Tuesday, October 23, 2007

The Direction of Spontaneous Change

We began the first lecture of Week 6 with an examination of the inadequacy of the First Law to sufficiently describe thermodynamic events. For example, it does not preclude a penny, say, from absorbing thermal energy from a table and turning it into gravitational work (that is, springing up off the table). The Boltzmann formula will demonstrate that the probability of this event is nonzero but exceedingly small (so unimaginably improbable that perhaps we should call it impossible?) But the point is that, macroscopically, we see a definite directionality to energy transfer:
First Law - limits the magnitude of energy transfer
Second Law - limits the direction of energy transfer
Before discussing how it was first discovered, we first needed to correct some misconceptions about entropy, one of the most thoroughly mangled concepts in all of science:
Entropy is not equal to disorder, nor is it a measure of disorder (whatever that means scientifically).
One of the most common [bad] examples demonstrating the alleged relationship between entropy and disorder is a deck of cards. When we shuffle an "ordered" deck of cards, we always see it become "disordered". The problem with this language is that there is no way to quantify order, especially since every outcome is equally probable. We have simply defined A,2,3,4 .. Q,K of each suit as being the ordered state -- but that definition is arbitrary. Nature should not -- and does not -- depend on such human definitions. Other [bad] examples include blaming messy desks and cluttered rooms on this "law of entropy."

Entropy is a measure of the tendency of energy to disperse, rather than being localized.
When we connect it directly to the number of accessible microstates (ala Boltzmann) we will understand the probabilistic basis of entropy more fully.

Through his theoretical work on heat engine efficiency, the French engineer Sadi Carnot was our first thermodynamicist. His memoirs, lost for twenty years and posthumously rescued by college friend Benoit Clapeyron, inspired the work of Clausius and Thomson [Kelvin], both of whom essentially triggered the thermodynamic revolution. Carnot's greatest achievement was to demonstrate that heat flow could be harnessed and transmogrified into usable work by engines, but with a maximum efficiency less than 100%. Indeed, this maximum efficiency is dependent only on the reservoir temperatures and not on the material used in the engine, nor on the actual steps of each cycle. It is a thermodynamic limit imposed on us by nature, who has decreed that heat is a form of energy rather than a transferred substance.

[Note: In today's lecture, I believe that I inadvertently flipped the subscripts for the temperatures in the adiabatic formulas. Consult Engel-Reid for consistency]

Looking further at the results of Carnot we see, as Clausius did, a hidden state function, one that sums to zero as we go around a cycle. From this fact we can back out the relationship dS = dq/T [the Clausius equation], introduced at the end of the hour but forming the basis of Wednesday's lecture to come.

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