dyn/cm, J/m²); tube radius a (6 length units: mm / µm / cm / m / nm / in); contact angle θ (degrees or radians); meniscus radius R (same 6 length units). The calculator handles two computation paths automatically — given (a, θ) it computes R = a / cos θ and ΔP = 2γ cos θ / a; given R directly it computes ΔP = 2γ / R; if all three are entered, it verifies consistency. 13 fluid presets with auto-filled surface tensions at 20-25 °C: water (0.07294 N/m), mercury (0.4865), ethanol (0.02239), methanol, acetone, benzene, glycerol, chloroform, diethyl ether, n-hexane, olive oil, blood plasma, seawater. Custom mode for any other fluid.

Output: capillary pressure ΔP in 6 unit systems (Pa, kPa, bar, atm, mmHg, psi) plus the capillary rise via Jurin's law (h = 2γ cos θ / (ρ g a)) computed for a water-density fluid (scale by 1000/ρ for other liquids). Smart warnings flag exceptional surface tensions (γ > 1 N/m), molecular-scale tube radii (< 1 nm where continuum mechanics breaks down), and contact angles ≥ 90° where the meniscus inverts (capillary depression instead of rise). Designed for surface chemistry / fluid mechanics coursework, microfluidics engineers designing chip channels, ink-jet printer designers tuning nozzle physics, soil scientists modeling water retention, biomedical engineers studying capillary blood flow, and educators teaching capillarity — runs entirely in your browser, no account, no data stored.

Pro Tip: Pair this with our Vapor Pressure Calculator for the Kelvin-equation effects on small droplets, our Molarity Calculator for surfactant solutions, our Molality Calculator for colligative-property work, or our Graham's Law Calculator for gas-phase kinetics.

How to Use the Young-Laplace Equation Calculator?

Pick a Fluid Preset (or Custom): 13 common fluids with auto-filled surface tensions at 20-25 °C — water, mercury, ethanol, methanol, acetone, benzene, glycerol, chloroform, diethyl ether, n-hexane, olive oil, blood plasma, seawater. Or pick "Custom" and enter γ manually for any other fluid.
Verify or Override Surface Tension γ: in N/m, mN/m, dyn/cm, or J/m² (all equivalent except for unit prefix; 1 N/m = 1000 mN/m = 1000 dyn/cm = 1 J/m²). Edit if you have a temperature-corrected or measured value (γ typically decreases ~0.2% per °C).
Enter Tube Geometry — Two Options: Option A: inner radius (a) of capillary tube + contact angle (θ); the calculator computes R = a / cos θ. Option B: meniscus radius R directly; bypasses the contact-angle calculation. Option C: enter all three for a consistency check.
Contact Angle Reference: water on clean glass θ ≈ 0° (perfect wetting); water on PTFE / Teflon θ ≈ 110° (non-wetting); mercury on glass θ ≈ 140° (capillary depression); paint/wax on metal θ ≈ 30-60°. Critical: contact angle is highly sensitive to surface preparation and contamination — clean vs slightly-oily glass differs by 10-30°.
Apply ΔP = 2γ cos θ / a (Young-Laplace): the calculator returns capillary pressure in 6 unit systems. Reference values: water in 0.1 mm tube → ΔP ≈ 1.5 kPa; water in 1 µm tube → ΔP ≈ 150 kPa; water in 10 nm tube (zeolite pore) → ΔP ≈ 15 MPa.
Read the Capillary Rise (Jurin's Law): h = 2γ cos θ / (ρ g a). The calculator computes h for water-density fluid (1000 kg/m³); for other liquids, scale by (1000 / ρ). Reference: water in 0.1 mm tube → h ≈ 149 mm; in 0.01 mm tube → h ≈ 1.5 m (xylem of tall trees); in 1 µm tube → h ≈ 15 m (impossible for tree-height water transport — actual mechanism involves transpiration tension, not pure capillarity).
For Negative cos θ (Non-wetting Liquids): cos θ < 0 when θ > 90° → ΔP and h become negative → meniscus is convex and the liquid is DEPRESSED below the bulk surface (e.g. mercury in glass; water in waxed/PTFE tubes). The calculator warns and explains the inversion.

How is the Young-Laplace equation calculated?

The Young-Laplace equation is the cornerstone of surface chemistry — derived from minimizing surface energy at a curved interface, it provides the link between molecular-scale surface tension and macroscopic capillary phenomena. The math is one short equation; the implications span natural and engineered systems from tree xylem to microfluidic chips.

References: Thomas Young, Phil. Trans. R. Soc. London 95 (1805) 65; Pierre-Simon Laplace, Mécanique céleste vol. 4 (1806); Adamson, Physical Chemistry of Surfaces (6th ed., 1997); de Gennes, Brochard-Wyart & Quéré, Capillarity and Wetting Phenomena (2004).

Core Formula — General (Any Curved Interface)

ΔP = γ × (1/R₁ + 1/R₂)

Where R₁ and R₂ are the two principal radii of curvature at any point on the interface. For a spherical meniscus, R₁ = R₂ = R, giving the simplified form ΔP = 2γ/R. For a cylindrical interface (e.g. liquid filament), R₁ = R, R₂ = ∞, giving ΔP = γ/R.

Capillary Tube Form

For a fluid with contact angle θ in a circular tube of inner radius a, the meniscus is approximately spherical with radius R = a / cos θ. Substituting:

ΔP = 2γ cos θ / a

When θ < 90° (wetting), cos θ > 0, ΔP > 0, and capillary rise occurs. When θ > 90° (non-wetting), cos θ < 0, ΔP < 0, and capillary depression occurs.

Capillary Rise — Jurin's Law

Equilibrium height h is reached when capillary pressure balances the hydrostatic head:

h = 2γ cos θ / (ρ g a)

Where ρ is liquid density, g = 9.81 m/s². For water (γ = 0.073, ρ = 1000): h(mm) ≈ 14.9 / a(mm) for θ = 0.

Worked Example — Water in 0.1 mm Glass Capillary

γ(water at 20 °C) = 0.07294 N/m; a = 0.1 mm = 1 × 10⁻⁴ m; θ ≈ 0° for clean glass.

  • cos 0° = 1.
  • R = a / cos θ = 0.1 mm (meniscus is hemispherical).
  • ΔP = 2 × 0.07294 × 1 / (1 × 10⁻⁴) = 1459 Pa = 1.46 kPa = 11 mmHg.
  • Capillary rise h = 2 × 0.07294 × 1 / (1000 × 9.81 × 1 × 10⁻⁴) = 0.149 m = 149 mm.
  • This is the classic introductory-physics capillary-rise demonstration.

Worked Example — Mercury in Glass (Capillary Depression)

γ(mercury at 20 °C) = 0.4865 N/m; a = 1 mm; θ = 140° (mercury does not wet glass).

  • cos 140° = −0.766.
  • ΔP = 2 × 0.4865 × (−0.766) / (1 × 10⁻³) = −745 Pa (negative → depression).
  • Capillary depression h = 2 × 0.4865 × (−0.766) / (13546 × 9.81 × 1 × 10⁻³) = −5.6 mm.
  • Mercury level INSIDE the tube is 5.6 mm BELOW the bulk surface — the foundation of mercury-barometer correction tables.

Reference Surface Tensions (γ at 20-25 °C, in N/m)

  • Mercury: 0.4865 (very high — mercury's tightly-held electrons).
  • Water (pure): 0.07294 — high among non-metallic liquids due to hydrogen bonding.
  • Glycerol: 0.0631.
  • Blood plasma (37 °C): 0.052.
  • Olive oil: 0.032.
  • Benzene: 0.0289.
  • Acetone: 0.0237.
  • Ethanol: 0.0224.
  • Methanol: 0.0223.
  • n-Hexane: 0.0184.
  • Diethyl ether: 0.0170 — very low; basis of ether's wetting/spreading characteristics.
  • Liquid nitrogen (77 K): 0.00875 (very low at cryogenic T).

Reference Contact Angles (Water on Various Surfaces)

  • Clean glass: 0° (perfect wetting; water spreads completely).
  • Slightly contaminated glass: 10-30° (typical lab condition).
  • Polyethylene: 88°.
  • PTFE (Teflon): 110°.
  • Paraffin wax: 110°.
  • Lotus leaf (super-hydrophobic): 160-170° (the famous "lotus effect" self-cleaning).
  • Engineered super-hydrophobic surfaces: > 170°.
  • Mercury on glass: 140° (non-wetting).
Real-World Example

Worked Example — Water Rise in Xylem (Tree Capillaries)

Question: Tree xylem (water-conducting tissue) consists of capillaries with typical radius 50-200 µm. How high can pure capillarity raise water in a 100 µm xylem capillary?

Step 1 — Set Up the Problem.

  • γ(water at 20 °C) = 0.07294 N/m.
  • a = 100 µm = 1 × 10⁻⁴ m.
  • θ ≈ 0° for water in cellulose-rich xylem (effectively perfect wetting).
  • ρ(water) = 1000 kg/m³; g = 9.81 m/s².

Step 2 — Compute Capillary Pressure.

  • ΔP = 2γ cos θ / a = 2 × 0.07294 × 1 / (1 × 10⁻⁴) = 1459 Pa ≈ 1.46 kPa.
  • This is roughly 11 mmHg, or 1/100 of atmospheric pressure.

Step 3 — Apply Jurin's Law.

  • h = 2γ cos θ / (ρ g a) = 2 × 0.07294 × 1 / (1000 × 9.81 × 1 × 10⁻⁴).
  • h = 0.14588 / 0.981 = 0.149 m = 14.9 cm.
  • By pure capillarity alone, water can only rise about 15 cm in a 100 µm tree capillary.

Step 4 — Compare to Real Tree Heights.

  • Coast redwood (Sequoia sempervirens) reaches > 100 m height.
  • For pure capillary rise to lift water 100 m would require xylem radius a = 2γ / (ρ g h) = 2 × 0.073 / (1000 × 9.81 × 100) = 0.15 µm — much smaller than actual xylem (50-200 µm).
  • Real mechanism: "cohesion-tension theory" — transpiration at leaves creates negative pressure (tension) that PULLS water up the stem under continuous water columns; xylem capillarity prevents air-bubble nucleation but doesn't push water up.
  • Measured xylem tension at canopy of tall trees: typically −1 to −3 MPa (extreme cases up to −6 MPa near desert plant roots).

Step 5 — When IS Capillarity the Dominant Mechanism?

  • Sub-micron pores (1-100 nm): capillary pressure 10⁵-10⁷ Pa — dominates over gravity at all heights. Examples: nano-porous membranes (RO desalination), zeolite catalysts, soil micropores.
  • Micron-scale (1-10 µm): capillary rise tens of meters. Examples: paper towels, wicking fabrics, soil mesopores, tree xylem (combined with transpiration tension).
  • 0.1-1 mm: capillary rise centimeters to a meter. Glass capillary tubes, blood capillaries (5-10 µm but with active circulation, not pure capillarity), oil-shale fractures.
  • > 1 mm: capillary effects negligible compared to gravity; pure pressure / pump-driven flow.

Who Should Use the Young-Laplace Equation Calculator?

1
Standard early-curriculum topic in p-chem and surface science. Calculator handles arithmetic; students focus on understanding capillarity, contact angles, and the link between molecular surface tension and macroscopic phenomena.
2
Microfluidic channels (10-500 µm) operate in the capillary-dominated regime. ΔP = 2γ/R sets droplet formation, channel filling, and surface-tension-driven flow design.
3
Ink-jet nozzles (10-50 µm) rely on Young-Laplace + Rayleigh-Plateau instability for droplet formation. Calculator quantifies the surface-tension contribution to drop ejection physics.
4
Soil water retention follows capillary-pressure curves derived from Young-Laplace (smaller pores hold water more tightly). Standard input for soil moisture modeling and irrigation scheduling.
5
Compute capillary contribution to plant water transport; understand limits of pure capillarity for tall trees (transpiration tension dominates at tree-height scales).
6
Oil/water capillary pressure curves in porous reservoir rocks govern water-flooding sweep efficiency, residual oil saturation, and enhanced oil recovery design.
7
Capillary blood flow, alveolar surface tension (with surfactant!), drug-tablet wetting, blister formation, contact-lens-tear-film dynamics — all involve Young-Laplace.

Technical Reference

Historical Origin. Thomas Young (1773-1829), British polymath, derived the curvature-pressure relationship in his 1805 paper "An Essay on the Cohesion of Fluids" (Phil. Trans. R. Soc. London 95, 65). Pierre-Simon Laplace (1749-1827), independently working on planetary mechanics, derived the same relationship from molecular-attraction considerations in his 1806 supplement to Mécanique céleste. The combined equation has been called Young-Laplace ever since. It is one of the foundational results of 19th-century physical chemistry, alongside Clausius-Clapeyron (1834), Raoult (1887), and van der Waals (1873).

Generalized Form for Any Curved Interface. For an arbitrary point on a curved interface with two principal radii of curvature R₁ and R₂:

ΔP = γ × (1/R₁ + 1/R₂)

where ΔP is the pressure difference across the interface (positive on the concave side), γ is surface tension, and the curvatures 1/R₁ + 1/R₂ define the local mean curvature. For special cases: spherical meniscus R₁ = R₂ = R → ΔP = 2γ/R; cylindrical meniscus / liquid jet R₁ = R, R₂ = ∞ → ΔP = γ/R; flat interface R = ∞ → ΔP = 0.

Capillary Tube Geometry. For a circular cylindrical tube of inner radius a, with a wetting liquid forming a contact angle θ at the wall, the meniscus is approximately spherical with radius R = a / cos θ. This geometry follows from: the meniscus is normal to the wall at the contact line; the contact angle is the angle between the meniscus and the wall measured through the liquid; for a hemispherical meniscus, R cos θ = a (geometry of the wetting circle).

Jurin's Law (Capillary Rise). James Jurin (1718) measured capillary rise as a function of tube diameter and discovered the inverse-radius dependence experimentally — over 80 years before Young-Laplace explained it theoretically. Derivation: at hydrostatic equilibrium, the capillary pressure ΔP at the meniscus equals the hydrostatic pressure of the liquid column ρgh. So 2γ cos θ / a = ρgh, giving h = 2γ cos θ / (ρ g a). For water at room T (γ = 0.073, ρ = 1000, g = 9.81, θ = 0°): h(mm) ≈ 14.9 / a(mm).

Surface Tension Temperature Dependence. Surface tension decreases with rising T because thermal motion disrupts the cohesive forces that create surface tension. Empirical relation (Eötvös rule): γ × V_m^(2/3) ≈ k_E × (T_c − T), where V_m is molar volume, T_c is critical temperature, and k_E ≈ 2.1 × 10⁻⁷ J/(K·mol^(2/3)) for "normal" liquids (deviation indicates association). Approximate water surface tension:

  • 0 °C: 0.0756 N/m.
  • 20 °C: 0.0728 N/m.
  • 25 °C: 0.0720 N/m.
  • 50 °C: 0.0679 N/m.
  • 100 °C: 0.0589 N/m.
  • 374 °C (critical T): 0 N/m.

Contact Angle and Surface Energy. The contact angle θ is determined by the balance of three interfacial energies via Young's equation: γ_SV = γ_SL + γ_LV cos θ, where γ_SV is solid-vapor, γ_SL is solid-liquid, and γ_LV is liquid-vapor surface tension. θ < 90° (wetting): cos θ > 0; liquid spreads. θ = 0°: complete wetting (water on clean glass). θ > 90° (non-wetting): cos θ < 0; liquid beads up. θ = 180°: complete non-wetting (idealized lotus-leaf surface).

Beyond Simple Young-Laplace — Modifications.

  • Gravity-induced curvature (large drops/bubbles): ΔP varies with height; full solution requires solving the Young-Laplace differential equation in cylindrical coordinates → gives shapes characterized by capillary length κ⁻¹ = √(γ/(ρg)).
  • Disjoining pressure (very thin liquid films): for thin films < 100 nm, additional disjoining-pressure terms (van der Waals, electrostatic, structural) modify Young-Laplace.
  • Marangoni effects: surface-tension gradients (from T or surfactant concentration variations) drive surface flows; standard Young-Laplace assumes uniform γ.
  • Dynamic contact angle: at moving contact lines (advancing or receding meniscus), θ_dyn differs from θ_eq — usually θ_advancing > θ_receding (contact-angle hysteresis).

Connection to Kelvin Equation. The Young-Laplace pressure across a curved interface drives the Kelvin equation for vapor-pressure modification: ln(P/P°) = 2γ V_m / (R T r), where r is droplet radius (positive for convex / droplets, negative for concave / capillaries). For water at 25 °C: a 1 nm droplet has 38% higher vapor pressure than a flat surface; a 10 nm droplet has 11% higher; a 100 nm droplet has 1% higher. This explains why small droplets evaporate preferentially (Ostwald ripening), why nucleation requires supersaturation, and why fog persists at > 100% relative humidity. References: Young (1805) Phil. Trans. R. Soc.; Laplace (1806); Adamson, Physical Chemistry of Surfaces (6th ed., 1997); Atkins' Physical Chemistry (12th ed., Chapter 16); de Gennes, Brochard-Wyart & Quéré, Capillarity and Wetting Phenomena (Springer, 2004); CRC Handbook of Chemistry and Physics (surface tension data).

Conclusion

The Young-Laplace equation is the cornerstone of capillarity and surface chemistry — one short equation (ΔP = 2γ/R or ΔP = 2γ cos θ / a) that explains capillary rise, capillary depression, microfluidic flow, ink-jet drop formation, soil moisture, plant water transport, and surface-tension-dominated phenomena across natural and engineered systems. Memorize the key reference values: water γ = 0.073 N/m, mercury γ = 0.487, water in 0.1 mm clean-glass tube → ΔP ≈ 1.46 kPa and h ≈ 149 mm capillary rise.

Two operational reminders: (1) Surface tension γ is temperature-sensitive (~0.2% per °C decrease), so use the value at your operating T; preset values are at 20-25 °C. (2) Contact angle θ is highly sensitive to surface preparation — clean glass gives θ ≈ 0° for water, but trace contamination raises it to 30° or more. For non-wetting systems (mercury on glass θ = 140°, water on PTFE θ = 110°, lotus-leaf θ ≈ 165°), cos θ is negative and the meniscus inverts (capillary depression). The calculator handles all these cases with smart warnings and a transparent calculation breakdown.

Frequently Asked Questions

What is the Young-Laplace Equation Calculator?
It implements the foundational Young-Laplace equation for capillary pressure across a curved liquid-vapor interface: ΔP = 2γ/R for spherical meniscus, or ΔP = 2γ cos θ / a for circular tube of radius a with contact angle θ. 13 fluid presets (water, mercury, ethanol, methanol, acetone, benzene, glycerol, etc.) with auto-filled surface tensions. Output: capillary pressure in 6 unit systems (Pa, kPa, bar, atm, mmHg, psi) plus capillary rise from Jurin's law.

Pro Tip: Pair this with our Vapor Pressure Calculator.

What is the Young-Laplace equation?
The fundamental equation of capillarity: ΔP = γ × (1/R₁ + 1/R₂) for a general curved liquid-vapor interface, simplifying to ΔP = 2γ/R for a spherical meniscus and ΔP = 2γ cos θ / a for a circular tube. Independently derived by Thomas Young (1805) and Pierre-Simon Laplace (1806). Explains capillary rise, capillary depression, microfluidic behavior, ink-jet drop formation, plant water transport, soil moisture retention, and dozens of other surface-tension-dominated phenomena.
What's the formula for capillary pressure?
Three equivalent forms. Spherical meniscus: ΔP = 2γ/R, where R is the radius of curvature. Circular tube: ΔP = 2γ cos θ / a, where a is the tube radius and θ is the contact angle. The two are related by R = a / cos θ. Capillary rise (Jurin's law): h = 2γ cos θ / (ρ g a), where ρ is liquid density and g = 9.81 m/s². Negative h indicates capillary depression for non-wetting liquids (cos θ < 0).
How high does water rise in a capillary tube?
h = 2γ cos θ / (ρ g a). For water (γ = 0.073 N/m, ρ = 1000 kg/m³, θ = 0° on clean glass): h(mm) ≈ 14.9 / a(mm). Reference values: 1 mm tube → 15 mm rise; 0.1 mm → 149 mm rise; 0.01 mm → 1.5 m rise; 1 µm → 15 m; 100 nm → 150 m; 10 nm → 1500 m (theoretical only — at this scale continuum mechanics breaks down). For non-zero contact angle, multiply by cos θ; for non-water fluid, scale by (γ_fluid / γ_water) × (1000 / ρ_fluid).
What is surface tension?
The energy per unit area required to expand a liquid surface, equivalently the force per unit length pulling tangentially along a surface. Units: N/m or J/m² (numerically equal). Caused by the asymmetry of intermolecular forces at a surface — molecules in the bulk feel attractive forces from all directions; surface molecules feel attractive forces only from below (no neighbors above), producing a net inward pull that minimizes surface area. Reference values at 20-25 °C: water 0.073 N/m (high due to H-bonding); ethanol 0.022; mercury 0.487 (very high — metallic bonding); diethyl ether 0.017 (very low).
What is contact angle?
The angle between the liquid surface and the solid surface at the three-phase contact line, measured through the liquid. Determined by the balance of three interfacial energies (Young's equation: γ_SV = γ_SL + γ_LV cos θ). θ = 0°: complete wetting (water on perfectly clean glass). θ < 90°: wetting (most water on most surfaces). θ = 90°: neutral. θ > 90°: non-wetting (water on PTFE 110°; mercury on glass 140°). θ > 150°: super-hydrophobic (lotus leaf, engineered surfaces). Highly sensitive to surface preparation — clean vs slightly contaminated glass differs by 10-30°.
What is the Young-Laplace pressure for water in a 1 mm tube?
~146 Pa = 0.146 kPa. Math: γ(water) = 0.07294 N/m, a = 1 mm = 0.001 m, θ = 0° on clean glass. ΔP = 2 × 0.07294 × 1 / 0.001 = 145.9 Pa. Capillary rise: h = 2γ cos θ / (ρga) = 0.14588 / 9.81 = 14.9 mm. So water rises about 15 mm in a 1 mm clean-glass tube — a classic demonstration of capillarity.
Why does mercury sink in a glass tube?
Because mercury's contact angle on glass is ~140°, giving cos θ = −0.766. The capillary pressure ΔP = 2γ cos θ / a is negative, meaning the meniscus is convex (bulges upward) and the liquid level INSIDE the tube is BELOW the bulk surface. Math for 1 mm tube: ΔP = 2 × 0.4865 × (−0.766) / 0.001 = −745 Pa. Capillary depression h = ΔP / (ρ_Hg × g) = −745 / (13546 × 9.81) = −5.6 mm. Mercury level is 5.6 mm BELOW outside level — the basis of the historical mercury barometer correction tables.
Why doesn't pure capillarity explain how tall trees transport water?
Because xylem capillary radius (50-200 µm) is far too large. For pure capillarity to lift water 100 m (coast redwood height), the capillary radius would need to be ~0.15 µm — but actual xylem is 100-1000× larger. The actual mechanism is the cohesion-tension theory: transpiration at the leaves creates a water-vapor pressure deficit that pulls water up the stem under continuous, cohesive water columns; xylem capillarity prevents air-bubble nucleation but doesn't push water up. Measured xylem tension at canopy of tall trees: typically −1 to −3 MPa.
How does temperature affect surface tension?
Surface tension decreases with rising temperature — typically ~0.2% per °C for most liquids; vanishes at the critical temperature (where liquid and vapor become indistinguishable). Empirical relations: Eötvös γ V_m^(2/3) ≈ k_E (T_c − T); linear approximation γ(T) = γ(20°C) × [1 − α × (T − 20)]. Water specifically: 0 °C → 0.0756; 20 °C → 0.0728; 25 °C → 0.0720; 50 °C → 0.0679; 100 °C → 0.0589; 374 °C (critical) → 0 N/m. Use temperature-corrected γ in the Young-Laplace equation for accurate work; the calculator presets are at 20-25 °C.
What's the difference between a and R in the equations?
a is the inner radius of the tube; R is the radius of the meniscus. They're related by R = a / cos θ. For a wetting fluid (θ = 0°): R = a (meniscus is hemispherical). For non-wetting fluid (θ approaches 90°): R approaches infinity (meniscus is flat). For more non-wetting (θ > 90°): R becomes negative (meniscus inverts/convex). The calculator accepts EITHER (a, θ) OR R as input; if you enter all three, it verifies consistency. Use (a, θ) for a known capillary tube and surface; use R for a measured spherical bubble or droplet.

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our ToolsACE chemistry team built this calculator to handle the foundational <strong>Young-Laplace equation</strong> for capillary pressure across curved liquid-vapor interfaces. Independently derived by <strong>Thomas Young</strong> (1805) and <strong>Pierre-Simon Laplace</strong> (1806), the equation describes the pressure difference ΔP across a curved fluid interface as <strong>ΔP = 2γ/R</strong> for a spherical meniscus, or <strong>ΔP = 2γ cos θ / a</strong> for a circular tube of radius a with contact angle θ (where R = a / cos θ). The calculator includes <strong>13 fluid presets</strong> with auto-filled surface tensions at 20-25 °C: water (0.07294 N/m), mercury (0.4865), ethanol (0.02239), methanol, acetone, benzene, glycerol, chloroform, diethyl ether, n-hexane, olive oil, blood plasma, seawater. <strong>4 surface-tension units</strong> (N/m, mN/m, dyn/cm, J/m²); <strong>6 length units</strong> for tube and meniscus radius (mm, µm, cm, m, nm, in); <strong>angle in degrees or radians</strong>. Output: capillary pressure in 6 systems (Pa, kPa, bar, atm, mmHg, psi) plus the capillary rise from Jurin's law (h = 2γ cos θ / (ρ g a)). Smart consistency checks for users who enter all three of a, θ, R.

Thomas Young (1805) and Pierre-Simon Laplace (1806) — Young-Laplace equationAtkins' Physical Chemistry; Adamson's Physical Chemistry of Surfaces (6th ed.)CRC Handbook of Chemistry and Physics — surface tension data

Disclaimer

The Young-Laplace equation assumes a smooth, well-defined liquid-vapor interface and continuum-scale fluid behavior; for tube radii below ~1 nm (molecular dimensions), continuum mechanics breaks down and molecular-dynamics simulations are required. Surface tension γ depends on temperature (~0.2% decrease per °C); preset values are at 20-25 °C. Contact angle θ depends strongly on surface preparation — clean vs contaminated/oily glass differs by 10-30°. For non-wetting liquids (mercury on glass θ ≈ 140°), cos θ is negative and the meniscus inverts (capillary depression). Capillary rise (Jurin's law) assumes hydrostatic equilibrium and a vertical tube; for inclined or horizontal capillaries, additional gravitational/dynamic terms apply. References: Young (1805); Laplace (1806); Adamson, Physical Chemistry of Surfaces (6th ed., 1997); Atkins' Physical Chemistry; de Gennes, Brochard-Wyart & Quéré, Capillarity and Wetting Phenomena (2004).