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The intermetallic μ-phase, which precipitates in steels and superalloys, can noticeably soften the mechanical properties of their matrix. Despite the importance of developing superalloys and steels, the thermodynamic properties and directions of thermal expansion of the μ-phase are still poorly studied. In this work, the thermal expansion paths, elastic, thermal and thermodynamic properties of the Fe23Mo16 and Fe7Mo6 μ-phases are studied using the first-principles-based quasi-harmonic Debye–Grüneisen approach. A method that avoids differentiation in many variables is used. The free energies consisting of the electronic, vibrational and magnetic energy contributions, calculated along different paths of thermal expansions, were compared among themselves. A path with the least free energy was chosen as the trajectory of thermal expansion. Negative thermal expansion of the Fe7Mo6 compound was predicted, while Fe23Mo16 exhibits conventional thermal expansion. The thermal expansions of both these compounds are not isotropic. The elastic constants, moduli, heat capacities, Curie and Debye temperatures were predicted. The obtained results satisfactorily agree with the available experimental data. Physical factors affecting the stability of Fe23Mo16 and Fe7Mo6 have been studied. This study presents an essential feature of thermal expansion of the μ-phase of the Fe–Mo system, which can provide an insight into future developments. [ABSTRACT FROM AUTHOR]