Today, most scientists and students of physics and engineering probably
believe that the heyday of thermodynamics was in the nineteenth century
with the work of Carnot, Clausius, Boltzmann, and Gibbs. Yet thermodynamics
has its twentieth-century heroes as well. In 1908 thermodynamics took
a giant step with the work of the German-born mathematician, Constantin
Carathéodory. Carathéodory developed
a proof that showed "entropy increase" is not so much the
general statement of the second law as it is its most fundamental observation—that
all natural phenomena are irreversible.
Unlike earlier definitions, Carathéodory's mathematically elegant
proof does not depend on the nature of the system, nor on the concepts
of entropy or temperature. And although it can be stated poetically,
Carathéodory's assertion (as it is called) is mathematically formidable
(Carathéodory 1976).
Until the work of Carathéodory, proofs of the second law were
based on thought and physical experiments associated with a perfect Carnot
engine, or on statistical mechanics. Although rigorous, their scope was
limited. Carathéodory's generalization focuses on the irreversibility
of thermodynamic processes, rather than on entropy increases per se.
By doing so, it gets around the difficult problem of measuring entropy
or entropy production in nonequilibrium situations. Remember that entropy
or entropy production can be calculated only in systems that are at,
or will go to, equilibrium. This ability to shake off the tight shackles
around the measurements of entropy gives us greater theoretical clarity
to explore thermodynamic systems further.
An irreversible process can be understood as a series of internal constraints
removed one by one until a system comes to equilibrium. The constraints
can be mechanical, for example, a series of doors that separate a system
into compartments. Imagine a box with four compartments with doors that
open and close . One of these compartments holds 10,000 molecules of a
gas, and the other three compartments contain nothing; they hold a vacuum.
What we have then is a gradient or potential gradient blocked by the doors.
Upon opening the first door, or constraint, the gas will spread into the
next compartment. This part of the system will reach a local equilibrium
of roughly 5,000 molecules in each of two boxes, with no perceivable gradient
between them. The process repeats as we open the remaining doors until
2,500 molecules or so settle in each of the four boxes. Each time a constraint
is removed, that part of the system approaches equilibrium. Once the whole
system comes to equilibrium, you cannot determine the order in which the
doors between the compartments were opened. Now if we replace the doors
with pistons, work could be extracted from the system along the way to
equilibrium. These principles of erasure of the path, or past, as work
is produced on the way to equilibrium hold for a broad class of thermodynamic
systems, from chemical kinetic reactions to a hot cup of tea reaching the
temperature of the room….
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