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Internal energy is defined in thermodynamics as a function of state, in such a way that, in an adiabatic process, the variation of internal energy equals to the work done, regardless of the way it has been carried out. Nonetheless, in mechanics, is defined as the total energy of the system minus the orbital energy and the potential energy due to external forces or, in other words, the sum of the kinetic energy relative to the center of mass and the potential energy due to internal forces.

These definitions of internal energy, in thermodynamics and mechanics, refer to the same concept? Can we say that they are the same magnitude?

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Go back and review the complete statement of the first law of thermodynamics for a closed system. You will find that it includes kinetic energy and potential energy, as well as internal energy. It is just that, in most problems we solve in thermodynamics, the changes in KE and PE are negligible. If these changes are not negligible, then in an adiabatic process, the sum of the changes in KE, PE, and U are equal to minus the work W done on the surroundings (not just that $\Delta U = -W$).

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