Capillary Rise Calculator
Calculate how far a liquid rises (or is depressed) in a capillary tube using Jurin's law. Enter surface tension, liquid density, tube radius, contact angle and gravity to get the capillary rise in mm, cm, m or inches.
Find the Capillary Rise from Surface Tension, Density and Tube Radius
Capillary Rise
Where:
- \(h\) — capillary rise or depression of the liquid column (m)
- \(\gamma\) — surface tension of the liquid against air (N/m)
- \(\theta\) — contact angle between the liquid and the tube wall (°)
- \(\rho\) — density of the liquid (kg/m³)
- \(g\) — gravitational acceleration (m/s²)
- \(r\) — inner radius of the capillary tube (m)
The first equation is Jurin's law: it follows from balancing the upward capillary force \(2\pi r\,\gamma\cos\theta\) with the weight \(\rho g \pi r^2 h\) of the liquid column. The second gives the capillary (Laplace) pressure jump across the curved meniscus that drives the motion.
Capillary Rise – Calculation Example
Example: water at 20 °C (surface tension 72.8 mN/m, density 998 kg/m³) rises in a clean glass capillary of inner radius 0.5 mm. Water perfectly wets clean glass, so the contact angle is 0°, and gravity is 9.81 m/s².
\[ \gamma = 72.8\ \text{mN/m} = 0.0728\ \text{N/m}, \quad r = 0.5\ \text{mm} = 5\times10^{-4}\ \text{m} \]
\[ h = \frac{2 \times 0.0728 \times \cos 0°}{998 \times 9.81 \times 5\times10^{-4}} = \frac{0.1456}{4.895} \approx 0.02974\ \text{m} \]
\[ h \approx 29.74\ \text{mm} \approx 2.97\ \text{cm} \approx 1.17\ \text{in} \]
Result: the water rises about 29.74 mm (≈ 2.97 cm) above the surrounding water level — exactly what the calculator shows with its default values.
How to Use the Capillary Rise Calculator
- Enter the Surface Tension (γ) of the liquid and pick its unit (mN/m, N/m or dyn/cm). Water at 20 °C is about 72.8 mN/m.
- Enter the Liquid Density (ρ) and its unit (kg/m³, g/cm³ or lb/ft³). Use the density of the liquid, not of the surrounding air.
- Enter the inner Tube Radius (r) — remember this is the radius, so a 1 mm inner diameter tube takes 0.5 mm. Choose mm, µm, cm, m or in.
- Set the Contact Angle (θ) in degrees: 0° for water in clean glass, around 140° for mercury in glass (which produces a depression instead of a rise).
- Adjust the Gravitational Acceleration (g) if needed; 9.81 m/s² is the Earth default, and 1.62 m/s² approximates the Moon.
- Choose the output unit under Show Capillary Rise in (mm, cm, m or in) — the rise grows quickly as tubes get narrower, so switch units if the number is unwieldy.
The result is signed: a positive value is a rise above the surrounding liquid level, and a negative value is a depression below it.
What Is Capillary Rise and Why Does It Happen?
Capillary rise is the phenomenon in which a liquid climbs up a narrow tube (a capillary) dipped into it, apparently defying gravity. It is driven by surface tension: the molecules at the liquid surface pull the meniscus outward along the tube wall, and if the liquid wets the wall (contact angle below 90°), the meniscus is concave and its curvature generates a pressure deficit that sucks liquid upward. The column stops rising when its weight exactly balances the capillary force — that balance is Jurin's law.
The most striking feature of the formula is the inverse proportionality to the tube radius: halving the radius doubles the rise. Water in a 0.5 mm tube rises only about 3 cm, but in a 10 µm pore it climbs roughly 1.5 m. This scaling explains why paper towels soak up spills, why wicks and sponges work, and how trees lift water tens of meters through microscopic xylem channels — capillarity, together with osmosis and transpiration, is one of the key transport mechanisms in plants.
The contact angle determines the direction of the effect. Wetting liquids (\(\theta < 90°\), water in glass) rise; \(\theta = 90°\) leaves the surface flat; and non-wetting liquids (\(\theta > 90°\), mercury in glass) are depressed below the surrounding level, because \(\cos\theta\) is negative. The calculator reports the sign automatically and tells you whether you have a rise or a depression. It also shows the radius of curvature of the meniscus (\(R = r/\cos\theta\)) and the capillary pressure \(\Delta p\), which is what a porous material or a wick actually "feels".
Assumptions, Limitations and Practical Tips
- Tube geometry: the formula assumes a vertical circular cylinder of uniform radius. Square channels, tapered bores or porous networks only follow it approximately.
- Small radius: the meniscus is treated as a perfect spherical cap, which is accurate when the radius is well below the capillary length of the liquid (about 2.7 mm for water). Around 1 mm, neglecting the meniscus volume introduces an error of a few percent; below 0.5 mm it is negligible.
- Constant contact angle: real surfaces are rarely perfectly clean or uniform, so the effective angle can differ from textbook values. Contamination, grease or surfactants can change \(\gamma\) and \(\theta\) substantially — adding detergent to water lowers both surface tension and rise.
- Static liquid: the result is the equilibrium height; it says nothing about how fast the liquid climbs (which depends on viscosity, via the Washburn equation).
- Temperature: both surface tension and density vary with temperature, so use values at the temperature of interest for accurate results.
If you are working with other liquid or gas properties, you may also find our Gas Density Calculator, the Antoine Equation Calculator for vapor pressures, or the Mean Free Path Calculator useful for related transport problems.
Frequently Asked Questions
How do I convert surface tension units: mN/m, dyn/cm and N/m?
The conversions are simple because 1 dyn/cm = 1 mN/m = 0.001 N/m. So water's 72.8 mN/m is the same as 72.8 dyn/cm or 0.0728 N/m. Older textbooks and tables often quote dyn/cm, so check which unit your source uses before entering the value — mixing them up shifts the answer by a factor of 1000. The calculator handles the conversion automatically once you select the right unit.
Why did I get a negative capillary rise?
A negative result means the liquid is depressed, not raised — the meniscus sits below the surrounding liquid level. This happens when the contact angle is above 90°, so \(\cos\theta\) is negative and capillary force pushes the liquid down instead of pulling it up. The classic case is mercury in glass (θ ≈ 140°), which drops several millimeters in a narrow tube. The effect gets stronger in smaller tubes, just like the rise does.
Should I enter the tube's diameter or its radius?
The formula needs the inner radius, which is half the inner diameter. A tube sold as "1 mm capillary" usually means 1 mm inner diameter, so you should enter 0.5 mm. Entering the diameter by mistake halves the calculated rise. Also make sure you use the inner bore, not the outer radius — wall thickness plays no role in Jurin's law.
What is a typical capillary rise for water in everyday tubes?
For water in clean glass (θ = 0°), a rough rule of thumb is h ≈ 14 mm per millimeter of tube diameter, or about 14.4/r² mm when r is in mm. That gives roughly 29 mm for a 1 mm bore, 5.7 cm for a 0.5 mm bore, and about 1.5 m in a 10 µm pore. If your result is far outside these magnitudes, double-check the radius unit — µm and mm are easily confused.
How does temperature affect the capillary rise?
Both inputs drift with temperature: water's surface tension falls from about 75.7 mN/m at 0 °C to 58.9 mN/m at 100 °C, roughly 0.15 mN/m per degree, while density drops only slightly. Net effect: warm liquid rises noticeably less than cold liquid in the same tube — about 4% lower from 20 °C to 40 °C. For precise work, look up γ and ρ at the actual measurement temperature instead of using the 20 °C defaults.
Does the calculator tell me how fast the liquid rises?
No — Jurin's law gives only the final equilibrium height, reached after the liquid settles. The speed of climbing is a separate problem described by the Washburn equation, where the rise distance grows roughly with the square root of time and depends on viscosity: the height reached scales as \( h(t) \sim \sqrt{\gamma r t \cos\theta / (2\eta) } \). A very viscous liquid like honey eventually reaches the same height as water, just far more slowly.
What contact angle should I use for my liquid and tube material?
Use tabulated values for your specific liquid–wall pair, since the angle depends on both. Common approximations: water on clean glass ≈ 0°, water on plastic (PTFE, polyethylene) ≈ 90–110°, mercury on glass ≈ 140°, and many organic liquids on glass ≈ 0–30°. Real tubes are rarely perfectly clean: grease, dust or surfactant films raise the contact angle and lower surface tension, so measured rises often come out 10–20% below theoretical predictions.