Simple relations between the first (or second) pressure derivative of the bulk modulus and first (or second) thermal Grüneisen parameter γTh (or qTh) are derived from equations that relate KT(P) to vibrational frequencies Vi as a function of pressure for cubic structures or those that have only one nearest neighbor distance, by assuming that the mode Grüneisen parameters, γi = −∂ ln vi/∂ ln V, are approximately equal. The free volume equation results, which holds within experimental uncertainties for about 70% for solids meeting the symmetry requirements. Agreement is poor for compounds with the rutile structure, which have the largest range of γi observed. Another result is a three-parameter equation of state, KoK″o = (5/3 - K′o)2 - (5/3 - K′o)qTh(0) - (K′o - 1)2. All available experimental determinations of K″o are bracketed by the spectroscopic limits for qTh. Use of the average value of zero for qTh (or qTh calculated from other thermodynamic parameters) gives the proper sign for K″o and nearly the same values as experiment, suggesting that this approximate equation can be used predictively. The equations involving K″o and K′o are probably valid for any structure, if the γi are roughly equal, although this most likely occurs for solids that compress nearly isotropically. The new formula, with qTh of zero, give values similar to that obtained from the Vinet-Ferrante-Rose-Smith equation at low K′o gives the same values as the Birch-Murnaghan (third-order finite strain) equation near K′o of 4, and gives nearly equal values as the Grover-Getting-Kennedy equation of state at high K′o. Physically reasonable positive values of K″o as K′o approaches zero are obtained only from the new equation of state and that of Vinet et al. Comparison of the new equation of state with third-order finite strain suggests that the latter is also likely to be incorrect at high K′o and thus should be cautiously applied to V(P) for liquids.