Rapid molecular-beam kinetics data are reported of thermal desorption of K atoms from pyrolytic graphite films over the temperature range 960-1800 K. By using these-called field reversal method (FR), the kinetics of desorption is studed at time constants down to the microsecond range, where bulk diffusion becomes rate-limiting. The neutral and ionic desorption rates are measured and shown to contain both a primary as well as a secondary rate. These measurements are combined with data on the steady-state and FR peak signals, revealing several states of K on the surface, similar to the previously studied case of Cs on pyrolytic graphite. Two covalently bound states sigma(4p) and sigma(3d) exist, which are 4.30 and 4.40 eV, respectively, below the corresponding atomic configurations 4p (a P-2(degrees) term) and 3d (a D-2 term) outside the surface. An ionic state is also found, which is 2.0 V below the corresponding desorbed ion K+ (with the electron at the Fermi level). The 4s and 5s states do not correlate with stable adsorbed states. The apparent neutral rate of desorption is only slowly temperature dependent in the range 960-1550 K, with a primary (fast) rate constant of the order of 300 s(-1) and a secondary (slow) rate of 1-10 s(-1). This is due to interconversion processes involving diffusion on the surface. In the range 1550-1760 K, processes with activation energies up to 6.07 eV and preexponential factors up to 10(21) s(-1) are observed for both the fast and the slow rates. Such large preexponential factors are indicative of thermal electronic excitation processes, implying a direct switch to a Rydberg state on the surface. The thermal emission and desorption of alkali atoms in Rydberg states is possible by two main mechanisms: by direct emission from the bulk into high Rydberg states over a thermal barrier of 7.4 eV and by excitation from the covalent states that are transferred to Rydberg states in collisions with the surface.