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We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher order Gr¨uneisen parameters using the generalized free volume theory, and the generalized Rydberg equation of state. The properties derived in the present study are directly related to the understanding of thermoelastic properties of solids. The third order Gr¨uneisen parameter (lambda λ) in the limit of in nite pressure has been found to approach a positive finite value for lambda in nity (λ∞) equal to 1/3. This is a result shown to be independent of the value of K-prime in nity, i. e., the pressure derivative of the bulk modulus at infinite pressure. The results based on other equations of state have also been reported and discussed. We find a relationship between λ∞ and pressure derivatives of bulk modulus at infinite pressure which is satisfied by different types of equations of state.