Trapping and Escape in a Turbid Medium

We investigate the absorption of diffusing molecules in a fluid-filled spherical beaker that contains many small reactive traps. The molecules are absorbed either by hitting a trap or by escaping via the beaker walls. In the physical situation where the number $N$ of traps is large and their radii $a$ are small compared to the beaker radius $R$, the fraction of molecules $E$ that escape to the beaker wall and the complementary fraction $T$ that eventually are absorbed by the traps depend only on the dimensionless parameter combination $\lambda =Na\mathrm{/}R$. We compute $E$ and $T$ as a function of $\lambda$ for a spherical beaker and for beakers of other three-dimensional shapes. The asymptotic behavior is found to be universal: $1-E\sim \lambda$ for $\lambda \to 0$ and $E\sim {\lambda }^{-1\mathrm{/}2}$ for $\lambda \to \mathrm{\infty }$.