Is thermodynamics equilibrium related to mechanical equilibrium in anyway? For eg if the net Torque on a system is not zero does that still mean the system can be in thermodynamic equilibrium?


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Unless the extensive or conjugate intensive parameters of the system in question are independent of each other, no constraint tying them together, thermodynamic equilibrium does not mean explicit individual mechanical, electrical, gravitational, etc., equilibrium. Here a constraint could be geometric or physical.

One such geometric constraint example is the relationship between differential volume change and the differential surface area. The geometric constraint results in the non-equality of the internal and external pressures of a spherical bubble where the apparent disequilibrium is balanced by the surface tension, so that $p_1-p_0=\frac{2\gamma}{R}$ (Kelvin's formula), in which $p_1$, $p_0$ are the inside and outside pressures, $r$ is the radius of the bubble, and $\gamma$ is the surface tension per unit area.

An example for a physical constraint is caused by the interplay between the gravitational mass of a gas particle and its contribution to the pressure its weight creates. Take a gas column in a gravitational field that varies vertically and will show both density and temperature variation in its equilibrium caused by the variation of the local gravitational potential. Another, possibly even more complicated coupling can occur when a stationary concentration gradient forms with particles carrying electric charge while are also subject to a chemical potential within the electric potential gradient.

The minimum of the free energy at a constant contact temperature with the environment will drive the system to equilibrium but this equilibrium does not necessarily mean a homogeneous spatial distribution of its extensive or intensive parameters.


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