Properties of liquids and solids

The surface of a liquid behaves like a “stretched skin”, while solids resist stretching and expand when heated. These properties explain why water rises in capillaries, how strong bridges are and why gaps are left between railway rails. They all have one cause — the forces between molecules. In a gas the molecules are far apart and hardly interact; in a liquid they are closely packed but can change places; in a solid they vibrate about fixed equilibrium positions. This topic links molecular kinetic theory with mechanics (Hooke’s law) and comes up often in exams.

§1. Surface tension

A molecule inside a liquid is pulled equally in all directions by its neighbours, so the resultant force is zero. A molecule at the surface is pulled only by neighbours beside and below it, so the resultant force points into the liquid. The surface therefore tends to shrink, taking the smallest possible area. For a given volume a sphere has the smallest surface, which is why drops are round and why a pond skater can walk on water without sinking.

Surface molecules have extra potential energy compared with those inside — surface energy. To increase the surface area by ΔS, work must be done:

The surface tension force acts along the surface and perpendicular to the boundary. For water σ ≈ 0.073 N/m, for soapy water ~0.04 N/m. Soap and other surfactants reduce σ, and so does a rise in temperature. In problems with a wire frame the film is two-sided, so l is doubled.

Worked example

What force does a soap film (σ = 0.04 N/m) exert on a 5 cm side of a frame?

1. The film has two surfaces: l = 2·0.05 = 0.1 m.
2. F = σ·l = 0.04·0.1 = 4·10⁻³ N.

Worked example

Water drops fall from a pipette with an inner diameter of 1 mm. What is the mass of one drop (g = 10 m/s²)?

1. At the moment of detachment the drop’s weight equals the surface tension force: mg = σ·πd.
2. m = σπd/g = 0.073·3.14·10⁻³/10 ≈ 2.3·10⁻⁵ kg = 23 mg.

§2. Wetting and capillarity

If liquid molecules are attracted to the molecules of a solid more strongly than to each other, the liquid spreads over the surface — it wets it; if less strongly, it gathers into drops — it does not wet it.

• A wetting liquid (water on glass) rises in a narrow tube (a capillary), with a concave meniscus.
• A non-wetting liquid (mercury on glass) falls, with a convex meniscus.

Why the liquid rises: the surface tension force σ·2πr along the edge of the meniscus pulls the column up until it is balanced by the column’s weight ρgπr²h. Hence (for complete wetting):

The narrower the tube, the higher the rise (h ∝ 1/r). Water rising in plants, towels and paper soaking up moisture, moisture rising through soil — these are all capillary effects. Loosening the soil breaks the capillaries and reduces evaporation.

Worked example

How high does water rise in a glass tube of radius 0.5 mm?

1. h = 2σ/(ρgr) = 2·0.073/(1000·10·0.5·10⁻³).
2. h = 0.146/5 ≈ 0.029 m ≈ 3 cm.

§3. Crystalline and amorphous solids

• In crystalline solids (table salt, metals, ice) the atoms form an ordered, repeating crystal lattice. They have a definite melting point and are often anisotropic — their properties (strength, thermal conductivity, optical properties) depend on direction. Mica, for example, splits easily into thin sheets in one direction.
• A single crystal is one continuous crystal; a polycrystal (most metals) consists of many small, randomly oriented crystals, so overall it is isotropic.
• Amorphous solids (glass, resin, plastics) have disordered atoms, like a “frozen liquid”; they have no definite melting point, soften gradually when heated and are isotropic.

The same substance can exist in both forms: quartz is a crystal, while fused quartz glass is amorphous.

§4. Deformation and Hooke’s law

An external force changes a body’s shape or size — this is deformation. Types: tension and compression, shear, bending, torsion. Deformation that disappears when the force is removed is elastic; deformation that remains is plastic (permanent).

Stress is the force per unit cross-sectional area; strain is the extension divided by the original length:

The Young modulus is a property of the material and does not depend on the body’s dimensions; for steel E ≈ 2·10¹¹ Pa. Writing Hooke’s law as F = kΔl, the stiffness of a rod is k = ES/l₀. As stress increases, first the limit of proportionality is reached, then the elastic limit, beyond which deformation becomes permanent; at the ultimate strength the body breaks. Structures are designed with a safety factor: the working stress is several times below the ultimate strength.

Worked example

A 100 N weight hangs from a steel wire 2 m long with a cross-section of 1 mm² (E = 2·10¹¹ Pa). What is the extension?

1. σ = F/S = 100/10⁻⁶ = 10⁸ Pa.
2. ε = σ/E = 10⁸/(2·10¹¹) = 5·10⁻⁴.
3. Δl = εl₀ = 5·10⁻⁴·2 = 10⁻³ m = 1 mm.

Worked example

What is the stiffness of this wire?

1. k = ES/l₀ = 2·10¹¹·10⁻⁶/2.
2. k = 10⁵ N/m — check: Δl = F/k = 100/10⁵ = 1 mm.

§5. Thermal expansion

When a body is heated, its atoms vibrate with larger amplitude and their average separation grows — the body expands.

α is the linear expansion coefficient (K⁻¹); for steel α ≈ 1.2·10⁻⁵ K⁻¹. That is why gaps are left between rails, bridges rest on sliding bearings and power lines sag in summer. A bimetallic strip (two different metals) bends when heated and is used in thermostats and circuit breakers.

Water is an exception: between 0 °C and 4 °C it contracts on heating. Its density is greatest at 4 °C, so in winter the water at the bottom of a lake stays at 4 °C while ice floats on top, and the lake does not freeze to the bottom.

Worked example

A steel rail 12.5 m long (α = 1.2·10⁻⁵ K⁻¹) warms by 50 K from winter to summer. What gap is needed?

1. Δl = αl₀Δt = 1.2·10⁻⁵·12.5·50.
2. Δl = 7.5·10⁻³ m = 7.5 mm.

Check yourself

1. What force does a soap film (σ = 0.04 N/m) exert on a frame side 5 cm long?

2. The radius of a capillary is halved. How does the height of rise change?

3. How does mercury behave in a glass capillary?

4. A wire’s cross-section is doubled with the same force. How does the stress change?

5. Is glass crystalline or amorphous?

6. How much does a 100 m steel bridge (α = 1.2·10⁻⁵ K⁻¹) lengthen when heated by 60 K?

7. A wire 1 m long with a cross-section of 2 mm² stretches by 1 mm under 400 N. What is its Young modulus?

8. At what temperature is water densest?

Common mistakes

• Forgetting that a soap film has two surfaces.
• Using the diameter instead of the radius in the capillary formula.
• Converting mm² to m² wrongly (1 mm² = 10⁻⁶ m²).
• Thinking all liquids rise in a capillary.
• Believing the Young modulus depends on a body’s length or cross-section: it depends only on the material.
• Taking the volume expansion coefficient equal to the linear one (β ≈ 3α).