Intensive quantity

Dividing two <extensive quantities> with the same replication factor produces an <intensive quantity>:
$$
\frac{\lambda X}{\lambda Y}=\frac XY,\qquad Y>0.
$$
Division by <volume (thermodynamics)> gives a density, such as <mass density> $\rho=m/V$ or <energy density> $U/V$. Division by <mass> gives a specific quantity, such as <specific internal energy> $u=U/m$ or <specific volume> $v=V/m$. Division by <amount of substance> gives a molar quantity, such as <molar volume> $V_m=V/n$. Their units are respectively $\mathrm{kg\,m^{-3}}$, $\mathrm{J\,m^{-3}}$, $\mathrm{J\,kg^{-1}}$, $\mathrm{m^3\,kg^{-1}}$ and $\mathrm{m^3\,mol^{-1}}$. The denominator must be stated: “per particle”, “per <mole>” and “per unit <mass>” are different normalizations.

For a hypothetical uniform parcel with <mass> $2\,\mathrm{kg}$, <volume (thermodynamics)> $0.002\,\mathrm{m^3}$ and <internal energy> $40\,\mathrm{kJ}$ relative to a specified reference, $u=20\,\mathrm{kJ\,kg^{-1}}$ and $\rho=1000\,\mathrm{kg\,m^{-3}}$. Three independent copies have $6\,\mathrm{kg}$, $0.006\,\mathrm{m^3}$ and $120\,\mathrm{kJ}$; both ratios remain unchanged. This permits comparison of materials without confusing the amount tested with the state tested.

If $X$ and $Y$ are additive but the parcels have different $r_i=X_i/Y_i$, their composite ratio is a <weighted arithmetic mean>:
$$
r_{\rm total}=\frac{\sum_i X_i}{\sum_i Y_i}
=\sum_i\frac{Y_i}{\sum_jY_j}r_i.
$$
The denominator supplies the weights: <specific internal energy> uses <mass> weights, while <mass density> uses <volume (thermodynamics)> weights. This identity is accounting for the specified subsystems; predicting a new state after mixing also needs the relevant <equation of state>.

\Image[/extensive-specific-quantities-weighted-density.png]
{title=Size changes extensive inventory; composite density weights each parcel by its volume}


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 Incoming links: Normalizing extensive quantities