Summary: We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schrödinger part and a noisy Poisson-timed scattering part. The dynamics describes the evolution of a test particle in \(\mathbb R^n, n = 1, 2, 3\), immersed in a gas, and the noisy scattering part is defined by the reduced effect of an individual interaction, where the interaction between the test particle and a single gas particle is via a repulsive point potential. In the limit that the mass ratio \(\lambda = m/M\) tends to zero and the collisions become more frequent as \(1/\lambda \), we show that our dynamics \(\Phi _{t,\lambda}\) approaches a limiting dynamics \(\Phi _{t,\lambda}^{\diamond} \) with second order error. Working in the Heisenberg representation, for \(G\in B(L^{2}(\mathbb R^n)) n = 1, 3\) we bound the difference between \(\Phi _{t,\lambda}(G)\) and \(\Phi _{t,\lambda} ^{\diamond}(G)\) in operator norm proportional to \(\lambda ^{2}\).