What is the relation between surface tension and capillary rise?

A liquid of density $\rho$ and surface tension $\sigma$ rises in a capillary of inner radius $r$ to a height \begin{align} h=\frac{2\sigma\cos\theta}{\rho g r} \end{align} where $\theta$ is the contact angle made by the liquid meniscus with the capillary’s surface.

The liquid rises due to the forces of adhesion, cohesion, and surface tension. If adhesive force (liquid-capillary) is more than the cohesive force (liquid-liquid) then liquid rises as in case of water rise in a glass capillary. In this case, the contact angle is less than 90 deg and the meniscus is concave. If adhesive force is less than the cohesive force then liquid depresses as in case of mercury in a glass capillary. In this case, the contact angle is greater than 90 deg and the meniscus is convex.

The formula for capillary rise can be derived by balancing forces on the liquid column. The weight of the liquid ($\pi r^2 h \rho g$) is balanced by the upward force due to surface tension ($2\pi r \sigma \cos\theta$). This formula can also be derived using pressure balance.

The capillary rise experiment is used to measure the surface tension of a liquid.

Solved Problems on Capillary Rise

Problem from IIT JEE 2014

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross-section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$, the value of $h$ will be (where $g$ is acceleration due to gravity),

1. $\frac{2S}{b\rho g}\cos(\theta-\alpha)$
2. $\frac{2S}{b\rho g}\cos(\theta+\alpha)$
3. $\frac{2S}{b\rho g}\cos(\theta-\alpha/2)$
4. $\frac{2S}{b\rho g}\cos(\theta+\alpha/2)$

Solution: Let $R$ be the radius of the meniscus formed with a contact angle $\theta$ (see figure). By geometry, this radius makes an angle $\theta+\frac{\alpha}{2}$ with the horizontal and, \begin{align} \label{oxb:eqn:1} \cos\left(\theta+\tfrac{\alpha}{2}\right)={b}/{R}. \end{align}

Let $P_0$ be the atmospheric pressure and $P_1$ be the pressure just below the meniscus. Excess pressure on the concave side of meniscus of radius $R$ is, \begin{align} \label{oxb:eqn:2} P_0-P_1={2S}/{R}. \end{align} The hydrostatic pressure gives, \begin{align} \label{oxb:eqn:3} P_0-P_1=h\rho g. \end{align} Eliminate $(P_0-P)$ from second and third equations and substitute $R$ from first equation to get, \begin{align} h=\frac{2S}{\rho g R}=\frac{2S}{b\rho g}\cos\left(\theta+\tfrac{\alpha}{2}\right).\nonumber \end{align}

Problem from IIT JEE 2018

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

1. For a given material of the capillary tube, $h$ decreases with increase in $r$.
2. For a given material of the capillary tube, $h$ is independent of $\sigma$.
3. If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases.
4. $h$ is proportional to contact angle $\theta$.

Solution: Consider the portion of water that rises in the capillary tube. In equilibrium, the upward force due to surface tension, $F=2\pi r\sigma\cos\theta$, is balanced by the weight of the water, $W=\pi r^2 h \rho g$, i.e., $2\pi r\sigma\cos\theta=\pi r^2 h \rho g$ which gives \begin{align} \label{bcc:eqn:1} h=\frac{2 \sigma\cos\theta}{r\rho g}. \end{align} From this equation, $h$ decreases with increase in $r$, it depends on $\sigma$ and it is proportional to $\cos\theta$. The contact angle $\theta$ depends on the material of the capillary tube.

In a lift going up with a constant acceleration $a$, the force due to surface tension remains same but weight of the water increases i.e., $W_\text{lift}=\pi r^2 h \rho (g+a)$. Thus, the height of the water rise in a capillary becomes \begin{align} h=\frac{2 \sigma\cos\theta}{r\rho (g+a)}=\frac{2 \sigma\cos\theta}{r\rho g_\text{eff}},\nonumber \end{align} where $g_\text{eff}$ is the effective $g$ in the lift. Hence, $h$ decrease if the experiment is performed in a lift going up with a constant acceleration. Can you guess value of $h$ if the experiment is performed in a lift falling freely?

Questions on Capillary Rise

Question 1: In an experiment to measure the surface tension of water using capillary rise method, a student uses a capillary whose length is shorter than the height of capillary rise (obtained theoretically for this capillary). Which of the following statement is correct?

Question 2: In an experiment to measure the surface tension of a liquid using capillary rise method, a student kept the capillary inclined instead of the vertical. The vertical height of liquid rise in tilted capillary?

Question 3: The capillary rise experiment was carried out in a space station under weightlessness conditions. The water will

Question 4: In capillary rise experiment, the work done by the force of surface tension is almost equal to the rise in gravitational potential energy of the liquid

Question 5: The meniscus surface in a fine capillary may be considered to be semispherical. In this case, the weight of the liquid above the lowest point of the meniscus is $\pi r^3\rho g/3$. Taking this into account, the formula for surface tension is

Question 6: Why is the meniscus of mercury convex?

Question 7: Water rises to a height of 3 cm in capillary of radius 0.48 mm. If angle of contact is zero then surface tension of the water is

Question 8: The inner radius of a barometer tube in 2.5 mm. The error introduced in the observed reading due to surface tension is (density of mercury is 13.6 g/cc, contact angle of mercury in glass is 135 deg)

Question 9: The height of liquid rise in a capillary tube depends on the size of the tube but it is independent of the tube material.

Question 10: Two capillary tubes of diameter 6 mm and 3 mm are joined together to form a U-tube open at both ends. If the U-tube is filld with water then water levels on the two limbs of the tube are different.

Related Topics

1. Surface Enery and Surface Tension

References and External Links

When you pour a glass of water, or fill a car with gasoline, you observe that water and gasoline flow freely. But when you pour syrup on pancakes or add oil to a car engine, you note that syrup and motor oil do not flow as readily. The viscosity of a liquid is a measure of its resistance to flow. Water, gasoline, and other liquids that flow freely have a low viscosity. Honey, syrup, motor oil, and other liquids that do not flow freely, like those shown in Figure \(\PageIndex{1}\), have higher viscosities. We can measure viscosity by measuring the rate at which a metal ball falls through a liquid (the ball falls more slowly through a more viscous liquid) or by measuring the rate at which a liquid flows through a narrow tube (more viscous liquids flow more slowly).

Figure \(\PageIndex{1}\): (a) Honey and (b) motor oil are examples of liquids with high viscosities; they flow slowly. (credit a: modification of work by Scott Bauer; credit b: modification of work by David Nagy)

The IMFs between the molecules of a liquid, the size and shape of the molecules, and the temperature determine how easily a liquid flows. As Table \(\PageIndex{1}\) shows, the more structurally complex are the molecules in a liquid and the stronger the IMFs between them, the more difficult it is for them to move past each other and the greater is the viscosity of the liquid. As the temperature increases, the molecules move more rapidly and their kinetic energies are better able to overcome the forces that hold them together; thus, the viscosity of the liquid decreases.

The various IMFs between identical molecules of a substance are examples of cohesive forces. The molecules within a liquid are surrounded by other molecules and are attracted equally in all directions by the cohesive forces within the liquid. However, the molecules on the surface of a liquid are attracted only by about one-half as many molecules. Because of the unbalanced molecular attractions on the surface molecules, liquids contract to form a shape that minimizes the number of molecules on the surface—that is, the shape with the minimum surface area. A small drop of liquid tends to assume a spherical shape, as shown in Figure \(\PageIndex{2}\), because in a sphere, the ratio of surface area to volume is at a minimum. Larger drops are more greatly affected by gravity, air resistance, surface interactions, and so on, and as a result, are less spherical.

Figure \(\PageIndex{2}\): Attractive forces result in a spherical water drop that minimizes surface area; cohesive forces hold the sphere together; adhesive forces keep the drop attached to the web. (credit photo: modification of work by “OliBac”/Flickr)

Surface tension is defined as the energy required to increase the surface area of a liquid, or the force required to increase the length of a liquid surface by a given amount. This property results from the cohesive forces between molecules at the surface of a liquid, and it causes the surface of a liquid to behave like a stretched rubber membrane. Surface tensions of several liquids are presented in Table \(\PageIndex{2}\).

Among common liquids, water exhibits a distinctly high surface tension due to strong hydrogen bonding between its molecules. As a result of this high surface tension, the surface of water represents a relatively “tough skin” that can withstand considerable force without breaking. A steel needle carefully placed on water will float. Some insects, like the one shown in Figure \(\PageIndex{3}\), even though they are denser than water, move on its surface because they are supported by the surface tension.

Figure \(\PageIndex{3}\): Surface tension (right) prevents this insect, a “water strider,” from sinking into the water.

The IMFs of attraction between two different molecules are called adhesive forces. Consider what happens when water comes into contact with some surface. If the adhesive forces between water molecules and the molecules of the surface are weak compared to the cohesive forces between the water molecules, the water does not “wet” the surface. For example, water does not wet waxed surfaces or many plastics such as polyethylene. Water forms drops on these surfaces because the cohesive forces within the drops are greater than the adhesive forces between the water and the plastic. Water spreads out on glass because the adhesive force between water and glass is greater than the cohesive forces within the water. When water is confined in a glass tube, its meniscus (surface) has a concave shape because the water wets the glass and creeps up the side of the tube. On the other hand, the cohesive forces between mercury atoms are much greater than the adhesive forces between mercury and glass. Mercury therefore does not wet glass, and it forms a convex meniscus when confined in a tube because the cohesive forces within the mercury tend to draw it into a drop (Figure \(\PageIndex{4}\)).

Figure \(\PageIndex{4}\): Differences in the relative strengths of cohesive and adhesive forces result in different meniscus shapes for mercury (left) and water (right) in glass tubes. (credit: Mark Ott)

If you place one end of a paper towel in spilled wine, as shown in Figure \(\PageIndex{5}\), the liquid wicks up the paper towel. A similar process occurs in a cloth towel when you use it to dry off after a shower. These are examples of capillary action—when a liquid flows within a porous material due to the attraction of the liquid molecules to the surface of the material and to other liquid molecules. The adhesive forces between the liquid and the porous material, combined with the cohesive forces within the liquid, may be strong enough to move the liquid upward against gravity.

Figure \(\PageIndex{5}\): Wine wicks up a paper towel (left) because of the strong attractions of water (and ethanol) molecules to the −OH groups on the towel’s cellulose fibers and the strong attractions of water molecules to other water (and ethanol) molecules (right). (credit photo: modification of work by Mark Blaser)

Towels soak up liquids like water because the fibers of a towel are made of molecules that are attracted to water molecules. Most cloth towels are made of cotton, and paper towels are generally made from paper pulp. Both consist of long molecules of cellulose that contain many −OH groups. Water molecules are attracted to these −OH groups and form hydrogen bonds with them, which draws the H2O molecules up the cellulose molecules. The water molecules are also attracted to each other, so large amounts of water are drawn up the cellulose fibers.

Capillary action can also occur when one end of a small diameter tube is immersed in a liquid, as illustrated in Figure \(\PageIndex{6}\). If the liquid molecules are strongly attracted to the tube molecules, the liquid creeps up the inside of the tube until the weight of the liquid and the adhesive forces are in balance. The smaller the diameter of the tube is, the higher the liquid climbs. It is partly by capillary action occurring in plant cells called xylem that water and dissolved nutrients are brought from the soil up through the roots and into a plant. Capillary action is the basis for thin layer chromatography, a laboratory technique commonly used to separate small quantities of mixtures. You depend on a constant supply of tears to keep your eyes lubricated and on capillary action to pump tear fluid away.

Figure \(\PageIndex{6}\): Depending upon the relative strengths of adhesive and cohesive forces, a liquid may rise (such as water) or fall (such as mercury) in a glass capillary tube. The extent of the rise (or fall) is directly proportional to the surface tension of the liquid and inversely proportional to the density of the liquid and the radius of the tube.

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

\[h=\dfrac{2T\cosθ}{rρg} \label{10.2.1}\]

where

• 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/s2.

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. We will not concern ourselves with calculating capillary height in this course.