understand that the scale of thermodynamic temperature does not depend on the property of any particular substance

Published by Patrick Mutisya · 14 days ago

Temperature Scales – Cambridge A‑Level Physics 9702

Temperature Scales

Learning Objective

Understand that the scale of thermodynamic temperature is defined independently of the properties of any particular substance.

1. What is Thermodynamic Temperature?

The thermodynamic temperature, denoted \$T\$, is a fundamental quantity that characterises the average kinetic energy of the microscopic particles in a system. It is defined through the second law of thermodynamics and does not rely on any specific material.

In statistical mechanics the relationship is expressed as

\$E{\text{avg}} = \frac{3}{2}k{\mathrm B}T\$

where \$E{\text{avg}}\$ is the average translational kinetic energy per molecule of an ideal gas and \$k{\mathrm B}=1.380\,649\times10^{-23}\,\text{J K}^{-1}\$ is the Boltzmann constant.

2. Defining the Kelvin Scale

  • Zero point: Absolute zero (\$0\ \text{K}\$) – the temperature at which the internal energy of a perfect crystal would be at its minimum, and all molecular motion ceases.

  • Unit size: The kelvin is defined such that the triple point of water occurs at exactly \$273.16\ \text{K}\$.

  • Formal definition (2022 SI): The kelvin is defined by fixing the numerical value of the Boltzmann constant to \$k_{\mathrm B}=1.380\,649\times10^{-23}\ \text{J K}^{-1}\$.

3. Historical Temperature Scales

  1. Celsius (°C) – based on the freezing point (0 °C) and boiling point (100 °C) of water at 1 atm.

  2. Fahrenheit (°F) – based on the freezing point (32 °F) and boiling point (212 °F) of water at 1 atm, with the interval between them divided into 180 equal parts.

  3. Rankine (°R) – an absolute scale using the Fahrenheit degree size; 0 °R corresponds to absolute zero.

4. Conversions Between Scales

The conversion formulas are derived from aligning the zero points and degree sizes of each scale with the Kelvin scale.

From / ToKelvin (K)Celsius (°C)Fahrenheit (°F)Rankine (°R)
K → °C\$T_{\text{K}}\$\$T_{\text{K}} - 273.15\$\$\displaystyle \frac{9}{5}(T_{\text{K}} - 273.15) + 32\$\$\displaystyle \frac{9}{5}T_{\text{K}}\$
°C → K\$T_{\text{C}} + 273.15\$\$T_{\text{C}}\$\$\displaystyle \frac{9}{5}T_{\text{C}} + 32\$\$\displaystyle \frac{9}{5}(T_{\text{C}} + 273.15)\$
°F → K\$\displaystyle \frac{5}{9}(T_{\text{F}} - 32) + 273.15\$\$\displaystyle \frac{5}{9}(T_{\text{F}} - 32)\$\$T_{\text{F}}\$\$T_{\text{F}} + 459.67\$
°R → K\$\displaystyle \frac{5}{9}T_{\text{R}}\$\$\displaystyle \frac{5}{9}T_{\text{R}} - 273.15\$\$\displaystyle \frac{5}{9}T_{\text{R}} - 459.67\$\$T_{\text{R}}\$

5. Why Thermodynamic Temperature Is Independent of Any Substance

Early temperature scales (e.g., Celsius, Fahrenheit) were anchored to the physical properties of water. However, the thermodynamic temperature scale is built on universal physical constants:

  • The Boltzmann constant \$k_{\mathrm B}\$ links temperature to energy, a property of all matter.

  • Absolute zero is defined as the point where the entropy of a perfect crystal approaches zero (Third Law of Thermodynamics), a condition that applies to any pure substance.

  • The triple point of water is used only to set the numerical value of the kelvin; the definition itself does not rely on water’s behaviour.

Consequently, the Kelvin scale provides a consistent measure of temperature that can be applied to gases, solids, plasmas, and even exotic states of matter without reference to a particular material’s phase changes.

6. Summary

  1. The thermodynamic temperature scale (kelvin) is defined by fundamental constants, not by any specific substance.

  2. Absolute zero (\$0\ \text{K}\$) is the universal lower bound of temperature.

  3. Historical scales (Celsius, Fahrenheit, Rankine) are related to the kelvin through simple linear transformations.

  4. Understanding the independence of the Kelvin scale is essential for accurate scientific measurement and for comparing temperatures across different physical systems.

Suggested diagram: A schematic showing the relationship between the four temperature scales, with arrows indicating conversion formulas and the location of absolute zero, the triple point of water, and the freezing/boiling points of water.