= Standard addition
{title2=$c_x=-x_0V_f/V_x$}
\b[<Standard addition>] is a <measurement calibration> procedure that measures an unknown sample both before and after adding known amounts of its <analyte>. The additions probe sensitivity in the sample's own <sample matrix>. IUPAC identifies a multiplicative <analytical matrix effect> as a setting in which this procedure gives an unbiased result.<Standard addition in the IUPAC Gold Book>[Entry 08030, definition and Note; originating Recommendations 2021, p. 1012]{ref}
For an original equal-final-volume design, place sample volume $V_x$ in each flask, add standard concentration $c_s$ at different volumes $V_s$, and dilute every flask to $V_f$. Suppose the background-corrected signal is linear:
$$
y=k\left(\frac{c_xV_x}{V_f}+\frac{c_sV_s}{V_f}\right).
$$
The unknown original <concentration> is $c_x$; the same sensitivity $k$ must apply to native and added <analyte>. Plotting $y$ against added final concentration $x=c_sV_s/V_f$ gives a <regression> intercept $a=kc_xV_x/V_f$ and slope $k$. Its extrapolated horizontal intercept is $x_0=-a/k$, so
$$
\boxed{c_x=-x_0\frac{V_f}{V_x}}.
$$
The negative intercept is an extrapolation to cancellation of the modeled native amount, not a physically prepared negative-concentration solution.
Take $V_x=5.00$ mL, $V_f=25.00$ mL and $c_s=10.0\,\mathrm{mg\,L^{-1}}$. Standard volumes 0, 1.00 and 2.00 mL produce final added concentrations 0, 0.400 and $0.800\,\mathrm{mg\,L^{-1}}$. Hypothetical corrected signals 0.120, 0.200 and 0.280 give $k=0.200\,\mathrm{L\,mg^{-1}}$, $x_0=-0.600\,\mathrm{mg\,L^{-1}}$ and original $c_x=3.00\,\mathrm{mg\,L^{-1}}$.
An unremoved constant background of 0.040 would instead shift the intercept and imply $4.00\,\mathrm{mg\,L^{-1}}$. <Standard addition> does not automatically eliminate an additive interference. It also cannot repair nonlinear response, changing <sample matrix>, or unequal chemical availability of native and added <analyte>. Matched volumes, suitable blanks and additions spanning a useful signal range test those assumptions. Since the result involves an extrapolated intercept and fitted slope, their covariance matters in <uncertainty propagation>.
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