 Spike equilibration time

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.