Adiabatic operation

Definition

Consider a thermodynamic system in an equilibrium state denoted by $(T; X)$ , where $T$ is the temperature and $X$ represents the set of extensive variables. We enclose this system within adiabatic walls and change the set of extensive variables from $X$ to $X^{'}$ through mechanical operations applied from the surrounding. After the operation, the system reaches a new equilibrium state $(T^{'}; X^{'})$ . Note that the final temperature $T^{'}$ is determined by the system itself. An operation that transforms an initial equilibrium state $(T; X)$ to a new equilibrium state $(T^{'}; X^{'})$ in this way is called an adiabatic operation, and it is symbolically represented as:

(T; X) \overset{a}{\to} (T^{'}; X^{'})

Existence of an Adiabatic Operation to Increase Temperature

Let $(T; X)$ denote an arbitrary equilibrium state of a thermodynamic system. For any temperature $T^{'}$ such that $T^{'} > T$ , there exists an adiabatic operation that transforms the system from $(T; X)$ to $(T^{'}; X)$ , while maintaining the values of the extensive variables $X$ .

During this operation, positive work must be done on the system by the surrounding.

Symbolically, this can be expressed as:

\begin{matrix} (T; X) \overset{a}{\to} (T^{'}; X^{'}) \\ where T^{'} > T and X^{'} = X . \end{matrix}

Example: Adiabatic Stirring of a Viscous Fluid

Consider a viscous fluid contained in a thermally insulated (adiabatic) chamber. A mechanical stirrer is introduced into the system, and work is done on the fluid by stirring. The internal friction dissipates the mechanical energy as heat within the fluid. As a result, the system reaches a new equilibrium state with a higher temperature.

Reference

Tasaki, Chapter 4, P58