Equilibrium in Non-isolated Systems
Non-isolated systems exchange work or heat with their surroundings. For the study of their equilibria, we use the first and second law in their global forms,
which are valid when the system exchanges work only via a piston. Here, pB is the pressure at the piston boundary, and the system exchanges heat only at boundary temperature TB ; we shall consider only cases with homogeneous pressure and temperature at the system boundary. For simplicity, we ignore kinetic energy, which can be incorporated as in the previous sections, with the same result that all elements of the system will be at rest in equilibrium.
For all systems discussed below, if single phase systems are considered, the respective maximization or minimization requirements are mathematically very similar to the maximization of entropy as discussed above.
In cases where the homogeneous boundary temperature TB is prescribed, the role of the Lagrange multiplier ΛE is assumed by the boundary temperature TB , and thus the homogeneous equilibrium temperature of the system is T = TB .
In cases where the piston pressure is prescribed, the pressure condition g (p, T ) = −T Λρ − γz must be compatible with the pressure prescribed at the piston. If gravity can be ignored, this gives g (p, T ) = g (pB ,T ) = −T Λρ, hence homogeneous pressure p = pB . In cases with gravity, since we have assumed homogeneous piston pressure, this implies horizontal piston andg (p, T ) = g (pB ,T ) − γ (z − zB ), where zB is the height of the piston.