15.1 Temperature, Heat, and Thermal Expansion

Temperature is a measure of how fast particles are moving. The hotter something is, the faster its particles move on average. This connection between heat and particle motion explains why materials expand when heated and contract when cooled.

Heat always flows from hot to cold until balance is reached. This process happens through direct contact, fluid movement, or electromagnetic waves. Understanding heat flow and thermal expansion helps you predict how materials behave as temperatures change.

Temperature and Kinetic Energy

Temperature and particle kinetic energy

Temperature quantifies how hot or cold something is, but at the molecular level, it's really measuring the average kinetic energy of particles in a substance. Kinetic energy is the energy of motion, and it's directly proportional to temperature.

• Higher temperature means faster particle motion and higher average kinetic energy (think of boiling water, where molecules are moving rapidly).
• Lower temperature means slower particle motion and lower average kinetic energy (think of an ice cube, where molecules mostly vibrate in place).

Particles move differently depending on the phase of matter:

• Gases: Molecules fly around freely in translational motion (air molecules bouncing off walls).
• Solids: Atoms vibrate back and forth around fixed positions in a crystal lattice.
• Liquids: Molecules have a mix of translational and vibrational motion, sliding past each other while still jostling around.

Three temperature scales come up in physics:

• Celsius (℃): Based on water's freezing (0℃) and boiling (100℃) points.
• Fahrenheit (℉): Used mainly in the U.S.; water freezes at 32℉ and boils at 212℉.
• Kelvin (K): The absolute scale used in physics. 0 K is absolute zero, the theoretical lowest temperature where particle motion reaches a minimum. To convert: $K = °C + 273.15$.

Heat Transfer and Thermal Expansion

Thermal equilibrium and heat transfer

Thermal equilibrium is the state where two objects reach the same temperature and no net heat flows between them. A cup of coffee left on a table eventually cools to room temperature because heat transfers from the hotter coffee to the cooler surroundings until both are equal.

Heat always flows from higher temperature to lower temperature. The greater the temperature difference, the faster the transfer occurs. There are three methods of heat transfer:

1. Conduction: Heat moves through direct contact between materials. A metal pot on a hot stove heats up because energy transfers from the burner through the metal.
2. Convection: Heat moves through the bulk motion of a fluid (liquid or gas). Hot air near a heater rises and cooler air sinks, creating a circulation pattern.
3. Radiation: Heat transfers through electromagnetic waves with no physical contact needed. This is how the Sun warms the Earth across the vacuum of space.

Thermal equilibrium is also the basis for how thermometers work. A thermometer placed under your tongue reaches thermal equilibrium with your body, and the reading reflects that shared temperature.

Thermal expansion in materials

When a material heats up, its atoms vibrate more vigorously and push slightly farther apart. This causes the material to grow in size, a phenomenon called thermal expansion. There are three types:

• Linear expansion: A change in length. Railway tracks have small gaps between sections to allow for lengthening on hot days.
• Area expansion: A change in surface area. A metal sheet will grow in both length and width as it heats.
• Volume expansion: A change in overall volume. A balloon left in the sun expands as the air inside heats up.

Different materials expand at different rates, characterized by their coefficient of thermal expansion:

• Metals generally expand more than ceramics or glass (aluminum expands roughly twice as much as glass per degree).
• Liquids typically expand more than solids, which is why mercury or alcohol rises in a thermometer.
• Gases expand the most dramatically with temperature increases, which is the principle behind hot air balloons.

Real-world applications rely on controlling or exploiting thermal expansion:

• Expansion joints in bridges and buildings provide gaps so structures can expand and contract without cracking.
• Bimetallic strips in thermostats are made of two metals bonded together. Since each metal expands at a different rate, the strip bends as temperature changes, triggering a switch.
• Precision instruments like telescopes need temperature compensation to maintain accurate measurements.

Calculations for thermal expansion types

Each type of expansion has its own formula. They all share the same structure: the change in size equals a coefficient times the original size times the change in temperature.

Linear expansion:

$\Delta L = \alpha L_0 \Delta T$

• $\Delta L$ = change in length
• $\alpha$ = coefficient of linear expansion (units: 1/℃ or 1/K)
• $L_0$ = original length
• $\Delta T$ = change in temperature

Area expansion:

$\Delta A = 2\alpha A_0 \Delta T$

• $\Delta A$ = change in area
• $A_0$ = original area
• The factor of 2 appears because area depends on two linear dimensions.

Volume expansion:

• For solids: $\Delta V = 3\alpha V_0 \Delta T$
• For liquids: $\Delta V = \beta V_0 \Delta T$
• $\Delta V$ = change in volume
• $V_0$ = original volume
• $\beta$ = coefficient of volume expansion (for solids, $\beta \approx 3\alpha$)

Steps for solving thermal expansion problems:

1. Identify the expansion type (linear, area, or volume).
2. Select the correct equation.
3. List your known values and convert units if needed. Make sure temperatures are in consistent units (a change of 1℃ equals a change of 1 K, so either works in $\Delta T$).
4. Plug in values and solve for the unknown.
5. Check that your answer makes physical sense. Expansion values are typically very small for solids.