A researcher monitors an isolated system consisting of two equal masses of water contained in separate compartments that are initially kept at different temperatures. The compartments are separated by a thin conducting wall that permits energy transfer but not mass transfer. At t=0 the wall is removed, and at a later time t=τ the external insulation around the whole apparatus is suddenly taken away, allowing the combined water to exchange energy with the environment at a constant ambient temperature Tamb. The researcher keeps a continuous record of the total entropy Ssys of the water as a function of time, generating a curve with two distinct regions of differing slope.
(a) Without performing numerical calculations, describe how the slope dtdSsys in the time interval 0<t<τ reflects the internal approach to equilibrium between the two water portions.
(b) After the insulation is removed at t=τ, predict qualitatively how the slope of the Ssys versus t curve changes and explain the physical reason for that change.
(c) Suppose instead that the insulation were removed much earlier, at a time t1<τ while the water portions were still far from mutual equilibrium. Sketch the modified Ssys versus t curve on the same axes as your original sketch and discuss the differences in shape and key features.