In physical sciences, any measurable property of a system is called a quantity. The most structuring classification distinguishes between quantities that depend on the size of the system and those that do not. Understanding this distinction allows for the correct formulation of a thermodynamic balance, verification of the consistency of an equation, or interpretation of a phase diagram.
Extensive and intensive quantities: the test of merging two systems
The most direct method for classifying a quantity is based on a simple thought experiment. We consider two strictly identical systems, then we merge them into one.
Some properties double: mass, volume, amount of substance, internal energy, entropy. These are extensive quantities. Their value is proportional to the amount of substance present in the system.
Other properties remain unchanged: temperature, pressure, density, concentration. These are intensive quantities. They characterize the local state of the system, regardless of its size.
This merging test provides a reliable operational criterion. If the quantity adds up when two disjoint subsystems are merged, it is extensive. If it remains constant, it is intensive. A useful resource for exploring the distinctions between extensive and intensive quantities details these mechanisms with concrete cases in thermodynamics.
The role of the extensive/intensive ratio in thermodynamics
Thermodynamics constantly exploits the link between these two categories. Dividing an extensive quantity by another extensive quantity produces an intensive quantity. This is the mechanism for constructing mass and molar quantities.
Mass and molar quantities
Specific heat (thermal energy per unit mass) does not change when a smaller sample of the same material is taken. The ratio of two extensive quantities is always intensive. This principle generates most of the useful intensive variables in engineering:
- Density (mass divided by volume) characterizes a material regardless of the amount considered.
- Molar entropy (entropy divided by amount of substance) allows for comparison of substances on a per-unit basis.
- Specific energy (energy divided by mass) is used to evaluate the energy density of a fuel or battery.
This mechanism explains why thermodynamic data tables almost always express properties in mass or molar form: these intensive values are transferable from one system to another, without size-related corrections.
State variables and balances
An energy or material balance relies on extensive quantities because they add up. We can state that the total energy of a composite system equals the sum of the energies of each subsystem.
Intensive quantities are used to describe equilibrium conditions. Two systems in thermal contact reach equilibrium when their temperature (intensive) equalizes, not when their thermal energy (extensive) equalizes. Thermodynamic equilibrium is reflected in intensive quantities.
Concrete examples to identify each type of quantity
Correctly classifying a quantity sometimes requires careful reasoning. Some properties may seem intensive at first glance but turn out to be extensive, and vice versa.
Common extensive quantities
- Mass: doubling the amount of substance doubles the mass.
- Volume: merging two identical containers gives a total volume equal to the sum of the two.
- Entropy: an additive quantity, central to the second law of thermodynamics.
- Electric charge: the total charge of a system is the sum of the charges of its parts.
- Internal energy: an additive property that is involved in the first law.
Common intensive quantities
Temperature, pressure, and chemical potential do not vary with the size of the system. Mixing two liters of water at the same temperature does not yield water that is twice as hot.
Density illustrates the intensive nature well: a ten-kilogram block of steel has the same density as a one-kilogram block of the same steel. Density depends only on the nature of the material and the conditions of pressure and temperature.
Limits of classification at the nanoscale
The extensive/intensive distinction is based on the assumption that the system is large enough for surface effects to remain negligible compared to volume effects. This assumption weakens at the nanoscale.
Recent work on metallic nanowires shows that some quantities typically considered extensive (strain energy, mechanical resistance) exhibit nonlinear size effects. Doubling the amount of substance by changing the morphology of a nanowire does not double the mechanical response. This phenomenon, sometimes summarized by the phrase “smaller is stronger,” challenges the characteristic proportionality of extensive quantities for nanostructured systems.
This observation does not eliminate the classification but clarifies its domain of validity. At the macroscopic scale, the distinction remains perfectly operational. At scales where the surface-to-volume ratio becomes significant, the boundary between extensive and intensive requires a more nuanced analysis, taking into account the geometry and morphology of the studied objects.
The classification of physical quantities into extensive and intensive has structured thermodynamics since its foundations. It conditions how to write a balance, define equilibrium, and compare materials. Its robustness at the macroscopic scale does not prevent a critical perspective when the size of the system approaches the nanometer, where surface physics takes over.