Conditions for equilibrium and spontaneity under constraints

Under the combinations of restraints usually imposed in the laboratory, relations (10.4) and (10.5) can be expressed in simple and convenient terms. We consider each set of constraints separately. 

For an isolated system, dU = 0,4W = 0,4Q = 0 thus relation (10.4) becomes directly 

This requirement for an isolated system was discussed in detail in Section 8.14, where it was shown that in an isolated system the entropy can only increase and reaches a maximum at equilibrium. 

From relation (10.6) it follows that an isolated system at equilibrium must have the same temperature in all its parts. Assume that an isolated system is subdivided into two parts, a and If a positive quantity of heat, dQ ey, passes reversibly from region a to region we have 

If this flow of heat is to occur spontaneously, then by relation (10.6) dS 0. Since i 

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