We have considered heat and work to be path-dependent. However, if all heat transfer with surroundings is performed using a reversible heat transfer device (some type of reversible Carnot-type device), work can be performed by the heat transfer device during heat transfer to the surroundings. The net heat transferred to the surroundings and the net work done will be independent of the path. Demonstrate this by calculating the work and heat interactions for the system, the heat transfer device, and the sum for each of the following paths where the surroundings are at T_surr = 273 K. The state change is from state 1, P₁ = 0.1 MPa, T₁ = 298 K and state 2, P₂ = 0.5 MPa and T2 which will be found in part (a).

(a) Consider a state change for an ideal gas in a piston/cylinder. Find T₂ by an adiabatic reversible path. Find the heat and work such that no entropy is generated in the universe. This is path a. Sketch path a qualitatively on a P-V diagram.

(b) Now consider a path consisting of step b, an isothermal step at T₁, and step c, an isobaric step at P2. Sketch and label the step on the same P-V diagram created in (a). To avoid generation of entropy in the universe, use heat engines/pumps to transfer heat during the steps. Calculate the W_EC and W_S as well as the heat transfer with the surroundings for each of the steps and overall. Compare to part (a) the total heat and work interactions with the surroundings.

(c) Now consider a path consisting of step d, an isobaric step at P1, and step e, an isothermal step at T₂. Calculate the W_EC and W_S as well as the heat transfer with the surroundings for each of the steps and overall. Compare to part (a) using this pathway and provide the same documentation as in (b).