Normalizing extensive quantities

Dividing two extensive quantities with the same replication factor produces an intensive quantity:Division by volume 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 $\mathrm{kg}{\mathrm{m}}^{-3}$, $\mathrm{J}{\mathrm{m}}^{-3}$, $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$ and ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. 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 $0.002{\mathrm{m}}^3$ and internal energy $40\mathrm{kJ}$ relative to a specified reference, $u=20\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$ and $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$. 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:The denominator supplies the weights: specific internal energy uses mass weights, while mass density uses volume weights. This identity is accounting for the specified subsystems; predicting a new state after mixing also needs the relevant equation of state.

Consider two hypothetical parcels at the same temperature $300\mathrm{K}$, kept separated with negligible partition mass and excluded container volume. Parcel A has mass $2\mathrm{kg}$ and volume $0.002{\mathrm{m}}^3$; parcel B has mass $3\mathrm{kg}$ and volume $0.006{\mathrm{m}}^3$. They may be different materials. Their individual mass densities and the composite mass density areThe composite mass density is total material mass divided by total occupied volume; it need not be the local mass density anywhere. Equal temperature does not make these parcels replicas. In particular, the unweighted average $750\mathrm{kg}{\mathrm{m}}^{-3}$ is wrong. The volume fractions are $1/4$ and $3/4$, givingWith mass fractions $2/5$ and $3/5$, the equivalent rule is a weighted harmonic mean:This follows directly by writing $V_i=m_i/{\rho }_i$ and dividing total volume by total mass; it is also the mass-weighted average of specific volumes. An unweighted average of mass densities is valid when the parcel volume values are equal, not merely when their temperatures are equal.

If a partition is removed and different substances mix, the final volume need not equal the sum of initial volume values. A new temperature, thermodynamic phase or composition may also change the local mass density. Use the actual final volume in $\rho =m/V$; the pre-mixing weighted identity cannot supply it. Even unchanged temperature alone does not guarantee unchanged volume.

A pycnometer makes the distinction between mass and mass density experimentally tangible. NIST's 1994 report describes a glass vessel with filling capillaries, wire plugs and a covered overflow bowl in a controlled water bath. The operators soaked a pure-water filling to establish the bath temperature, dried the outside, then left it overnight in the balance enclosure to reach thermal equilibrium before weighing. Empty and full force-restoration balance readings, together with known water mass density, calibrated its capacity at the bath temperature. A tin-oxide coating suppressed stray electric charge; an electrometer detected none during use. The authors distinguish bath and weighing temperatures and correct for buoyancy and changes of displacement volume. Capacity and water-reference uncertainties carry into later liquid mass density determinations. Their pycnometer data were not yet analyzed, so this account provides no measured density or achieved precision.^[ref]

The ideal accounting behind this measurement calibration can be derived without numerical observations. Let $m_0,m_w,m_x$ denote corrected masses of the empty vessel, water-filled vessel and sample-filled vessel. At the same filling temperature, with known reference water mass density ${\rho }_w$ and reproducibly filled volume $V_c$,The empty-vessel subtraction removes vessel mass; the water filling sets volume. Doubling $V_c$ for the same uniform sample doubles the sample mass increment but preserves ${\rho }_x$. Uncorrected balance indications, trapped bubbles, incomplete filling or unequal filling temperatures break the simple identification with corrected mass and a common $V_c$. This explains why replication is a statement about comparable states rather than arbitrary instrument readings.