 Density of a composite system

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.