= Intensive and extensive thermodynamic quantities
An <extensive quantity> measures how much of a <thermodynamic system> there is; an <intensive quantity> describes its state independently of that amount. The clean comparison is replication: take independent identical systems at the same <thermodynamic equilibrium>, with the same substance, composition and <thermodynamic phase>. Combining their accounting boundaries multiplies <internal energy> $U$, <entropy> $S$, <volume (thermodynamics)> $V$, <mass> $m$ and <amount of substance> $n$, while leaving <temperature> $T$, <pressure> $p$ and <chemical potential> $\mu$ unchanged.<Statistical Physics by David Tong>[Statistical Physics, chapter 1 The Fundamentals of Statistical Mechanics, §1.4.3 Extensive and Intensive Quantities]{ref}
For $\lambda$ copies the ledger is
$$
(U,S,V,m,n)\longmapsto\lambda(U,S,V,m,n),
\qquad (T,p,\mu)\longmapsto(T,p,\mu).
$$
An integer $\lambda$ counts actual copies; a positive real $\lambda$ describes the corresponding bulk scaling law. Copies can remain separated: replication need not involve removing a partition, changing a container or performing any <work>. Distinguish $n$, the <amount of substance> measured in <moles>, from $N=nN_A$, the dimensionless count of specified entities, where $N_A$ is the <Avogadro constant> in $\mathrm{mol}^{-1}$. For example, $2\,\mathrm{mol}$ contains $1.204428152\times10^{24}$ specified entities; tripling it gives $6\,\mathrm{mol}$ and $3.613284456\times10^{24}$ entities. This uses the exact <Avogadro constant> $N_A=6.02214076\times10^{23}\,\mathrm{mol}^{-1}$, which does not triple. In the differential $dU=T\,dS-p\,dV+\mu\,dn$, $\mu$ is a molar <chemical potential> in $\mathrm{J\,mol^{-1}}$; the coefficient of $dN$ instead has units of <energy> per entity.
\Image[/thermodynamic-replication-ledger.png]
{title=Replication multiplies the extensive ledger while preserving the equilibrium state}
As a direct check, the <ideal gas law> $pV=nRT$ gives
$$
p'=\frac{(\lambda n)RT}{\lambda V}=p.
$$
For a monatomic <ideal gas> with only translational <degrees of freedom> active, <internal energy of an ideal gas> gives $U=3nRT/2$, so $U'=\lambda U$ at unchanged <temperature>. Here $R$ is the <molar gas constant>. The scaling distinction lets a small sample supply an <equation of state> for a much larger amount: material state and inventory become separate questions.
Replication does not mean multiplying every linear dimension by $\lambda$: a geometrically similar container then has $\lambda^3$ times the <volume>. Nor does it mean combining samples at different states. Equal <temperature> establishes <thermal equilibrium> but does not imply equal <pressure>, composition or <mass density>. <Density of a composite system> makes this distinction concrete.
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