<b>The Grüneisen Parameter and Higher Order Thermo Elastic Properties of Fe, Co at Higher Pressures</b> Expressions have been obtained for the higher order Thermo elastic properties of the Grüneisen parameter in the limit of infinite pressure The usefulness of these expressions has been demonstrated by considering a relationship between reciprocal Grüneisen parameter and the pressure bulk modulus ratio at finite pressure. The results have been obtained using the shanker reciprocal gamma relationship. The results have also been determined using the Stacey Davis formulations for the third order Grüneisen parameters. The calculations have been performed in case of transition metals Fe and Co for the wide range of compressions. It has been found that both the models yield close agreement with each other. The obtained results in the present paper satisfy the thermodynamic constraints in the limit of infinite pressure. Grüneisen parameters, Thermoelastic properties, infinite pressure behavior, transition metal 994-996 Issue-4 Volume-4 Dr. Sheelendra Kumar