The uniqueness of the integration factor associated with the exchanged heat in thermodynamics

State functions play important roles in thermodynamics. Different from the process function, such as the exchanged heat $δQ$ and the applied work $δW$, the change of the state function can be expressed as an exact differential. We prove here that, for a generic thermodynamic system, only the inverse of the temperature, namely $1/T$, can serve as the integration factor for the exchanged heat $δQ$. The uniqueness of the integration factor invalidates any attempt to define other state functions associated with the exchanged heat, and in turn, reveals the incorrectness of defining the entransy $E_{vh}=C_VT^2 /2$ as a state function by treating $T$ as an integration factor. We further show the errors in the derivation of entransy by treating the heat capacity $C_V$ as a temperature-independent constant.

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