One mole of a monoatomic ideal gas (C = 3R) Consider the general M step irreversible expansion (see diagram). P PO T=constant M=4 PO/2 VO 2V0 AV=VO/M M i. Derive expressions for the kth step of the M step expansion for AUX,Qk, WK, ASR, and, ASsurr,k. Find the dependence of AS = EX-1 ASK on the number of steps. (Hint: you should find that AS is independent of M, as it is a state function.) ii. Show that ASuniv = 0 in the limit M +0. (Hint: you will need to take the limit of a sum to be an integral and do the integral.) Recall: - (a) The internal energy for an ideal gas U(T) depends only on T. (b) The equation of state of an ideal gas is PV = NRT. (C) The work in a quasi-static process can be given as W = -PdV. (d) Some useful logarithmic identities, i. In A +In B = ln(AB). ii. In A In B = lng (e) Useful summation trick: DkM = 2221 1 - EXIT (f) In the limit of M +00, an integral can replace the discrete summation since each step Ak 0 becomes infinitesimally narrow, where Ak = kmar-kimin i. limM600 D Btm = limM=o0 D 1 , Ak = SMdk E > = M M = = Id2M - N M 1 k=1 KUM = =