Spike equilibration

Spike equilibration is the process by which added analyte comes to share the chemical forms or physical environments relevant to recovery of the native analyte. Mixing a solution uniformly is not sufficient if native material is bound, trapped or otherwise inaccessible to the added material. The spike-recovery estimate is representative only to the extent that the two undergo comparable processing.

IUPAC's recovery guidance makes this equivalence a condition for using an added surrogate to correct native recovery. It notes that even an isotopically modified analyte can fail to equilibrate with native material bound to the sample matrix.^[ref]

An original two-environment model describes a fast-extractable fraction $a(t)$ approaching $a_{\mathrm{eq}}$:Let the fresh spike start entirely accessible, $a(0)=1$, while native material has accessible fraction $a_{\mathrm{eq}}=0.20$. With stipulated spike equilibration time $\tau =5\min$, a 15-minute wait leaves $a\approx 0.240$, not 0.20. If extraction measures only the accessible material, correcting a native recovered amount 20 by the spike's apparent recovery 0.240 gives approximately 83.4, below the true 100 amount units. The mismatch persists despite a high initial spike recovery.

A useful test varies the waiting time before extraction while keeping preparation and added analyte amount fixed. Compare the behavior of the spike with independent native-material or reference material evidence where available. A stable time series supports constancy over that window, but cannot prove that an inaccessible native compartment has been sampled.

More vigorous processing may change chemical forms or destroy the analyte; longer waiting alone does not guarantee equivalence. State addition timing, equilibration conditions and the extraction stage to which the measured extraction recovery applies. These choices are part of the measurement model, rather than details that can be recovered from the final percentage alone.

A spike equilibration time characterizes how rapidly added analyte approaches the distribution relevant to native-material recovery. A model's time constant differs from the waiting time needed to meet a specified analytical tolerance. Neither is a universal property independent of sample matrix, chemical form and preparation conditions.

For the illustrative first-order spike equilibration modelthe deviation from equilibrium falls to $1/e$ of its initial value after $\tau$. Suppose acceptance requires $|a(t)-a_{\mathrm{eq}}|/a_{\mathrm{eq}}\le ϵ$, with $a_{\mathrm{eq}}>0$. If the initial mismatch exceeds the tolerance, solving givesFor original stipulated values $a(0)=1$, $a_{\mathrm{eq}}=0.20$, $\tau =5\min$ and $ϵ=0.05$, the waiting time is at least $5\ln 80\approx 21.9\min$. After three time constants, the residual deviation is only5% of its initial value but still approximately20% of the equilibrium accessible fraction. The denominator in the acceptance criterion matters.

At known equilibrium value, two positive deviations at $t_1<t_2$ estimateIf $a_{\mathrm{eq}}$ is fitted from the same observations, its uncertainty and covariance must also be considered. A time series with few late observations can make the plateau and time constant difficult to distinguish.

Experimentally, prepare matched portions with different waiting times before the same extraction. IUPAC warns that added and native material may fail to reach equivalent states, especially when native analyte is bound to the sample matrix.^[ref] A stable signal alone cannot distinguish complete equilibration from an inaccessible pool or a competing degradation process. State the observed quantity, waiting conditions and acceptance criterion when reporting an equilibration time.