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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]

The gravimetric dilution factor is the ratio of final mixture mass to the stock mass used in a gravimetric dilution:Here $m_d$ is added diluent mass and $f$ is the retained stock mass fraction. For a diluent containing none of the component being measured, mass conservation gives final component mass fraction $w_f=w_s/F$. This mass-based relation avoids assuming that liquid volumes are additive. Its uncertainty follows from the same input masses and their covariance as the uncertainty of a gravimetric dilution.^[ref]

For an original two-step example, combine $2.000\mathrm{g}$ of stock with $18.000\mathrm{g}$ of diluent, giving $F_1=10$. After complete mixing, transfer $1.000\mathrm{g}$ of that mixture and add $4.000\mathrm{g}$ of diluent, giving $F_2=5$. The combined gravimetric dilution factor isIf the original stock has component mass fraction $0.2000$, the first mixture has $0.02000$ and the final mixture $0.004000$. The transferred gram contains $0.02000\mathrm{g}$ of the component; dividing by the final $5.000\mathrm{g}$ checks the result. This calculation requires a representative, homogeneous transferred portion and no component loss.

State whether a dilution number denotes $F$ or its reciprocal $f$: here a tenfold dilution means $F=10$ and $f=0.1$, not that ten parts of diluent were added per part of stock. The latter recipe gives $F=11$.