Unit 9: Heat and Temperature

Introduction

Heat and temperature are everyday experiences, yet their microscopic origins lie in the motion of molecules. Understanding how energy is stored, transferred, and measured forms the cornerstone of thermodynamics. This unit builds a clear picture of thermal energy as the total kinetic energy of particles, distinguishes it from heat as energy in transit, and defines temperature as a measure of average molecular motion.

1. Molecular Concept of Thermal Energy, Heat, and Temperature

Thermal Energy

Thermal energy (U) is the sum of the kinetic energies of all molecules in a substance. For an ideal monatomic gas, each molecule contributes ½ m v² to the kinetic energy. Using the equipartition theorem, the average kinetic energy per degree of freedom is ½ k_B T, where k_B is Boltzmann’s constant and T is absolute temperature. Hence, for a gas with f degrees of freedom per molecule and N molecules:

U = (f/2) N k_B T

In solids and liquids, potential energy from intermolecular forces also contributes, but the kinetic part dominates temperature‑related changes.

Heat

Heat (Q) is energy transferred between two bodies solely because of a temperature difference. It is not a property of a system but a process quantity. The sign convention: Q > 0 when heat enters the system, Q < 0 when it leaves.

For a substance of mass m and specific heat capacity c, the heat required to change its temperature by ΔT is:

Q = m c ΔT

Specific heat capacity depends on the material and its phase; water’s high c ≈ 4.18 J g⁻¹ K⁻¹ makes it an excellent coolant.

Temperature

Temperature is a macroscopic measure of the average kinetic energy of the molecules. For an ideal gas:

⟨E_k⟩ = (3/2) k_B T

Thus, doubling the absolute temperature doubles the average kinetic energy per molecule. Temperature determines the direction of spontaneous heat flow: heat always flows from the body at higher temperature to the one at lower temperature until thermal equilibrium is reached.

Cause and Direction of Heat Flow

• Cause: A temperature difference (ΔT ≠ 0) creates a driving force for heat transfer.
• Direction: Heat flows spontaneously from higher temperature to lower temperature. This is a statement of the second law of thermodynamics for heat conduction.

2. Thermal Equilibrium and the Zeroth Law of Thermodynamics

Thermal Equilibrium

Two bodies are in thermal equilibrium when they are in thermal contact and there is no net exchange of heat between them. At this point, their temperatures are equal.

Zeroth Law of Thermodynamics

The Zeroth Law provides the foundation for temperature measurement:

If body A is in thermal equilibrium with body B, and body B is in thermal equilibrium with body C, then body A is in thermal equilibrium with body C.

This transitive property allows us to assign a numerical value (temperature) to a body by comparing it with a reference thermometer. When the thermometer reads the same value for two bodies, they are in thermal equilibrium and thus share the same temperature.

The Zeroth Law justifies the use of thermometers as reliable temperature sensors and underpins the construction of temperature scales.

3. Mercury Thermometer

Principle of Operation

A mercury thermometer operates on the principle of thermal equilibrium: the mercury column reaches the same temperature as the surrounding medium, and its volume changes predictably with temperature.

Why Mercury?

• Mercury is a liquid metal with a high coefficient of thermal expansion, providing a visible, linear change in volume over a wide temperature range.
• It does not wet glass, ensuring a smooth meniscus.
• It remains liquid from –39 °C to 357 °C, covering most everyday temperature measurements.

Fixed Points and Scale Division

To create a reproducible scale, two fixed points are defined:

The interval between these points is divided into 100 equal parts, giving the Celsius scale:

1 °C = (Steam Point – Ice Point) / 100 = (100 °C – 0 °C) / 100 = 0.01 °C per division

Thus, each division on the thermometer corresponds to a 1 °C change.

Construction Details

1. A narrow, uniform glass capillary tube is sealed at one end and attached to a bulb containing mercury.
2. The bulb and tube are evacuated of air, then filled with mercury.
3. When placed in contact with a body, heat flows until thermal equilibrium is reached; mercury expands or contracts, moving the meniscus.
4. The length of the mercury column is read against a calibrated scale.

Limitations and Corrections

• Non‑linearity at extremes: Near the freezing and boiling points, the expansion deviates slightly from linearity; corrections are applied using higher‑order terms.
• Environmental pressure: Changes in atmospheric pressure affect the boiling point of water used for calibration; corrections are made using the Clausius‑Clapeyron relation.
• Toxicity: Mercury is hazardous; modern alternatives (alcohol, digital sensors) are preferred in many settings.

Applications and Problem‑Solving Examples

Calorimetry

When a hot metal block is immersed in water, heat flows from the block to the water until equilibrium. Using Q_lost = Q_gained:

m_metal c_metal (T_initial,metal – T_final) = m_water c_water (T_final – T_initial,water)

Solving for the final temperature yields the equilibrium temperature of the mixture.

Thermal Expansion in Engineering

The linear expansion of a solid is given by:

ΔL = α L₀ ΔT

where α is the coefficient of linear expansion, L₀ the original length, and ΔT the temperature change. This formula is essential for designing bridges, railways, and pipelines to accommodate temperature‑induced length changes.