The dynamics of energy exchange between gas atoms and crystal surfaces, with consequent trapping and desorption, is discussed in terms of a simple model for the gas-surface potential involving motion in a conservative attractive potential and fast energy-exchanging collisions between gas and surface atoms. Integral recurrence relations for the velocity distributions of atoms leaving the surface after N hops are derived and used to study the approach to the asymptotic steady state for large N. For a Boltzmann incoming distribution this steady-state distribution is non-Boltzmann, as are the velocity distributions for small N, but in such a way that the total velocity distribution for all atoms leaving the surface is Boltzmann, as required by detailed balance. Residence-time distributions are also calculated, both for Boltzmann and for mono-energetic incident beams. For high incident-beam energies these show a pronounced peak at short residence times (prompt inelastic scattering) together with a decay at long-residence times which is ultimately exponential. If the gas-surface collisions are 'perfectly equilibrating' the residence-time distribution is purely exponential in form. For high incident-beam energies the probability of trapping is zero at very low and at very high surface temperatures, with a maximum in between. For low incident-beam energies the probability of trapping decreases monotonically from one at low temperatures to zero at high temperatures. Arguments are presented indicating that many of these qualitative features are independent of the simplifications of our model. A general relation between mean residence time and equilibrium adsorption is stated. The implications of these results for the kinetics of physisorption and chemisorption are discussed.