A surface tension calculator is really two calculations that meet in the middle. One is measurement: a tensiometer pulls a plate or a ring out of a liquid, reads a force, and that force divided by the length of the contact line is the surface tension. The other is prediction: given a surface tension, how high does the liquid climb inside a narrow tube. This page does both, and it will also run the second one backwards to extract a surface tension from a rise height you have measured with nothing more than a capillary and a ruler.
Arb Digital publishes free physics calculators that each own one job, and this page owns getting to a value of γ from an experiment. The live Young-Laplace calculator is the neighbouring tool and it works in the other direction: it takes γ as a given input and returns the pressure jump across a curved interface, plus the rise that follows from it. Use that page when you already know the surface tension and want the pressure inside a drop or a bubble; use this one when the surface tension is the unknown. The surface tension converter handles units alone and computes nothing.
What This Surface Tension Calculator Does
In force mode it converts a tensiometer reading into γ. You give it the force and the probe geometry, it works out the length of the three-phase contact line, and it divides one by the other. Two standard probes are built in: the Wilhelmy plate, which wets on both faces so the contact line is twice the width plus twice the thickness, and the Du Noüy ring, which wets inside and outside so the contact line is twice the circumference. You can also enter a wetted length directly for any other probe.
In rise mode it inverts Jurin's law. Measure how far the liquid climbs in a capillary of known radius, tell it the density and the contact angle, and it returns the surface tension that produced that height. This is the classic undergraduate method and it is surprisingly good when the tube is clean and genuinely circular.
In predict mode it runs Jurin's law forwards, returning the height for a γ you supply. In every mode it also reports the capillary length for the liquid and the Bond number for the tube, which together tell you whether capillarity or gravity is in charge.
How to Use It
- Pick the mode that matches what you actually measured. A force from an instrument, a height from a tube, or neither if you are predicting.
- Get the wetted length right. This is where most force-mode errors live. Both faces of a plate count, and both surfaces of a ring count. Halving the contact line doubles the answer.
- Enter the contact angle honestly. Zero is the usual assumption for clean water on clean glass and for a properly wetted platinum probe, but it is an assumption, not a fact.
- Use the true internal radius of the capillary, not the nominal bore. A ten per cent radius error is a ten per cent error in γ straight through.
- Check the Bond number. If it is not small, the meniscus is not the spherical cap the simple relation assumes and the result needs a correction.
The Formulas
Surface tension is defined as force per unit length of contact line, and OpenStax states it directly in section 11.8, Cohesion and Adhesion in Liquids:
γ = F / (L cosθ)
with L the wetted perimeter and θ the contact angle, taken as zero for a fully wetting probe. For a Wilhelmy plate of width w and thickness t, L = 2(w + t). For a Du Noüy ring of radius R, L = 4πR.
Capillary rise follows Jurin's law, which the same OpenStax section gives as
h = 2γ cosθ / (ρ g r)
and rearranging it gives the measurement form γ = ρ g r h / (2 cosθ). The capillary length, √(γ/ρg), is the scale at which surface tension and gravity balance, and the Bond number ρg r2/γ is the square of the tube radius divided by that length. HyperPhysics covers the molecular origin of the effect and gives 72.8 dyn/cm for water at twenty degrees Celsius.
Work the default by hand. A plate 20 mm wide and 0.2 mm thick has a contact line of 2(20 + 0.2) = 40.4 mm, or 0.0404 m. A force of 2.94 mN is 0.00294 N, so γ = 0.00294 ÷ 0.0404 = 0.07277 N/m, which is 72.77 mN/m — clean water. Feeding that into Jurin's law with a 0.2 mm radius and a density of 998 kg/m3 gives h = 2 × 0.07277 ÷ (998 × 9.80665 × 0.0002) = 0.14554 ÷ 1.9574 = 0.0744 m, or 74.4 mm. The capillary length is √(0.07277 ÷ 9787) = 2.73 mm.
Why Clean Water Is Almost Impossible To Measure
Surface tension is a surface property, so it responds to whatever is sitting on the surface, and almost everything wants to sit there. Surfactants concentrate at the interface by definition, and a quantity far too small to detect chemically in the bulk will change γ measurably. A fingerprint on a glass rod, a trace of detergent left in a beaker, or a film of oil from laboratory air will all drop the reading.
This is the single reason surface tension measurements disagree. A student measuring 65 mN/m for tap water has not found a different physics; they have measured slightly soapy water. The value is genuinely 65 for the liquid in front of them. Glassware for this work is cleaned aggressively and used immediately, and the reading is taken as quickly as the instrument allows because contamination accumulates with time at the surface.
Temperature matters too, and in a predictable direction. Surface tension falls as temperature rises, because the cohesive forces holding the surface together weaken, and it goes to zero at the critical point. For water the fall is roughly 0.15 mN/m per degree near room temperature, so a five-degree difference between your sample and the reference value in your textbook is nearly a full unit.
The Du Noüy Ring Correction Nobody Mentions
The ring method looks simpler than it is. As the ring lifts, it drags a volume of liquid with it, and the shape of that hanging column is not the simple cylinder the basic formula assumes. The raw force therefore overestimates γ, typically by a few per cent but by considerably more for small ring radii or dense liquids.
Published correction factors exist, tabulated against the ratio of ring radius to wire radius and against the volume lifted. Commercial tensiometers apply them automatically; a hand calculation using the raw force does not. This page returns the uncorrected value from the contact line you specify, which is the right base number to apply your instrument's own correction to, and it is worth stating which convention your reported figure uses.
The Wilhelmy plate avoids the problem by design. Because the plate is held so that it just touches the surface rather than being pulled out of it, there is no lifted column and no correction, which is why it is preferred for careful work. Its own weakness is that it assumes complete wetting, so a plate that is not perfectly clean introduces a contact angle nobody accounted for.
When Jurin's Law Stops Working
The capillary rise relation assumes the meniscus is a spherical cap that meets the wall at the contact angle, and that the tube is narrow enough for gravity to be negligible across its width. That is exactly the condition that the Bond number is small. Once the radius approaches the capillary length — about 2.7 mm for water — the meniscus flattens under its own weight, the spherical assumption fails, and the simple formula over-predicts the rise.
Two other cases break it in a more interesting way. If the contact angle exceeds ninety degrees the cosine goes negative and the liquid is pushed down rather than pulled up. Mercury in glass is the standard example, and the calculator reports a depression rather than refusing to answer. And if the tube is not circular, the relation needs the mean curvature of the actual cross-section, not a radius; square and rectangular channels have their own published forms.
Related fluid work is covered elsewhere on the site: the hydrostatic pressure calculator handles the gravity side of the balance, the Reynolds number calculator and the viscosity converter deal with flow, and the capillary tube sizing calculator is a refrigeration restrictor tool that shares only the word capillary.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Counting the contact line once instead of twice — a plate wets on both faces and a ring wets inside and out. Missing that doubles the reported surface tension.
- Using the tube's nominal bore — capillary radius enters the rise relation linearly, so a radius quoted to one significant figure gives a γ good to one significant figure.
- Assuming a zero contact angle — it is close to true for clean water on clean glass and false for almost everything else. A sixty-degree angle halves the rise.
- Ignoring temperature — γ falls with temperature, for water by roughly 0.15 mN/m per degree, so a reading and a textbook value taken at different temperatures will never match.
- Applying the ring formula without its correction — the raw force overestimates γ because the ring lifts a column of liquid. Instruments correct for it; a hand calculation does not.
Related Free Tools From Arb Digital
Once you have a value of γ, the Young-Laplace calculator turns it into the pressure jump across a drop, a bubble or a meniscus, and the surface tension converter moves it between mN/m, dyn/cm and pounds per inch. The hydrostatic pressure calculator covers the column of liquid the tension is holding up. For flow problems use the Reynolds number calculator and the viscosity converter, and note that the capillary tube sizing calculator is a refrigeration tool rather than a surface-physics one. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the force per unit length acting along a liquid surface, arising because molecules at the surface have fewer neighbours pulling on them than molecules in the bulk. It is measured in newtons per metre, and in practice in millinewtons per metre.
Divide the force on the probe by the length of the three-phase contact line, and by the cosine of the contact angle if it is not zero. The contact line for a Wilhelmy plate is twice its width plus twice its thickness; for a Du Noüy ring it is twice the circumference.
Because the conversion factors cancel exactly. One dyne is ten micronewtons and one centimetre is a hundredth of a metre, so one dyn/cm equals one mN/m. Older literature quoted in dyn/cm can be read straight across.
It is the height a liquid climbs inside a narrow tube because the wall attracts it more strongly than the liquid attracts itself. The height is inversely proportional to the tube radius, so water in a 0.2 mm radius glass capillary rises roughly 74 mm.
It is the square root of surface tension divided by density times gravity, and it is the length scale at which surface tension and gravity balance. For water it is about 2.7 millimetres, which is why drops much bigger than that flatten out rather than staying spherical.
Almost always contamination. Surfactants concentrate at the surface, so quantities far too small to detect in the bulk lower the reading noticeably. Temperature is the second cause, since surface tension falls as temperature rises.
Yes. When the contact angle is greater than ninety degrees the liquid does not wet the wall, the cosine term goes negative, and the level inside the tube sits below the reservoir. Mercury in glass is the standard example.
No, that is the Young-Laplace relation and it has its own calculator on this site. This page is about obtaining a value of surface tension from a measurement, and about the capillary rise that follows from one.
This tool is provided for educational and laboratory study use. It applies the ideal contact-line and Jurin relations and does not apply Du Noüy ring corrections, meniscus-weight corrections or temperature adjustments, so results from real apparatus should be checked against the instrument's own documented correction procedure.
