Surface Tension, Capillarity, Drops and Bubbles

Surface Tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is defined as the force acting per unit length perpendicular to the line in a планар direction, on the surface of a liquid. Alternatively, it can be defined as the energy required to increase the surface area of a liquid by a unit amount. This phenomenon arises from the cohesive forces between liquid molecules. At the surface, molecules are attracted to the molecules below and to the sides, but not to the molecules above, resulting in a net inward force. This inward pull causes the surface molecules to be packed more tightly, behaving like a stretched elastic membrane.

Consider a liquid surface. Molecules in the bulk of the liquid are attracted equally in all directions by their neighbors. However, molecules at the surface experience a net inward pull because there are no molecules above them to exert an outward pull. This imbalance of forces causes the surface to contract to the smallest possible area, which is why liquids tend to form spherical droplets.

The SI unit of surface tension is Newtons per meter (N/m). In terms of energy, it is Joules per square meter (J/m²). Surface tension is represented by the symbol 'γ' (gamma) or 'T'.

Mathematically, if 'F' is the force acting perpendicular to a line of length 'L' on the surface of a liquid, then the surface tension γ is given by:

γ = F / L

If 'W' is the work done to increase the surface area of a liquid by an amount 'ΔA', then the surface tension γ is given by:

γ = W / ΔA

This energy definition is often more useful. It represents the potential energy stored in the surface per unit area.

Factors Affecting Surface Tension

Surface tension is primarily affected by temperature and the presence of impurities. It is also influenced by the nature of the liquid and the medium in contact with the liquid surface.

• Temperature: As temperature increases, the kinetic energy of the molecules increases, weakening the cohesive forces between them. Consequently, surface tension decreases with an increase in temperature. At the critical temperature, the surface tension becomes zero.
• Impurities: Soluble impurities like salt increase the cohesive forces between liquid molecules, thereby increasing surface tension. Insoluble impurities like oil or soap, on the other hand, reduce surface tension by interfering with the cohesive forces.
• Nature of the Liquid: Different liquids have different cohesive forces. For instance, water has a high surface tension due to strong hydrogen bonding between its molecules. Liquids like alcohol or ether have lower surface tension.

Molecular Interpretation of Surface Tension

Surface tension can be understood at a molecular level. Consider a molecule 'A' deep inside the liquid. It is surrounded by other liquid molecules and experiences attractive forces from all sides, resulting in no net force. Now consider a molecule 'B' at the surface. It is surrounded by liquid molecules on its sides and below, but has fewer molecules above it (or it might be in contact with a gas). This leads to a net inward pull on molecule 'B'. These inward forces cause the surface molecules to be in a state of higher potential energy than the molecules in the bulk. To minimize this potential energy, the liquid tries to minimize its surface area. This tendency to minimize surface area is what we perceive as surface tension.

The surface of the liquid acts like a stretched membrane because the molecules at the surface are pulled inwards, creating a tension along the surface. This tension is equal to the force per unit length acting perpendicular to any line drawn on the surface.

Surface Energy

Surface energy is the excess energy possessed by the molecules at the surface of a liquid compared to the molecules in the interior. When a liquid surface is expanded, work must be done against the cohesive forces to bring molecules from the bulk to the surface. This work done is stored as potential energy in the surface molecules. Thus, surface energy is defined as the work done per unit increase in surface area.

Surface energy is numerically equal to the surface tension (γ). If we increase the surface area of a liquid by ΔA, the work done dW is given by:

dW = γ dA

The total surface energy is the integral of this work over the entire surface. For a liquid film of area A, the surface energy is γA.

Consider a rectangular frame of wire with one movable side of length 'L'. If this frame is dipped in a soap solution and lifted, a film of liquid is formed. Let the surface tension of the soap solution be γ. If the movable side is pulled outwards by a distance 'dx', the increase in the area of the film is 2L dx (since there are two surfaces, front and back). The work done to stretch the film is:

dW = Force × distance = (γ × 2L) × dx = γ × (2L dx)

Here, 2L is the total length of the free surface, and γ × 2L is the force required to pull the movable side.

The increase in surface area is dA = 2L dx. So, dW = γ dA, which confirms that surface energy per unit area is equal to surface tension.

Drops and Bubbles

The tendency of liquids to minimize their surface area due to surface tension leads to the formation of spherical shapes for drops and bubbles. A sphere has the smallest surface area for a given volume.

Formation of a Liquid Drop

Consider a small liquid drop of radius 'r'. The surface tension γ acts along the surface of the drop. Inside the drop, there is a pressure P_in, and outside the drop, there is atmospheric pressure P_out. The excess pressure inside the drop, ΔP = P_in - P_out, is responsible for balancing the inward pull of surface tension.

Imagine cutting the drop into two hemispheres. The force due to the excess pressure acting on the cross-sectional area (πr²) tends to push the hemispheres apart. This force is (P_in - P_out) × πr² = ΔP × πr².

The force due to surface tension acts along the circumference of the cut (2πr) and tends to pull the hemispheres together. This force is γ × 2πr.

For equilibrium, the outward force due to pressure must balance the inward force due to surface tension:

ΔP × πr² = γ × 2πr

Solving for ΔP, we get:

ΔP = 2γ / r

This equation shows that the excess pressure inside a liquid drop is inversely proportional to its radius. Smaller drops have a larger excess pressure.

Formation of a Bubble (e.g., Soap Bubble)

A bubble, like a soap bubble, has two surfaces: an inner surface and an outer surface. Both surfaces contribute to the surface tension effect.

Consider a soap bubble of radius 'r'. The pressure inside the bubble is P_in, and the atmospheric pressure outside is P_out. The excess pressure is ΔP = P_in - P_out.

Similar to the drop, the outward force due to excess pressure on the cross-sectional area is ΔP × πr².

However, now the surface tension force acts along the circumference on *both* the inner and outer surfaces of the bubble film. The total force due to surface tension pulling inwards is γ × (2πr) + γ × (2πr) = 2 × (2πrγ) = 4πrγ.

For equilibrium:

ΔP × πr² = 4πrγ

Solving for ΔP, we get:

ΔP = 4γ / r

Notice that the excess pressure in a bubble is twice that in a drop of the same radius. This is because the bubble has two surfaces (inner and outer) contributing to the tension.

If we consider a bubble formed by a liquid (like a steam bubble in boiling water) where there is only one surface, then the excess pressure inside would be 2γ/r, similar to a liquid drop. However, common soap bubbles have two surfaces.

Capillarity

Capillarity is the phenomenon by which a liquid in a narrow tube rises or falls. This rise or fall is due to the combined effect of surface tension and the adhesive forces between the liquid molecules and the walls of the tube.

When a narrow tube (called a capillary tube) is immersed in a liquid, the liquid surface inside the tube takes a curved shape (meniscus). The shape of the meniscus depends on the relative strengths of cohesive forces (between liquid molecules) and adhesive forces (between liquid molecules and the tube walls).

Adhesive and Cohesive Forces

Adhesive forces are the attractive forces between molecules of *different* substances. For example, the attraction between water molecules and glass molecules.

Cohesive forces are the attractive forces between molecules of the *same* substance. For example, the attraction between water molecules themselves.

Angle of Contact (θ)

The angle of contact is the angle between the tangent to the liquid surface at the point of contact with the solid surface and the solid surface itself. It is measured within the liquid. The angle of contact depends on the nature of the solid and the liquid.

• For liquids that wet the solid (e.g., water on glass): The adhesive forces are stronger than cohesive forces. The liquid surface tends to spread out on the solid. The meniscus is concave upwards, and the angle of contact is acute (θ < 90°). For pure water and clean glass, θ ≈ 0°.
• For liquids that do not wet the solid (e.g., mercury on glass): The cohesive forces are stronger than adhesive forces. The liquid tends to form a spherical shape and minimize contact with the solid. The meniscus is convex upwards, and the angle of contact is obtuse (θ > 90°). For mercury and glass, θ ≈ 140°.

Rise or Fall of Liquid in a Capillary Tube

Consider a capillary tube of radius 'r' immersed in a liquid. The surface tension γ acts along the line of contact between the liquid and the tube wall. The resultant force due to surface tension is tangential to the meniscus and makes an angle θ with the vertical (where θ is the angle of contact).

The vertical component of the surface tension force is γ cos θ, acting along the circumference of the contact line (2πr). So, the total upward force due to surface tension is F_up = (γ cos θ) × 2πr.

This upward force supports the weight of the liquid column that has risen in the tube. Let 'h' be the height of the liquid column and ρ be the density of the liquid. The volume of the liquid column is approximately πr²h (assuming a cylindrical shape for simplicity, ignoring the small volume in the meniscus). The mass of this column is m = ρ × (πr²h). The weight of this column is W = mg = (ρπr²h)g.

For equilibrium, the upward force must balance the weight of the liquid column:

2πrγ cos θ = ρπr²hg

Solving for the height 'h':

h = (2γ cos θ) / (ρrg)

This is the Jurin's Law for capillary rise.

Interpretation of the Formula

• h is directly proportional to γ: Liquids with higher surface tension will rise higher (if they wet the surface).
• h is directly proportional to cos θ: If the angle of contact is smaller (more wetting), the liquid rises higher. If θ = 0°, cos θ = 1, and h is maximum. If θ > 90° (non-wetting), cos θ is negative, and 'h' becomes negative, meaning the liquid falls or depresses in the tube.
• h is inversely proportional to ρ: Liquids with higher density will rise to a lesser height.
• h is inversely proportional to r: The narrower the capillary tube, the higher the liquid will rise.

Example: Water vs. Mercury

Water on clean glass has θ ≈ 0° (cos θ ≈ 1) and relatively high surface tension. Thus, water rises in a glass capillary tube.

Mercury on glass has θ ≈ 140° (cos θ ≈ -0.64) and a higher surface tension than water. Because cos θ is negative, mercury is depressed in a glass capillary tube, meaning it falls below the general liquid level.

The weight of the liquid column is more accurately given by considering the volume of the meniscus. However, for narrow tubes, the approximation πr²h is usually sufficient.

Capillary Depression

When the adhesive forces are weaker than the cohesive forces (θ > 90°), the liquid does not wet the solid. In this case, the surface tension pulls the liquid downwards, causing it to depress in the capillary tube. The formula for capillary depression is the same as capillary rise, but 'h' will be negative because cos θ is negative.

For mercury in a glass tube, the depression 'h' is given by:

h = -(2γ cos θ) / (ρrg)

The negative sign indicates depression.

Applications of Capillarity

Capillarity plays a crucial role in many natural and industrial processes:

• Absorption of water by blotting paper: The pores in the paper act as narrow capillary tubes.
• Movement of water from roots to leaves in plants: Capillary action in the xylem vessels helps transport water against gravity.
• Wicking action of cloth: It allows fabrics to absorb and spread liquids.
• Formation of dew: Water vapor condenses on surfaces, and capillary action helps it spread.
• Lubrication: Oil rises into the narrow gaps between moving parts of machinery due to capillary action.
• Operation of a fountain pen: Ink is drawn into the pen's nib through capillary action.

Summary of Key Points

This section covered the fundamental concepts of surface tension, surface energy, and capillarity. Here's a quick recap:

• Surface Tension (γ): Force per unit length or energy per unit area of a liquid surface. Arises from cohesive forces. Decreases with temperature.
• Surface Energy: Work done to increase surface area by a unit amount. Numerically equal to surface tension.
• Liquid Drop: Excess pressure ΔP = 2γ/r.
• Soap Bubble: Excess pressure ΔP = 4γ/r (due to two surfaces).
• Angle of Contact (θ): Angle between liquid surface tangent and solid surface. Acute for wetting, obtuse for non-wetting.
• Capillary Rise (h): h = (2γ cos θ) / (ρrg).
• Capillary Depression: Occurs when θ > 90° (h is negative).
• Applications: Plant water transport, blotting paper, fountain pens, etc.