An irreversible Stirling cycle containing 5.0 mol of a diatomic ideal gas goes through the following series of steps, moving through states a, b, c, d, and back to state a to complete the cycle: • Step 1 (a → b): Starts in state a: (Pa, Va, Ta) = (40. atm, 5.23 L, 510. K). Expands isothermally in three irrerversible steps: first against a constant external pressure of Pext = 26.67 atm to an intermediate volume of V ′a = 1.5Va; then further expansion against a (new) constant external pressure of Pext = 20.0 atm to another intermediate volume of V ′′a = 2Va; and finally a further expansion against a (new) constant external pressure of Pext = 10.0 atm to the final volume of Vb = 4Va. • Step 2 (b → c): Starts in state b: (Pb, Vb, Tb) = (10.0 atm, 20.92 L, 510. K). Undergoes constant-volume cooling to a final temperature of Tc = 255 K. • Step 3 (c → d): Starts in state c: (Pc, Vc, Tc) = (5.0. atm, 20.92 L, 255. K). Com- presses isothermally in three irrerversible steps: first against a constant external pressure of Pext = 7.50 atm to an intermediate volume of V ′c = Vc/1.5; then further
An irreversible Stirling cycle containing 5.0 mol of a diatomic ideal
gas goes through the following series of steps, moving through states a, b, c, d, and back to
state a to complete the cycle:
• Step 1 (a → b): Starts in state a: (Pa, Va, Ta) = (40. atm, 5.23 L, 510. K). Expands
isothermally in three irrerversible steps: first against a constant external pressure of
Pext = 26.67 atm to an intermediate volume of V ′a = 1.5Va; then further expansion
against a (new) constant external pressure of Pext = 20.0 atm to another intermediate
volume of V ′′a = 2Va; and finally a further expansion against a (new) constant external
pressure of Pext = 10.0 atm to the final volume of Vb = 4Va.
• Step 2 (b → c): Starts in state b: (Pb, Vb, Tb) = (10.0 atm, 20.92 L, 510. K). Undergoes
constant-volume cooling to a final temperature of Tc = 255 K.
• Step 3 (c → d): Starts in state c: (Pc, Vc, Tc) = (5.0. atm, 20.92 L, 255. K). Com-
presses isothermally in three irrerversible steps: first against a constant external
pressure of Pext = 7.50 atm to an intermediate volume of V ′c = Vc/1.5; then further
expansion against a (new) constant external pressure of Pext = 10.0 atm to another
intermediate volume of V ′′c = Vc/2; and finally a further expansion against a (new)
constant external pressure of Pext = 20.0 atm to the final volume of Vd = Vc/4.
• Step 4 (d → a): Starts in state d: (Pd, Vd, Td) = (20.0 atm, 5.23 L, 255. K). Undergoes
constant-volume heating to a final temperature of Ta = 510. K.
(a) Sketch the cycle on a Pext vs V diagram, marking the pressures and volumes
(either as numbers or using the above-defined labels). [It may be a little tricky to fit the
expansion and compression steps onto a single diagram since they partially overlap; you
can use a single diagram, or split it into two (one for expansion+cooling and the other for
compression+heating).]
(b) Find the net work around this cycle, |wcyc|. [Work in J, to three s.f.] [Hint:
two of the steps make no contribution to the work.]
(c) Find the efficiency of this cycle, ǫ = |wcyc| /qhot. [Dimensionless efficiency, to
three s.f.] [Note that for a Stirling cycle, the heat absorbed from the hot side (qhot) includes
contributions from both the isothermal expansion (step 1) and the constant-volume heating
(step 4): qhot = qab + qda
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