A new theoretical formula has been proposed for the study of the collective effect of pressure and size on Debye temperature, based on size dependence of Grüneisen parameter and bulk modulus. This approach enables us to explore the sensitivity of thermodynamic properties to the lattice vibrations at the nano level. This study reveals that the Grüneisen parameter and Debye temperature are the size dependent quantities and decreases with the decrement of size. The trend of theoretical prediction agrees well with the available experimental results of Debye temperature. At high-pressure, Debye temperature show opposite behaviour. For a fixed size of nanomaterials, Debye temperature increases with increase in pressure due to the unit cell parameter.

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