  We study diffusion-limited coalescence, A+A<-->A, in one dimension, in the
presence of a diffusing trap. The system may be regarded as a generalization of
von Smoluchowski's model for reaction rates, in that: (a) it includes reactions
between the particles surrounding the trap, and (b) the trap is mobile -- both
considerations which render the model more physically relevant. As seen from
the trap's frame of reference, the motion of the particles is highly
correlated, because of the motion of the trap. An exact description of the long
-time asymptotic limit is found using the IPDF method, and exploiting a
"shielding" property of reversible coalescence that was discovered recently. In
the case where the trap also acts as a source -- giving birth to particles --
the shielding property breaks down, but we find an "equivalence principle":
Trapping and diffusion of the trap may be compensated by an appropriate rate of
birth, such that the steady state of the system is identical with the
equilibrium state in the absence of a trap.
