The height to which a liquid will rise in a capillary tube is determined by several factors as shown in the following equation:

$$h\,=\frac{2Tcosθ}{rρg}$$In this equation, *h* is the height of the liquid inside the capillary tube relative to the surface of the liquid outside the tube, *T* is the surface tension of the liquid, *θ* is the contact angle between the liquid and the tube, *r* is the radius of the tube, *ρ* is the density of the liquid, and *g* is the acceleration due to gravity, 9.8 m/s^{2}. When the tube is made of a material to which the liquid molecules are strongly attracted, they will spread out completely on the surface, which corresponds to a contact angle of 0°. This is the situation for water rising in a glass tube.

Capillary Rise

At 25 °C, how high will water rise in a glass capillary tube with an inner diameter of 0.25 mm?

For water, *T* = 71.99 mN/m and *ρ* = 1.0 g/cm^{3}.

Solution

The liquid will rise to a height *h* given by: $h\,=\frac{2Tcosθ}{rρg}$

The Newton is defined as a kg m/s^{2}, and so the provided surface tension is equivalent to 0.07199 kg/s^{2}. The provided density must be converted into units that will cancel appropriately: *ρ* = 1000 kg/m^{3}. The diameter of the tube in meters is 0.00025 m, so the radius is 0.000125 m. For a glass tube immersed in water, the contact angle is *θ* = 0°, so cos *θ* = 1. Finally, acceleration due to gravity on the earth is *g* = 9.8 m/s^{2}. Substituting these values into the equation, and cancelling units, we have:

$$h\,=\frac{2(0.07199\,kg/s^2)}{(0.000125\,m)(1000\,\frac{kg}{m^3})(9.8\frac{m}{s^2})}=\,0.12\,m\,=\,12\,cm$$

Check Your Learning

Water rises in a glass capillary tube to a height of 8.4 cm. What is the diameter of the capillary tube? Answer:

diameter = 0.36 mm