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Re: entropy - motivation for definition



Mark Sylvester wrote:
In Classical Thermodynamics, what is the motivation for the
definition of entropy as Q/T?

First a philosophical point which is more important than
it may at first appear: I would rather say we _calculate_
S in terms of Q/T rather than "defining" S as Q/T.
Remember that entropy is primary and fundamental.
Everything has an entropy, while not everything has
a temperature ... which is a sufficient reason for
deciding that Q/T is unacceptable as a definition of S.

So let me answer the modified question, how do we motivate
the correspondence between S and Q/T (when the latter exists)?

The usual motivation involves a "totem pole" of heat engines
stacked on top of one another, as shown in
http://www.av8n.com/physics/img48/heat-totem.png

Entropy is primary and fundamental. A certain amount of
entropy is withdrawn from the upper heat reservoir, then
flows through each of the heat engines, and finally is
discharged into the lower heat reservoir. No entropy
is lost during the process, because entropy cannot be
destroyed, and no entropy (just work) flows out through
the horizontal arrows. No entropy is created, because
we are assuming the heat engines are 100% reversible.

You can measure the energy flowing through each junction
in the totem pole, for instance Q23. Now ... we simply
_define the temperature_ at this point to be
T23 := S / Q23

This is called the thermodynamic definition of temperature.

To finish the job, you need to show that the notion of
temperature defined thereby has the properties you want
a temperature to have, notably that two objects are in
thermal equilibrium with each other if and only if they
have the same temperature. To do this, rig up two
totem-poles side by side, and consider two points that
have the same S/Q value. If they did not have the same
temperature, you could rig up a heat engine between
the two points, creating a perpetual motion machine.

So you see it is T that is defined in terms of S, not
vice versa.

For more on this, see Feynman volume I chapter 44. His
totem pole is figure 44-8.

====================

This is the correct physics ... but to make it work as
good pedagogy requires laying the proper foundation.
Remember: learning proceeds from the known to the
unknown. Students have 'some' prior notion of what
temperature is, and typically not the slightest prior
notion of what entropy is, so it is necessary to spend
some time explaining what entropy is before trying to
connect it to temperature.

Entropy is defined in terms of probability. It can
be taught using examples such as a tray full of coins,
which has a quite direct connection to the spin-entropy
of a spin-1/2 system; both have 1 bit of entropy per
particle. For details on this, see
http://www.av8n.com/physics/thermo-laws.htm#sec-entropy

Another illustrative application of the ideas of entropy
is the twelve-coins puzzle,
http://www.av8n.com/physics/twelve-coins.htm

Once the customers understand what entropy is, you can
define temperature in terms of entropy.