I.G Approach to Equilibrium and Thermodynamic Potentials

Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For example, in the previous section we showed that for an adia­ batically isolated system entropy must increase in any spontaneous change and reaches a maximum in equilibrium. What about out of equilibrium systems that are not adiabati­ cally isolated and which may also be subject to external mechanical work? It is usually possible to define other thermodynamic potentials that are extremized when the system is in equilibrium. Enthalpy is the appropriate function when there is no heat exchange (dQ ̄ = 0), and the system comes to mechanical equilibrium with a constant external force. The minimum enthalpy principle merely formulates the observation that stable mechanical equilibrium is obtained by minimizing the net potential energy of the system plus the external agent. For example, consider a spring of natural extension L0 and spring constant K, subject to the force exerted by a particle of mass m. For an extension x = L−L0, the internal energy of the spring is Kx/2, while there is a change of −mgx in the potential energy of the particle. Mechanical equilibrium is obtained by minimizing Kx/2−mgx at an extension xeq = mg/K. The spring at any other value of the displacement initially oscillates before coming to rest at xeq due to friction. For general displacements x, at constant generalized forces J, the work input to the system is dW ̄ ≤ J · δx. (Equality is achieved for a reversible change, but there is generally some loss of the external work into friction.) Since dQ = 0, ̄ using the first law, δE ≤ J · δx, and