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 $\delta Q$ and the applied work $\delta W$, 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 $1\mathrm{/}T$, can serve as the integration factor for the exchanged heat $\delta Q$. 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 $E_{vh}=C_V{T}^2\mathrm{/}2$ 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.