Intensive Variables

Intensive variables: characterize state without depending on size. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Intensive variables are thermodynamic magnitudes that describe the state of a portion of matter without directly depending on how much matter has been chosen. If a uniform sample is split into two halves, each half keeps the same temperature $T$, pressure $P$, and density $\rho$ as the original sample. If two identical copies are joined, those variables are not added. This idea separates physical state from total system amount.

The difference from an extensive variable is operational. An extensive quantity counts inventory; an intensive variable characterizes local condition, equilibrium, or concentration. Mass $m$ and volume $V$ grow when a sample is duplicated, but $\rho$ can remain constant because it is a normalized ratio. Intensive variables therefore allow samples of different size to be compared without automatically favoring the larger sample.

🟢 Essential level

An intensive variable answers the question: what state does the substance have here? It does not answer how much system there is. Two cups of water at the same $T$ do not have twice the temperature when placed together. Two vessels containing the same gas in the same state do not have twice the $P$ when considered as one set. What changes when they are joined is total amount; what remains is local condition.

The simplest test is to imagine two identical copies of the system. If the magnitude remains equal when the copies are joined, it behaves as intensive. The $T$, $P$, and $\rho$ pass this test when the state is uniform. By contrast, $m$ and $V$ do not pass it, because a second copy contributes more matter and occupies more space.

Intensive variables are also signals of imbalance. If two bodies in thermal contact have different $T$, there is a tendency to transfer energy as heat. If two connected fluid regions have different $P$, mechanical motion may appear. Equality of compatible intensive variables indicates that the corresponding macroscopic tendency has disappeared.

It is important to remember that intensive does not mean small, universal, or always independent of position. A property can be intensive and vary inside a nonuniform system. In an atmosphere, $P$ changes with height, but it is still local pressure, not total amount. In a solid heated at one end, $T$ changes from point to point, but each temperature reading is still intensive.

🔵 Formal level

Basic formulas

The formal criterion for intensivity is written as a scaling test:

Scaling criterion for intensivity

The notation says that multiplying the system size by $\lambda$ does not change $X_{\int}$ if the same local state is kept. This is a degree-zero homogeneity test: the variable does not increase with scale. Physically, $X_{\int}$ describes state rather than inventory.

At equilibrium, many conditions are written as equality of intensive variables:

Intensive equality at equilibrium

The equality does not mean that extensive quantities in the two parts are equal. Two bodies with different masses can reach the same $T$. Two regions with different volume can share the same $P$. What becomes equal is the variable controlling an exchange tendency, not the amount stored.

An important case is density:

Density as an intensive variable

Although it is computed from $m$ and $V$, $\rho$ is intensive when both extensive quantities scale together. If mass and volume of a uniform sample are both doubled, the ratio remains unchanged. This example shows that the algebraic origin of a magnitude is not enough for classification; its response to scaling matters.

Real student questions

If density is computed with mass, why is it not extensive? Because $\rho$ compares $m$ with $V$. When both increase in the same proportion, the ratio keeps the same value. Mass alone grows; density describes concentration.

Does the temperature of a large object contain more heat? No. The $T$ indicates thermal state, not total energy. A large object at a given temperature may store more energy than a small one at the same temperature, but the intensive reading is the same.

Can pressure depend on volume? It may depend on volume in a specific process, but as a state variable it is not added when identical copies are duplicated. An ideal gas may change its $P$ if $V$ changes while other conditions are controlled, but that does not make pressure extensive.

Final synthesis

An intensive variable remains unchanged when a uniform sample is scaled. The $T$, $P$, and $\rho$ describe state condition; $m$ and $V$ help construct or weight those readings, but they do not replace them. The practical rule is clear: use intensive variables to compare states and extensive variables to count amounts.