Cambridge Syllabus Notes

Thermal Equilibrium

Learning Objective

Students will be able to:

State that regions of equal temperature are in thermal equilibrium.
Describe the direction of spontaneous heat flow.
Explain the zero‑law of thermodynamics and how it underpins the use of thermometers.
Convert reliably between the Kelvin and Celsius temperature scales (Kelvin is the SI base unit for temperature).

Key Concepts

Temperature (T) – a state variable that reflects the average kinetic energy of the particles in a system. Measured in kelvin (K) or degree Celsius (°C).
Heat (Q) – energy transferred *because* of a temperature difference; it is not a property possessed by a system.
Thermal contact – a condition that allows two or more regions to exchange heat.
Temperature gradient – the spatial rate of change of temperature, ∇T. When ∇T = 0 the system is in thermal equilibrium.
Zero‑law of thermodynamics – the logical foundation for defining temperature and for comparing temperatures of different systems.

Temperature Scales

Scale	Symbol	Reference point	Conversion to the other scale
Kelvin	K	Absolute zero (0 K = –273.15 °C)	°C = K – 273.15
Celsius	°C	Freezing point of water (0 °C = 273.15 K)	K = °C + 273.15

Kelvin is the SI base unit for temperature; all other temperature scales are defined relative to it.

Worked conversion example

Convert 25 °C to kelvin:

\[ K = 25\;°\text{C} + 273.15 = 298.15\;\text{K} \]

Convert 310 K to Celsius:

\[ °C = 310\;\text{K} - 273.15 = 36.85\;°\text{C} \]

Direction of Heat Flow

Heat always flows spontaneously from a region of higher temperature to a region of lower temperature.
The flow continues until the temperature difference (ΔT) disappears, i.e. ΔT = 0, at which point the temperature gradient is zero and no net heat transfer occurs.

Definition of Thermal Equilibrium

Two or more regions are in thermal equilibrium when they are in thermal contact and there is no net heat flow between them. This condition is satisfied when all the regions have the same temperature:

\[ T_1 = T_2 = \dots = T_n \qquad \Longleftrightarrow \qquad abla T = 0 \]

Zero‑Law of Thermodynamics

If region A is in thermal equilibrium with region B, and region B is in thermal equilibrium with region C, then region A is also in thermal equilibrium with region C. The zero‑law guarantees that temperature is a transitive property and therefore can be measured with a single thermometer.

Mathematical Expression

For two regions 1 and 2:

\[ T_1 = T_2 \;\; \Longleftrightarrow \;\; \text{no net heat flow (thermal equilibrium)} \]

If a temperature difference exists (ΔT = T₁ – T₂ ≠ 0), a heat flow Q occurs from the hotter to the cooler region.

Illustrative Example

A copper block (300 K) is placed on a wooden board that is also at 300 K.
The temperatures are already equal, so ∇T = 0.
Consequently, no net heat flows and the block and the board remain in thermal equilibrium.

Worked Example: Reaching Equilibrium

Two beakers contain water at 20 °C and 80 °C respectively. They are placed in contact through a thin metal rod.

Initially ΔT = 60 °C, so heat flows from the hot beaker to the cold beaker.
As heat leaves the hot beaker, its temperature falls; as heat enters the cold beaker, its temperature rises.
The process continues until both beakers reach the same temperature (≈ 50 °C if the masses and specific heats are equal). At that point ΔT = 0 and the system is in thermal equilibrium.

Common Misconceptions

“Equal temperature means no energy is present.” – Incorrect. Internal kinetic energy still exists; it is simply the same per particle in each region.
“Heat flow stops only when the temperature reaches zero.” – Incorrect. Heat flow stops as soon as the temperatures become equal, regardless of the absolute value.
“Heat is the same as temperature.” – Incorrect. Heat is energy in transit due to a temperature difference; temperature is a measure of the average kinetic energy of particles.

Check Your Understanding

Two cups of water, one at 20 °C and the other at 80 °C, are placed in contact. Describe qualitatively how their temperatures change until thermal equilibrium is reached.
Explain how the zero‑law justifies the use of a single thermometer to assign a temperature to any system.
Objects A, B and C have temperatures T_A = 250 K, T_B = 250 K and T_C = 250 K. State whether they are in thermal equilibrium and support your answer using the zero‑law.

Summary Table

Concept	Condition	Result
Thermal equilibrium	`T₁ = T₂ = … = Tₙ` (or ∇T = 0)	No net heat flow between the regions.
Zero‑law	If `T_A = T_B` and `T_B = T_C`	Then `T_A = T_C`; all three are mutually in equilibrium.
Direction of heat flow	`ΔT = T_{hot} – T_{cold} > 0`	Heat flows from hot to cold until ΔT = 0.
Temperature scales	Kelvin (K) – SI base unit; Celsius (°C) – relative scale	Conversions: °C = K – 273.15, K = °C + 273.15.

understand that regions of equal temperature are in thermal equilibrium