1 ΔS on rapid decompression
Steam at 400C, 100 bar in an insulated cylinder is suddenly decompressed to 10 bar, by unlocking the piston and allowing the gas to expand, with a constant external pressure of 10 bar applied to the piston. Find the final temperature T2, and the entropy change ΔS.
To answer the problem, ask yourself the following questions:
1) How much work is done as the steam expands?
2) What does energy conservation tell you in this situation?
2 Reversible heating with reservoirs
a) A system with constant heat capacity CP and initial temperature T_{1} is heated by contacting a reservoir at T_{f} . Find the entropy change of the system, reservoir, and system plus reservoir. Evaluate the total entropy change assuming T_{f} = 2T_{1}.
b) The same system is now heated in two stages, by first contacting with a reservoir at T_{2} halfway between T_{1} and Tf , then by contacting with the reservoir at T_{f }. Again find the entropy change of system, reservoirs, and system plus reservoirs, and again evaluate the total entropy change.
c) The system is now heated in n stages, by contacting with reservoirs at n equally spaced
T_{i} between T_{1} and T_{f} . Write an expression for the entropy change of the system, reservoirs, and system plus reservoirs. How does this compare to the limit of reversible heating as n becomes large?
3 Carnot cycle in steam
Consider a Carnot heat engine operating with steam as the working uid, between reservoirs at
TL = 200C and TH = 500C.
Do the following, using the steam tables:
1) Sketch the cycle on a PV diagram. Label the state points 1, 2, 3, 4 starting with the low-pressure, low-temperature state.
2) Let the lowest pressure in the cycle (P_{1}) be 1 bar, and the highest (P_{3}) be 80 bar. Find the values of P_{2} and P_{4}.
3) Compute the heat Q and work W on each leg of the cycle; tabulate your results (SI units).
4) Compute the efficiency of this cycle as a heat engine. Compare to the ideal-gas result for the Carnot efficiency, and briey comment on the comparison.
4 van der Waals U, S
For a substance described by the van der Waals equation of state with a constant heat capacity CV :
1) Find the internal energy U(T; V ), relative to a reference state at some T0; V0.
2) Find the entropy S(T; V ), likewise.