A microcanonical unimolecular rate theory (MURT) model incorporating quantized surface vibrations and Rice - Ramsperger- Kassel -Marcus rate constants is applied to a benchmark system for gas-surface reaction dynamics, the activated dissociative chemisorption and associative desorption of hydrogen on Cu(111). Both molecular translation parallel to the surface and rotation are treated as spectator degrees of freedom. MURT analysis of diverse experiments indicates that one surface oscillator participates in the dissociative transition state and that the threshold energy for H-2 dissociation on Cu(111) is E-0 = 62 kJ/mol. The spectator approximation for rotation holds well at thermally accessible rotational energies (i.e., for E-r less than similar to 40 kJ/mol). Over the temperature range from 300 to 1000 K, the calculated thermal dissociative sticking coefficient is S-T = S-0 exp(-E-a/k(B)T) where S-0 = 1.57 and E-a = 62.9 kJ/mol. The sigmoid shape of rovibrational eigenstate-resolved dissociative sticking coefficients as a function of normal translational energy is shown to derive from an averaging of the microcanonical sticking coefficient, with threshold energy E-0, over the thermal surface oscillator distribution of the gas-surface collision complexes. Given that H-2/Cu(111) is one of the most dynamically biased of gas-surface reactive systems, the simple statistical MURT model simulates and broadly rationalizes the H-2/Cu(111) reactive behavior with remarkable fidelity.