Temperature control total-immersion tests

Consider a process in a system that we will call the test tube, immersed in a heat bath. The system need not be a real test tube in a water jacket. It could be molecules in a solvent or air in the atmosphere. Heat bath refers to any surroundings of a system that hold the temperature of the system constant. This arrangement controls the temperature T, not the energy U, at the boundary around the subsystem. If the test tube plus heat bath are isolated from the greater surroundings, equilibrium will be the state of maximum entropy for the total system. However, we are not interested in the state of the total system. We are interested in what happens in the test tube itself. We need a new extremum principle that applies to the test tube, where the independent variables are (T,V,N). 

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