Why Are Intensive Properties Not Additive?

Intensive properties, such as temperature, pressure, and density, describe a system’s state without depending on how much material it contains. Extensive properties, such as mass, volume, and internal energy, measure totals and normally add when you combine separate subsystems. The distinction explains why adding two temperatures or two densities does not give the temperature or density of the combined system.

The scaling test

Imagine duplicating a system in the same state. The combined system has twice the mass and, for ordinary short-range systems, twice the volume and internal energy. But it has the same temperature, pressure, and density as either copy.

In thermodynamics, this is often expressed as a scaling rule. If a system is enlarged by a factor of $\lambda$ while kept in the same state, an extensive property $X$ scales with its size, whereas an intensive property $x$ stays fixed:

$$X(\lambda S)=\lambda X(S), \qquad x(\lambda S)=x(S).$$

If an intensive property were additive, combining two identical copies would make its value $2x$. But scaling says that the value remains $x$. The proposed sum would describe neither the original state nor the combined state. For a nonzero intensive property, additivity is therefore the wrong rule.

This argument concerns the property of the combined system, not whether you can perform arithmetic on two measured values. You can add numbers representing temperatures, but their sum usually has no thermodynamic meaning as a temperature.

What happens when systems have different values?

A combined intensive value can be calculated, but the calculation generally uses extensive quantities and a physical model. For temperature, an insulated mixture reaches a final equilibrium temperature determined by energy conservation—not by adding the initial temperatures.

For example, put 2 kg of water at 20 °C in contact with 1 kg of water at 80 °C in an insulated container. Assume both samples have the same constant specific heat and that no water evaporates or freezes. Energy conservation gives

$$m_1cT_1+m_2cT_2=(m_1+m_2)cT_f.$$

Since the specific heat $c$ is the same in both samples, it cancels:

$$T_f=\frac{(2)(20)+(1)(80)}{2+1}=40\ ^\circ\mathrm{C}.$$

The result is a mass-weighted average, not $20+80=100$ °C. If the samples have different heat capacities or undergo a phase change, the energy balance must include those differences as well.

Density is a ratio, not a total

Density is mass divided by volume: $\rho=m/V$. For two materials whose volumes are additive, the density of the mixture is

$$\rho_{\text{mix}}=\frac{m_1+m_2}{V_1+V_2} =\frac{\rho_1V_1+\rho_2V_2}{V_1+V_2}.$$

That is a volume-weighted average, not $\rho_1+\rho_2$. If mixing changes the total volume—for example, because the liquids contract—the actual final volume must be used instead.

The same general idea applies to many intensive quantities: they are ratios or state variables, while the totals used to determine a combined value are extensive. Pressure, for instance, is force per area; it is not obtained by adding the pressures of two containers. The final pressure depends on how the systems are joined and on the resulting state.

When the simple rules need care

Additivity is a useful thermodynamic idealization for subsystems with negligible interactions across their boundaries. It can need correction when surface effects are important, when interactions are long-range, or when mixing changes volume or energy. Likewise, two systems with different initial temperatures or pressures do not necessarily retain those values after they are joined; they may exchange heat or matter until they reach equilibrium.

So “intensive properties are not additive” does not mean they cannot be combined or calculated. It means that their combined value is not the sum of their separate values. Find that value from the final state and the relevant conservation laws.