Premed · Premed · Physics 1
Lecture 20: Temperature and Thermal Expansion
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define temperature and explain its microscopic interpretation
- Convert between Celsius, Fahrenheit, and Kelvin temperature scales
- Explain the concept of thermal equilibrium and the zeroth law of thermodynamics
- Calculate linear, area, and volume thermal expansion
- Describe the anomalous expansion of water and its consequences
- Apply thermal expansion concepts to practical engineering and biological situations
Lecture Content
I. Temperature and Thermal Equilibrium
Temperature is a measure of the average kinetic energy of the particles in a substance. Higher temperature corresponds to faster average molecular motion. At the microscopic level, for an ideal gas, (1/2)m v_rms^2 = (3/2)k_B T, where k_B = 1.38 x 10^-23 J/K is the Boltzmann constant.
Thermal equilibrium is the state reached when two objects in contact arrive at the same temperature. Heat flows from the hotter object to the cooler one until equilibrium is reached, at which point there is no net heat transfer. The zeroth law of thermodynamics formalizes an essential logical foundation: if object A is in thermal equilibrium with object C, and object B is in thermal equilibrium with object C, then A and B are in thermal equilibrium with each other. This law justifies the use of thermometers, since the thermometer (C) mediates the comparison, and it establishes temperature as a well-defined, transitive property.
II. Temperature Scales
The Celsius (C) scale sets 0 C at the freezing point of water and 100 C at the boiling point, both at 1 atm. The Fahrenheit (F) scale assigns 32 F to freezing and 212 F to boiling, with conversions T_F = (9/5)T_C + 32 and T_C = (5/9)(T_F - 32).
The Kelvin (K) scale is the SI unit of temperature, related to Celsius by T_K = T_C + 273.15. Absolute zero, 0 K = -273.15 C, is the lowest possible temperature. At absolute zero, molecular motion reaches its minimum, though quantum mechanical zero-point energy persists. All gas law and thermodynamic equations require temperatures in Kelvin. It is worth noting that temperature differences are the same in Celsius and Kelvin: Delta T_K = Delta T_C.
III. Thermometers
A thermometer exploits a physical property that varies predictably with temperature. Liquid-in-glass thermometers (mercury, alcohol) use the thermal expansion of a liquid. Thermocouples measure the voltage generated at the junction of two different metals. Resistance thermometers track the change in electrical resistance of a metal with temperature. Infrared thermometers measure thermal radiation without contact. The constant-volume gas thermometer, in which the pressure of a gas at constant volume is proportional to T in Kelvin, was historically used to define the Kelvin scale experimentally.
IV. Linear Thermal Expansion
Most materials expand when heated and contract when cooled. For a solid rod, bar, or wire, the linear expansion is given by Delta L = alpha L_0 Delta T, or equivalently L = L_0 (1 + alpha Delta T), where alpha is the coefficient of linear expansion (units: 1/K or 1/C), L_0 is the original length, and Delta T is the change in temperature.
Typical values of alpha include aluminum at 23 x 10^-6 /C, steel at 12 x 10^-6 /C, glass at 9 x 10^-6 /C, and Invar (a nickel-iron alloy designed for minimal expansion) at 1.2 x 10^-6 /C. Although expansion is small for typical temperature changes, it has important engineering consequences. Bridges require expansion joints, railroad tracks need gaps between rails or anchored continuous welded rail, hot water loosens metal lids on glass jars because metal expands more than glass, and bimetallic strips (two metals with different alpha values bonded together) bend when heated, forming the basis of many thermostats.
<image>Panel A: A metal rod of original length L_0 at temperature T_1. When heated to T_2, the rod expands to length L = L_0 + Delta L. The expansion Delta L = alpha L_0 Delta T is exaggerated for visibility. Panel B: A bimetallic strip at room temperature (straight) and when heated (curved), with the higher-alpha metal on the outside of the curve. The strip is labeled with the two metals and their alpha values. An application showing a bimetallic strip in a thermostat is sketched beside it.</image>
V. Area and Volume Expansion
Area expansion follows from linear expansion. Since area scales as the square of length, Delta A = 2 alpha A_0 Delta T, giving A = A_0 (1 + 2 alpha Delta T). Volume expansion for solids uses the coefficient of volume expansion beta, with Delta V = beta V_0 Delta T and V = V_0 (1 + beta Delta T). For solids, beta is approximately equal to 3 alpha.
For liquids, which have no fixed shape, only volume expansion is relevant. Typical values of beta include water at 207 x 10^-6 /C (at 20 C), mercury at 182 x 10^-6 /C, and ethanol at 1100 x 10^-6 /C. An important and sometimes counterintuitive result is that a hole in a material expands as if it were filled with the same material. A ring heated uniformly gets larger, with both the inner and outer diameters increasing.
VI. Anomalous Expansion of Water
Most liquids contract as they cool, but water behaves anomalously near 4 C. Water reaches its maximum density at 4 C (rho = 1000 kg/m^3). Below 4 C, water expands as it cools further, and ice at 0 C is about 9% less dense than liquid water, so it floats.
This anomaly has profound consequences. Lakes freeze from the top down, with the ice layer insulating the water below and allowing aquatic life to survive through winter. Pipes can burst when water freezes and expands. The underlying cause is the hydrogen-bonding structure of water molecules, which form an open crystalline lattice in ice that occupies more volume than the disordered liquid phase.
VII. Thermal Stress
When a material is constrained and cannot expand or contract freely, temperature changes create internal thermal stress given by stress = Y alpha Delta T, where Y is the Young's modulus of the material. The corresponding force per unit area is F/A = Y alpha Delta T. This stress can be enormous, sufficient to buckle railroad tracks or crack concrete.
Engineering solutions to thermal stress include expansion joints in bridges, roads, and buildings; loops in pipelines; and gaps between concrete slabs on highways. Thermal stress is also an important consideration in dental fillings and prosthetic joints, where materials with different thermal expansion coefficients meet.
<image>Panel A: A straight railroad track on a hot day buckling into a curve because the rails expanded but had insufficient room. Panel B: A bridge expansion joint shown in cross-section — interlocking teeth allow the bridge deck to slide as it expands or contracts with temperature. Panel C: A graph of water density vs. temperature from 0 to 10 degrees C, showing the maximum density at 4 degrees C and the anomalous decrease in density below 4 degrees C. An annotation explains why lakes freeze from the top.</image>
