The elastic scattering of low-energy neutrons by the nuclei of a monatomic gas, which have an isotropic Maxwellian velocity distribution, is examined in detail within the framework of classical physics. A unified mathematical treatment, which fully preserves the three-dimensional aspects of the scattering process, is employed to study the dynamics of the neutron-nuclear elastic collision. A new form of the scattering probability function in velocity space is derived under the assumption of isotropic scattering in the center-of-mass system. Unique single-integral expressions, which are valid for any analytical or numerical representations of σs(υr) and σa(υr), the microscopic scattering and absorption cross section as functions of the relative neutron-nuclear speed, are developed for the velocity scattering kernel, its spherical-harmonics weighted moments, and the total scattering and absorption probabilities. These formulations are tested by explicitly evaluating them in closed form for certain analytical cross-section representations and comparing these solutions with known results. The utility of the collision kernels for new solutions of the transport equation under conditions of variable scattering cross section is discussed.