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 T₁ 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 = 2T₁.

b) The same system is now heated in two stages, by first contacting with a reservoir at T₂ halfway between T₁ 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 T₁ 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 (P₁) be 1 bar, and the highest (P₃) be 80 bar. Find the values of P₂ and P₄.

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.