With the characteristic lengths of electronic and thermal devices approaching the mean free paths of the pertinent energy carriers, thermal transport across these devices must be characterized and understood, especially across interfaces. Thermal interface conductance can be strongly affected by the strength of the bond between the solids comprising the interface and the presence of an impurity mass between them. In this work, we investigate the effects of impurity masses and mechanical adhesion at molecular junctions on phonon transmission via non-equilibrium Green’s functions (NEGF) formalisms. Using NEGF, we derived closed form solutions to the phonon transmission across an interface with an impurity mass and variable bonding. We find that the interface spring constant that yields the maximum transmission for all frequencies is the harmonic mean of the spring constants on either side of the interface, while for a mass impurity, the arithmetic average of the masses on either side of the interface yields the maximum transmission. However, the maximum transmission for each case is not equal. For the interface mass case, the maximum transmission is the transmission predicted by a frequency dependent form of the acoustic mismatch model, which we will refer to as the phonon mismatch model (PMM), which is valid for specular phonon scattering outside the continuum limit. However, in the interface spring case, the maximum transmission can be higher or lower than the transmission predicted by the PMM.