2) From the lecture (compare also text page 193-194) we learn that the maximum amount of work which can be extracted from a system interacting with a single heat reservoir is given by Wx=-AF where AF = F, -F, is the change of the Helmholtz free energy of the system. Specifically for water at initial temperature To and constant heat capacity Cp interacting with a heat reservoir at temperature TR this maximum amount of work reads Wx=-AF=C(T-TR)+CT In(T/TO). Perform now an explicit calculation to confirm the above expression. Let a Carnot cycle run between the water and the reservoir (see sketch). The Carnot engine is small taking in only infinitesimal small amounts of heat per cycle. The water of constant heat capacity Cp is initially at temperature To. The reservoir is at constant temperature TR. The Carnot cycle runs until equilibrium is reached (hint: take into account that the temperature of the reservoir initially at To is changing and thus the changes the Carnot efficiency continuously). Calculate the total amount of work produced by the Carnot cycle. Express your result in terms of To, TR and Cp and compare with W from above. max To-To(t) with lim T₁(t) = T 1- Carnot TR=const W (5points)