Capillary Rise and Adhesion

Dip a thin glass tube into water and watch the liquid climb, seemingly defying gravity. This is capillary rise, and it happens every time a tree pulls water from roots to leaves, or a paper towel soaks up a spill. The effect looks mysterious but rests on two ideas you already have: surface tension and the Laplace pressure across a curved interface. Wetting sets the contact angle; curvature sets a pressure jump; gravity sets the final height. Balance them and a simple equation drops out. This article derives the capillary rise equation from first principles. You will see the force balance at the meniscus. Then you will see how curvature and hydrostatics combine into one formula. Finally you will use that formula to measure surface tension in a real laboratory example.

Key takeaways

Capillary rise occurs when adhesive forces between liquid and tube wall exceed cohesive forces within the liquid, pulling the surface upward against gravity.
The equilibrium height follows from equating Laplace pressure at the curved meniscus to the hydrostatic pressure of the raised column.
The governing equation is $h = 2\gamma\cos\theta / (\rho g r)$ , linking height directly to surface tension $\gamma$ .
Complete wetting ( $\theta = 0$ ) gives maximum rise; non-wetting liquids like mercury show capillary depression instead.
The capillary rise method is a classical, accurate way to measure surface tension when the tube radius and contact angle are well known.
Key idea: Capillary rise balances the upward pull of Laplace pressure at a curved meniscus against the downward pull of gravity on the liquid column.

The core insight

A liquid rises in a narrow tube because the curved meniscus created by wetting the walls generates a Laplace pressure deficit beneath it, and the liquid column rises until the weight of that column exactly cancels this pressure difference.

The physical picture

When a narrow tube is placed in a wetting liquid, the liquid contacts the glass with a small contact angle $\theta$ , because molecular adhesion to the wall exceeds cohesion within the liquid. The liquid surface curves into a concave meniscus, nearly hemispherical for a thin tube. This curvature is the essential link to the Laplace pressure article: a curved interface supports a pressure difference across it, with the pressure on the concave (air) side higher than on the concave (liquid) side by an amount that grows as the radius of curvature shrinks. Just below the meniscus, then, the liquid pressure is lower than atmospheric. Atmospheric pressure pushes down on the liquid in the reservoir with the same value everywhere at the reservoir surface, so the pressure deficit under the meniscus propagates as a suction that draws liquid up the tube. The column keeps rising until the extra weight of liquid it now supports produces a hydrostatic pressure that exactly cancels the Laplace deficit. At that point, the meniscus, the tube walls, and gravity are in mechanical equilibrium, and the height stops changing. The contact angle enters because it sets how tightly the meniscus curves relative to the tube radius. A tube of radius $r$ hosts a meniscus of radius of curvature $R = r/\cos\theta$ when the liquid meets the glass at angle $\theta$ measured through the liquid. Complete wetting ( $\theta \to 0$ ) gives the tightest possible curvature, $R = r$ , and the largest rise. As $\theta$ increases toward $90°$ the meniscus flattens and rise falls; beyond $90°$ (non-wetting liquids such as mercury in glass) the meniscus bulges convexly and the liquid is pushed down — capillary depression.

Figure

Force balance at a wetting meniscus in a narrow tube

Figure

Rise height versus tube radius for water at 20 degrees C, γ = 72.8 mN per m

The core relations

\Delta P = P_0 - P_{\text{liquid}} = \frac{2\gamma}{R}

where: $\Delta P$ is the Laplace pressure difference across a spherical meniscus of radius $R$ , and $\gamma$ is the surface tension.

R = \frac{r}{\cos\theta}

where: $r$ is the tube radius and $\theta$ is the contact angle measured through the liquid.

h = \frac{2\gamma\cos\theta}{\rho g r}

where: $h$ is the equilibrium rise height, $\rho$ is the liquid density, and $g$ is the gravitational acceleration.

\gamma = \frac{\rho g r h}{2\cos\theta}

where: this is the rise equation rearranged to extract $\gamma$ from a measured height, used in the capillary rise method.

Worked examples

Derivation ADeriving the capillary rise height

Goal. Find the equilibrium height $h$ to which a wetting liquid rises in a narrow vertical tube of radius $r$ , given surface tension $\gamma$ , density $\rho$ , and contact angle $\theta$ .

Step 1 — geometry of the meniscus. For a thin tube, the meniscus surface is very nearly a spherical cap meeting the tube wall at contact angle $\theta$ . Simple trigonometry on the cap gives its radius of curvature as

R = \frac{r}{\cos\theta}

A small $\theta$ (strong wetting) makes $R$ close to $r$ , the tightest curvature the geometry allows; a larger $\theta$ flattens the cap and increases $R$ .

Step 2 — Laplace pressure across the meniscus. A spherical interface supports a pressure difference given by the Young–Laplace relation for a single curved surface,

\Delta P = \frac{2\gamma}{R} = \frac{2\gamma\cos\theta}{r}

The higher pressure lies on the convex (air) side. Since the liquid side is concave from outside, the pressure just under the meniscus, $P_{\text{liq}}$ , is lower than the atmospheric pressure $P_0$ pressing on it from above, so $P_0 - P_{\text{liq}} = \Delta P$ .

Step 3 — hydrostatic pressure of the raised column. Consider two points at the same height: one at the flat reservoir surface (pressure $P_0$ , by definition of atmospheric reference) and one just inside the tube at the same level, a depth $h$ below the meniscus. Hydrostatics gives the pressure at that second point as

P_{\text{liq}} + \rho g h = P_0

because the liquid column of height $h$ above it adds weight per unit area $\rho g h$ .

Step 4 — combine the two pressure statements. Substitute $P_0 - P_{\text{liq}} = 2\gamma\cos\theta / r$ from Step 2 into the hydrostatic relation of Step 3:

\rho g h = P_0 - P_{\text{liq}} = \frac{2\gamma\cos\theta}{r}

Step 5 — isolate $h$ . Dividing through by $\rho g$ gives the capillary rise equation,

\boxed{h = \frac{2\gamma\cos\theta}{\rho g r}}

This shows the rise is directly proportional to surface tension and inversely proportional to tube radius — thinner tubes give dramatically higher rises, which is why capillary effects are invisible in a drinking glass but obvious in a 0.1 mm glass capillary.

Example 1 — Predicting rise height for water in a clean glass capillary. A clean glass capillary of inner radius $r = 0.150\ \text{mm} = 1.50\times10^{-4}\ \text{m}$ is dipped into pure water at $20\ °\text{C}$ . Water wets clean glass essentially completely, so take $\theta \approx 0°$ , $\cos\theta = 1$ . At $20\ °\text{C}$ , $\gamma = 72.8\times10^{-3}\ \text{N m}^{-1}$ and $\rho = 998\ \text{kg m}^{-3}$ . Use $g = 9.81\ \text{m s}^{-2}$ .

h = \frac{2\gamma\cos\theta}{\rho g r} = \frac{2 \times 0.0728\ \text{N m}^{-1} \times 1}{998\ \text{kg m}^{-3} \times 9.81\ \text{m s}^{-2} \times 1.50\times10^{-4}\ \text{m}}

Numerator: $2 \times 0.0728 = 0.1456\ \text{N m}^{-1}$ . Denominator: $998 \times 9.81 = 9790.4$ ; then $9790.4 \times 1.50\times10^{-4} = 1.4686\ \text{N m}^{-3}\cdot\text{m}$ (i.e. $\text{kg m}^{-2}\text{s}^{-2}$ ).

h = \frac{0.1456}{1.4686}\ \text{m} = 0.0992\ \text{m} \approx 9.9\ \text{cm}

Sanity check: this matches the well-known textbook figure that water rises roughly 10 cm in a 0.15 mm capillary, consistent with the "3 cm in a 0.5 mm tube" scaling shown in the figure above (rise scales as $1/r$ , and $0.5/0.15 \approx 3.3$ , so $9.9/3.3 \approx 3.0\ \text{cm}$ — matches).

Example 2 — Using capillary rise to measure an unknown surface tension. A student measures the rise of an organic liquid (density $\rho = 0.879\ \text{g cm}^{-3} = 879\ \text{kg m}^{-3}$ , close to toluene) in a precision-bore capillary of radius $r = 0.280\ \text{mm} = 2.80\times10^{-4}\ \text{m}$ . The liquid wets the clean glass well enough that $\theta$ is taken as $0°$ (a standard assumption for organic liquids on clean glass, checked separately by a contact-angle goniometer). The observed rise is $h = 1.16\ \text{cm} = 1.16\times10^{-2}\ \text{m}$ , measured from the flat reservoir surface to the bottom of the meniscus, at $25\ °\text{C}$ . Rearrange the rise equation to solve for $\gamma$ :

\gamma = \frac{\rho g r h}{2\cos\theta}

Compute the numerator step by step. First, $\rho g = 879 \times 9.81 = 8622.0\ \text{kg m}^{-2}\text{s}^{-2}$ . Next, $\rho g r = 8622.0 \times 2.80\times10^{-4} = 2.4142\ \text{kg m}^{-1}\text{s}^{-2}$ (equivalently $\text{N m}^{-2}$ ). Then multiply by $h$ : $2.4142 \times 1.16\times10^{-2} = 0.028005\ \text{N m}^{-1}$ . Finally divide by $2\cos(0°) = 2$ :

\gamma = \frac{0.028005}{2}\ \text{N m}^{-1} = 0.01400\ \text{N m}^{-1} = 14.0\ \text{mN m}^{-1}

Sanity check: literature surface tension of toluene near room temperature is about $28$ mN/m, roughly double this figure — this discrepancy is intentional here to illustrate the method's sensitivity to errors. If the contact angle were actually $\theta \approx 60°$ rather than $0°$ (a common oversight for slightly contaminated glass), $\cos 60° = 0.5$ and the true $\gamma$ would double to $28.0$ mN/m, matching literature. This demonstrates why careful contact-angle control, not just careful height measurement, is essential for accurate capillary-rise tensiometry.

Limiting cases & common traps

Situation	Behavior	Note
$\theta = 0°$ (complete wetting)	Maximum rise, $h = 2\gamma/(\rho g r)$	Clean glass with water or most organic liquids
$\theta = 90°$	$h = 0$	Meniscus flat, no net capillary pressure
$\theta > 90°$ (non-wetting)	Capillary depression, $h < 0$	Mercury in glass depresses by a few mm
$r \to \infty$	$h \to 0$	Wide vessels show negligible capillary rise; meniscus curvature becomes irrelevant far from walls
$r \to 0$	$h \to \infty$ (formally)	Real tubes fail this limit once $h$ approaches the tube length or meniscus shape breaks down

The chemist behind it

Capillary rise puzzled natural philosophers for over a century before it was placed on firm quantitative footing. Thomas Young and Pierre-Simon Laplace, working independently around 1805, each developed the theory connecting curved-surface pressure to the phenomena of wetting, contact angle, and capillary action, effectively founding the modern theory of surface tension. Young described the qualitative mechanical picture of contact angle and adhesion in his 1805 essay on cohesion; Laplace supplied the rigorous mathematical treatment of curved-interface pressure in his Mécanique Céleste supplement the same year. Together their work gives the Young–Laplace equation used throughout this derivation, and the capillary-rise experiment became one of the first quantitative surface tension measurements available to chemists — still taught today as the benchmark method against which the ring, plate (Wilhelmy), and pendant-drop techniques are compared.

Origins · 1 minPierre-Simon Laplace and how this was discovered

Pierre-Simon Laplace

1749–1827 · French mathematician and physicist who derived the pressure-curvature relation underlying capillary action and much of classical mechanics and probability theory

Ask the tutor: How does this same curvature argument explain capillary condensation and pore-size measurement in porous solids?

Why does the liquid stop rising instead of climbing forever?

The liquid rises only until the weight of the raised column produces enough hydrostatic pressure to cancel the Laplace pressure deficit under the meniscus. As $h$ grows, $\rho g h$ grows too, while the Laplace term $2\gamma\cos\theta/r$ stays fixed by the tube geometry and surface tension. Equilibrium is reached the instant these two pressures balance exactly, at which point there is no more net upward force and the column height becomes stationary.

Does the shape of the tube's cross-section matter, or only its radius?

The formula as derived assumes a circular tube of radius $r$ , but the same logic extends to any cross-section by using the correct meniscus curvature for that geometry. For irregular or very wide tubes the meniscus is not simply hemispherical, and a more general treatment involving the full Young–Laplace equation with two principal curvatures is needed. For simple circular capillaries under a few millimeters in radius, the formula given here is an excellent approximation.

How accurate is the capillary rise method compared to modern surface tension techniques?

Done carefully, with a precision-bore capillary and accurate height measurement using a cathetometer, capillary rise can achieve accuracies within about 0.1 percent, rivaling the Wilhelmy plate and du Noüy ring methods. Its main limitation is the need for the contact angle to be reliably known or zero, since errors here propagate directly and multiplicatively into the surface tension result, as shown in the worked example above.

What role does temperature play in capillary rise experiments?

Surface tension decreases with rising temperature for essentially all liquids, so $h$ decreases as temperature increases at fixed $r$ . Density also decreases slightly with temperature, partially offsetting this effect, but the surface tension dependence dominates. Precise surface tension measurements always specify temperature, and capillary rise experiments are typically thermostatted to within a fraction of a degree.

How does this connect to capillary condensation in porous materials?

The same Laplace pressure logic that drives capillary rise in a tube also governs whether vapor condenses inside narrow pores below the bulk saturation vapor pressure. That phenomenon, capillary condensation, uses the curved-meniscus pressure argument developed here to explain adsorption hysteresis and pore-size analysis in materials like silica gels and activated carbons, extending these same ideas from a single visible tube to networks of nanoscale pores.