## V

Here, the system moves from state 1 to state 2 by the lower path and returns by the upper path. No matter what the shapes of these two trajectories, AU for the cycle 1 ^ 2 ^ 1 must be zero because U is a path-independent thermodynamic property. The form of the first law for the cycle, which follows from integrating Eq (1.12) with (bdU = 0, is:

where the integrals traverse the complete cycle. The cyclical work and heat depend on the path chosen but their path-integrals are always equal.

The forms of the First Law given by Eqs (1.4) and (1.7) apply to a system and do not involve the surroundings. However, the surroundings, which supply the Q to the system and receive W from it, must also obey the first law. The equation:

is at once the law of Conservation of Energy and a form of the First Law of Thermodynamics. As in Eq (1.4), kinetic energy and potential energy changes have been neglected in Eq (1.8).

Since AUsurr = Qsurr - Wsun, Eqs (1.4) and (1.8) imply that Q + Qsurr = W + W^. This equation does not mean that Qsurr = -Q and Wsurr = -W; these individual equalities hold only for reversible processes such as the compression of the ideal gas in Fig. 1.12.

 System 1 System 2