We; analyze the transient kinetics of a mixture of two radicals (M and P), one of which (P) has very low, or zero self-reactivity (i.e., P is a "persistent" radical). For the case where the self-reactivity of the persistent radical is zero, we find a critical value for the initial concentration ratio r=P-0/M-0, above which the persistent radicals are isolated at long reaction times after all of the M's react. If r is below its critical value the surviving fraction of P is zero. An experiment employing virtually zero self-reactivity nitroxide radicals is proposed to test the predicted critical behavior If the self-reactivity of the M is stronger than the;cross-coupling reactivity, then the critical value of the ratio r is negative, and a finite fraction of the persistent P radicals are always isolated, regardless of the initial concentration ratio r. For the case where the self-reactivity of the persistent radicals is finite but small, we find similar behavior, except now the P radicals react with one another on a much; longer time scale following their isolation. The isolation process facilitates :the measurement of P-P reaction kinetics in situations where it is difficult to generate the P radicals by themselves. An experiment to measure fundamental polymer-polymer reaction kinetics in concentrated solutions of melts is proposed based on these ideas.