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In the recent paper by José I. Belandria, J. Chem. Educ. 1995, 72, 116118, halfway down the second column on p 118 one is arrested by the claim that "the compression process occurring in tank A is more efficient than a reversible compression process for the same change of state." This cannot be true when, as here, the change of state is nothing but the isothermal compression of an ideal gas from 1 atm to 4 atm. On the author's engineering convention, which makes w negative for a work input, the cited isothermal compression must satisfy the familiar relation
w < -Delta A
The inequality figures in the general case, while the equality applies to the special case of reversibility, where
-Delta A = wrev
In the given isothermal change of state, the
alteration (Delta A) of the Helmholtz function is a
constant eliminated by summing the last two relations to obtain the
restrictive condition
w < wrev
For the reversible isothermal compression from 1 to
4 atm, the author (correctly) gives as the requisite
work wrev = -17.3 kJ. But the author's mysterious
irreversible "theoretical process" purports to achieve the same
isothermal compression with work w = -14.05 kJ.
These work terms stand, alas, in the forbidden relationship
w > wrev
The author's theoretical process is thus a
thermodynamically impossible process. To see that it involves a
transparent violation of the second law, imagine tank
A as the cylinder of a Carnot engine in thermal contact with an
immense heat reservoir at 1500 K. Beginning at 1 atm, let
the author's notional irreversible isothermal compression
proceed to 4 atm with work input of -14.05 kJ. Let the gas
then resume its original state by a reversible isothermal
expansion yielding a work output of +17.3 kJ. Drawing on only
a single heat reservoir, this indefinitely renewable
closed cycle thus delivers a net work output of over 3 kJ per
cycle. Thus, a reductio ad absurdum.
One wonders where the author got his figure for
the final pressure (Pf) in tank A. That
Pf = 4 atm is nowhere derived or justified in any way. It is simply
announcedfirst in the diagram on p 116. The author calculates the
heat required to warm tank B from 373 to 1500 K, and
concludes (correctly) that this 14.05 kJ must also represent the
magnitude of the work input to the isothermal system
A. Unhappily, a -14.05 kJ work input just won't suffice to
compress the gas in A to Pf = 4 atm. Using this work input in the most effective possible way (i.e., reversibly) we
calculate Pf for the isothermal compression from
8.314 (1500) ln(1/Pf) = -14,054
ln(Pf /1) = +1.127: Pf = 3.086 atm
Hence this reversible isothermal compression
proceeds with
Delta SA = 8.314 ln (1/3.086) = -9.369 J/K
Alternatively, we can calculate Delta SA from the heat
expelled in that reversible isothermal compression, finding
Delta SA = -14,054/1500 = -9.369 J/K
Perfectly concordant with each other, these figures
differ significantly from the author's Delta SA =
-11.53 J/K, which he derives from a supposed
Pf = 4 atm. The difference is just the -2.16 J/K he alleges to be "destroyed" in tank A. Need we now credit any such entropy destruction?
The 2.16 J/K discrepancy in Delta SA propagates itself
in the author's conclusion that Delta Suniv = +5.83 J/K, as
against the present finding that Delta Suniv =
Delta SA + Delta SB = +7.99 J/K. Whatever the numerical value, need we follow the author in attributing the rise in Suniv to a creation of entropy in the diathermal membrane that separates A from B? A more prosaic alternative view finds the increase in Suniv a simple consequence of the flow of heat from a hotter body A to a cooler B.
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