= Pycnometer density measurement
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.<The Electronic Balance and Some Gravimetric Applications (The Density of Solids and Liquids, Pycnometry and Mass) by Randall M. Schoonover and others>[The Electronic Balance and Some Gravimetric Applications (The Density of Solids and Liquids, Pycnometry and Mass), NISTIR 5375, printed pp. 313–316, “Pycnometer Calibration”, Figure 5 and “Pycnometer Application”; p. 315 states that pycnometer data had not yet been analyzed]{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 (thermodynamics)> $V_c$,
$$
V_c=\frac{m_w-m_0}{\rho_w},\qquad
\rho_x=\frac{m_x-m_0}{V_c}
=\rho_w\frac{m_x-m_0}{m_w-m_0}.
$$
The empty-vessel subtraction removes vessel <mass>; the water filling sets <volume (thermodynamics)>. 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.
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