= Gravimetric dilution
\b[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> $F$ by
$$
f=\frac{m_s}{m_s+m_d},\qquad F=\frac1f.
$$
If the stock has solute <mass fraction> $w_s$, then
$$
w_f=w_sf.
$$
For 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> gives
$$
\left(\frac{u(w_f)}{w_f}\right)^2\simeq
\left(\frac{u(w_s)}{w_s}\right)^2+
\left(\frac{m_d}{M}\right)^2
\left[\left(\frac{u(m_s)}{m_s}\right)^2+
\left(\frac{u(m_d)}{m_d}\right)^2\right].
$$
Here $u$ 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Section5 and AppendixA: first-order combination, sensitivity coefficients and covariance terms]{ref}
\Image[/gravimetric-dilution-uncertainty.png]
{title=Original first-order uncertainty model for gravimetric dilution: at a fixed nominal final mass, strong dilution increases the relative contribution of independently uncertain component masses}
If input errors have <covariance>, their cross terms must be retained. A useful exact check is
$$
\frac{\lambda m_s}{\lambda m_s+\lambda m_d}=f.
$$
A 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.<Massic Activity Ratios of the NBS NIST Tritiated-Water Standards Issued Between 1954 and 1999 by L. L. Lucas>[Abstract and Section3: measured ratios compared with gravimetric dilution factors]{ref}
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