Double substitution weighing

Double substitution weighing determines the difference between an unknown mass and a reference through repeated substitutions on the same comparator, with a small known sensitivity mass used to determine response under load. NIST's SOP4 compares standard and unknown twice, rather than relying on the balance to establish their entire masses from its zero.^[ref]

For the source's sequence A, let S denote the standard, X the unknown and sw the added sensitivity mass. With optional tare masses omitted, the four pan loads areproducing measurement indications $O_1,O_2,O_3,O_4$. The standard remains the reference; the small addition probes transducer sensitivity close to the actual working load.^[ref]

An original simplified model neglects buoyancy, tare differences and drift, and assumes a common linear response coefficient $a$. Write $\Delta m=m_X-m_S$ and $m_{\mathrm{sw}}$ for the added mass. Relative to the standard load, the readings areThe two comparison differences estimate the same quantity; averaging them gives $D=[(O_2-O_1)+(O_3-O_4)]/2=a\Delta m$. The sensitivity step is $O_3-O_2=a{m}_{\mathrm{sw}}$, soFor original $m_{\mathrm{sw}}=10\mathrm{mg}$ and readings $(100,106,126,120)$, this gives $\Delta m=3\mathrm{mg}$. The offset $b$ and the unknown response coefficient cancel under the model.

Practical weighing records timing and environmental conditions, keeps settings fixed through the comparison, and controls handling and equilibration. The full NIST procedure supplies buoyancy corrections, uncertainty contributions and a comparison using a check standard.^[ref] For equally spaced readings, a linear additive drift cancels from $D$, but can still affect the sensitivity step $O_3-O_2$. Neither the simplified equation nor the method's name guarantees immunity to drift, contamination, nonlinear response or uncertainty in the measurement standard.

A sensitivity mass is a small calibrated mass added to a loaded weighing instrument to determine its response to a known mass increment. NIST's double-substitution procedure calls this a sensitivity weight and uses it to establish sensitivity under the conditions of the comparison.^[ref] It supplies a local input step rather than a new determination of the entire load from zero.

Let an original comparator give indications $r_0$ before and $r_1$ after adding mass $m_{\mathrm{sw}}$. For a stable linear response over that increment,where $\Delta r$ is another indication difference near that working load. A constant additive zero cancels from each difference; instrumental drift or load-dependent response need not cancel.

For original indications $r_0=860$, $r_1=885$ and assigned sensitivity mass $10.020\mathrm{mg}$, the local response is about $2.495$ indication increments per milligram. A difference $\Delta r=7.5$ increments gives $\Delta m=3.006\mathrm{mg}$. Using only a nominal $10\mathrm{mg}$ label instead would give $3.000\mathrm{mg}$. The assigned mass and its measurement uncertainty, rather than the nominal label alone, enter the comparison.

The added step must be large enough to resolve reliably but remain within the response region relevant to the unknown/reference comparison. NIST specifies selection criteria and regularly spaced observations; it also requires the standards to have valid calibration information.^[ref] Keep handling, settling and environmental conditions consistent. A sensitivity mass does not by itself cancel buoyancy, temperature effects or uncertainty in the main measurement standard. Those contributions remain in the full double substitution weighing model.