 Gravimetric dilution

Gravimetric dilution determines a dilution from measured masses. A measured amount of a stock solution is combined with material containing none of the quantity being diluted. For a closed transfer and a homogeneous mixture, the conserved amount divided by the final mass determines the new mass-based concentration.

Let $m_s$ be the stock's net mass, $m_d$ the added net mass, and $M=m_s+m_d$. Define the stock fraction $f$ and the gravimetric dilution factor $F$ byIf the stock has solute mass fraction $w_s$, thenFor stipulated $m_s=2.000\mathrm{g}$, $m_d=18.000\mathrm{g}$ and $w_s=0.2000$, the final mass is $20.000\mathrm{g}$, $F=10$, and $\boxed{w_f=0.02000}$: 2% by mass. The solute mass is $0.4000\mathrm{g}$ before and after dilution. The net masses exclude their containers. This mass basis does not require an assumption that liquid volumes add.

The uncertainty of a gravimetric dilution depends on which readings share errors. For independently evaluated inputs $w_s,m_s,m_d$, first-order propagation of uncertainty givesHere $u$ denotes standard uncertainty. The sensitivity factors follow from the partial derivatives $\partial f/\partial m_s=m_d/M^2$ and $\partial f/\partial m_d=-m_s/M^2$. With a chosen independent $u(m_s)=u(m_d)=0.001\mathrm{g}$ at the example's nominal masses, the relative mass-ratio contribution is approximately 0.0453%. Adding an independent 0.5% relative stock-value contribution in quadrature gives approximately 0.502%. These are illustrative inputs, rather than a balance specification.^[ref]

If input errors have covariance, their cross terms must be retained. A useful exact check isA common multiplicative mass-scale factor therefore cancels from $f$. Counting it as two independent contributions would lose that correlation. Container subtraction, transfer losses and changes in composition require their own assessment; this cancellation establishes only the common-scale property.

For massic activity, the same conservation argument applies after expressing all activities at a common reference time. A 2000 NIST study compared massic activity ratios inferred from gravimetric dilution with independent liquid scintillation counting. Their agreement within reported uncertainties provided an experimental check on transferring standards through mass ratios.^[ref]