Extensive Variables

Extensive variables: grow with system size. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Extensive variables are thermodynamic magnitudes that depend on how much system is considered. If two equal copies of a sample are joined, total mass $m$, volume $V$, internal energy $U$, and entropy $S$ double. This dependence on size is not a flaw: it is what allows balances of matter, energy, and entropy in composite systems.

The difference from intensive variables is operational. An intensive variable characterizes the local or mean state of the substance; an extensive variable counts total amount. Temperature and pressure are not added when two equal bodies are joined, but $m$ and $V$ are. Recognizing an extensive variable prevents scale errors when comparing different samples.

🟢 Essential level

The essential idea is the duplication test. If you duplicate the system without changing substance or intensive state, an extensive magnitude doubles. Two liters of water have twice the volume of one liter; two identical blocks have twice the mass of one block; two equal copies of a gas have twice the total internal energy.

The generic symbol $X$ represents any extensive amount. The unit may be kg, m3, J, or J/K; the question is whether the value grows when the amount of system grows. The scaling factor $\lambda$ summarizes how many equivalent copies are joined.

An everyday way to see this is to think of inventories. If you buy two identical packages, you have twice as much product, but not twice the "kind of product". An extensive counts how many packages, how much mass, or how much total energy there is. By itself it does not describe the local quality of the state.

[!NOTE] Extensive does not mean "large"; it means proportional to system size.

🔵 Formal level

Basic formulas

The scaling criterion is:

Extensive scaling criterion

Subsystem additivity is:

Additivity of extensive magnitudes

Dividing compatible extensives can produce a normalized property:

Density as a ratio of extensives

A common additional normalization is the specific magnitude:

Specific magnitude per mass

These four expressions form a logical progression: first defining what scaling means, then how subsystems add, and finally how to normalize that total to compare samples of different size.

Real student questions

Q: Is internal energy always extensive?

A: In the usual macroscopic thermodynamic model, yes. It adds when compatible subsystems are joined, except for interaction or surface corrections.

Q: Density depends on mass; why is it not extensive?

A: Because duplicating a homogeneous sample also duplicates volume. The ratio $\rho$ remains constant.

Q: Can entropies be added?

A: Yes, if they are total entropies of compatible subsystems. Specific entropies are not added as totals.

Final synthesis

An extensive variable grows with system size and adds when compatible subsystems are joined. $m$, $V$, $U$, and $S$ are physical inventories. Dividing by mass, volume, or amount produces normalized properties such as $\rho$ or $x$, useful for comparing systems of different size.

The practical rule is twofold: use extensives for balances and normalized properties for comparisons. This prevents confusing a larger system with a more intense state.