First part lost
Processes like these are
called "reversible" processes. Remember, reversible processes are postulated
to simplify the calculation of the entropy change in a system; it is NOT
necessary that they be capable of being achieved experimentally.

It should not be assumed that equation (1) requires that q, the heat
absorbed, must necessarily be absorbed reversibly. The concept of
reversibility is merely a means to an end: the calculation of entropy change
accompanying an irreversible process.

The following example will illustrate the calculation of a reversible
restoring process and at the same time develop the equation which is the
basis for the thermodynamical relationship between probability and the
second law.

We will postulate a system consisting of an "ideal" gas contained in a tank
connected to a second tank that has been completely evacuated, with the valve
between the two tanks closed. The temperature of the system and its
surroundings is postulated to be the same.  An ideal gas is one whose
molecules are infinitely small and have no attractive or repulsive forces on
each other. (Under ordinary conditions hydrogen and helium closely approximate
the properties of an ideal gas.)

An ideal gas is chosen in order to develop the basic relationship without
introducing complicating correction factors to account for the size of the
molecules and the forces they exert on each other.

When the valve is opened the gas expands irreversibly from V1 (its original
volume) to V2 (the volume of both tanks). There is no work of compression by
or upon the surroundings. Because the gas is ideal there is no temperature
change, and hence no heat flow takes place.

After expanding irreversibly from V1 to V2, the gas is restored to its
original condition by reversibly compressing it back to V1. This compression
requires work (force applied through a distance) which in turn generates heat
in the gas, heat that is absorbed by the surroundings so that there is no
increase in the gas temperature. In our mathematical model of this reversible
restoring process, the surroundings are postulated to be so large that they
also do not undergo any temperature increase. The temperature T remains 
unchanged during the entire irreversible expansion and subsequent reversible
restoration process.

The work of compressing the gas during restoration is equal to the pressure
of the gas times the volume change due to compression. Because the pressure
increases during compression, the work of compression must be determined by
the calculus integral:

   compression work = |PdV
                            where: P = pressure
                                    V = volume
                                   dV = the small change in volume taking 
                                        place at the corresponding pressure P
                                        
The integral sign | indicates the summation of all the individual values of
PdV.              

The equation relating temperature, pressure, and volume of an ideal gas is:

   PV = RT    (2)            where: P = pressure
                                    V = volume
                                    T = absolute temperature
                                    R = a constant which depends only on the
                                        amount of gas present
                                   
In the case of a reversible, isothermal compression of an ideal gas we may
substitute P from equation (2) into the equation for compression work. When
this is done, we have:

   compression work = RT|dV/V    (3)
                        

Although it is not necessary that our postulated reversible restoration
process be capable of being carried out in a practical sense, it is
nevertheless sometimes helpful to be able to visualize the process. To this
end, the reader may consider the restoring compression process being brought
about by a piston fitted into the end of the second tank. On compression from
V2 to V1, the piston moves down the length of the second tank, and with no 
mechanical friction forces all the gas contained therein back into the first
tank V1.

Since the work of compression is equal to q, the heat absorbed by the 
surroundings, q may be substituted in equation (3) to give:

   q = RT|dV/V    (4)
         

From equation (1) the entropy gained by the surroundings during restoration
from V2 to V1 is:

   #S = q/T     (1)

Substituting from equation (4):

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

Upon integrating (a calculus procedure for summing up the individual values
of dV/V) we have:

   #S = Rln(V2/V1)      (5)     

Where ln(V2/V1) is the natural logarithm of the ratio of expanded volume to
the initial volume, and #S is equal to the entropy increase in the 
surroundings upon restoration compression from V2 to V1. As we have seen, #S
is also equal to the entropy increase of the gas caused by its original
expansion from V1 to V2. This is because V1 is the same volume both before
expansion and after restoration compression, and therefore has the same 
entropy content. Therefore the entropy transferred to the surroundings during
restoration is equal to that gained by the system in expanding from V1 to V2.

                          ENTROPY AND PROBABILITY

The ratio of the probability that all the gas molecules are evenly 
distributed between the two tanks to the probability that all the molecules,
of their own accord and by random motion, would be in tank V1 is equal to 
(V2/V1)(exp)N, where N is the number of molecules.

(If V2/V1 were equal to 2.0, for example, and N were equal to 10, the
probability ratio would be 2 to the tenth power, or 1024. For N = 100, the
ratio would be approximately 1.27 times ten to the 30th power. It is clear
that the random motion of trillions of gas molecules heavily favors a 
uniform distribution.)
    
    let X1 = the probability of all the gas molecules, after the valve is
             opened, remaining in the first tank V1

    let X2 = the probability of all the gas molecules, after the valve is
             opened, being uniformly distributed in V2, the volume of both
             tanks.

From the probability equation:

   X2/X1 = (V2/V1)(exp)N

Taking the natural logarithm of both sides, and then multiplying both side
by R, the gas constant:

   R ln(X2/X1)   = RN ln(V2/V1)
   R/N ln(X2/X1) = R ln(V2/V1)

Substituting in equation (5):

   #S = R/N ln(X2/X1)       (6)

Equation (6) represents the fundamental relationship between probability and
the second law of thermodynamics. It states that the entropy of a gaseous
system increases when its molecular distribution changes from a lower
probability to a higher probability (X2 greater than X1).

Based on the belief that the laws of thermodynamics are universal, this
equation has been assumed to apply to all systems, not just gaseous.
In other words, any entropy change is proportional to the logarithm of the
ratio of probabilities. Therefore, for the general case equation (6) can
be written:

   #S = K ln(X2/X1)          (7)

Where K is a constant depending on the particular change involved.

However, individual values of K, X1, or X2 are seldom, if ever, known for
non-gaseous systems.

As we have seen before, #S can be either positive or negative. When #S
is negative equation (7) can be written:

   -#S = -K ln(X2/X1)
       = K ln(X1/X2)

Therefore a system can go from a more probable state (X2) to a less probable
state (X1), providing #S for the system is negative.

In cases where the system interacts with its surroundings, #S can be negative
PROVIDING the over-all entropy of the system and its interacting surroundings
is positive; the over-all change can be positive if the entropy increase of
the surroundings is numerically greater than the entropy decrease of the
system. 

In the case of the formation of the complex molecules characteristic of
living organisms, creationists raise the point that when living things
decay after death, the process of decay takes place with an increase in
entropy. They also point out, correctly, that a spontaneous change in a 
system takes place with a high degree of probability. They fail to realize,
however, that probability is relative, and a spontaneous change in a system
can be reversed, providing the system interacts with its surroundings in such
a manner that the entropy increase in the surroundings is more than enough
to reverse the system's original entropy increase.

The application of energy can reverse a spontaneous, thermodynamically
==========

"irreversible" reaction. Leaves will spontaneously burn (combine with oxygen)
to form water and carbon dioxide. The sun's energy, through the process of
photosynthesis, will produce leaves from water vapor and carbon dioxide, and
form oxygen.

If we unplug a refrigerator, heat will flow to the interior from the 
surroundings; the entropy increase inside the refrigerator will be greater
than the entropy decrease in the surroundings, and the net entropy change
is positive. If we plug it in, this spontaneous "irreversible" change is
reversed. Due to the input of electrical energy to the compressor, the
heat transferred to the surroundings from the condenser coils is greater
than the heat extracted from the refrigerator, and the entropy increase of
the surroundings is greater than the entropy decrease of the interior, in
spite of the fact that the surroundings are at a higher temperature. Here
again, the net entropy change is positive, as would be expected for any
spontaneous process.

In a similar manner, the application of electrical energy can reverse the
spontaneous reaction of hydrogen and oxygen to form water: when a current
is passed through a water solution, hydrogen is liberated at one electrode,
oxygen at the other.

As can easily be confirmed experimentally, agitating water raises its
temperature. When water falls freely from a higher elevation to a lower
elevation, its energy is changed from potential to kinetic, and finally
to heat as it splashes at the end of its fall. The second law of thermo-
dynamics states that the water will not spontaneously raise itself to its
original elevation using the heat produced on splashing as the sole source
of energy. To do so would require a heat engine that would convert all of
the heat of splashing to mechanical energy.

The efficiency of a heat engine is thermodynamically limited by the Carnot
cycle, which limits the efficiency of any heat engine to #T/T, where
#T is the temperature increase due to splashing, and T is the absolute
temperature. Since #T is only a small fraction of T, there is no device that
could be constructed which would allow all of the water to spontaneously jump
back to its former elevation.

We can, at least in theory, calculate the entropy increase of the water
resulting from its irreversible change in falling. In a manner analogous to
that used in the previous example, the entropy increase would be equal to
the heat generated by splashing agitation, divided by the absolute
temperature. If some of the energy of the falling water is extracted by a
water wheel, there will be less heat of splashing and hence less entropy
increase.
 
A properly designed turbine could extract most of the water's kinetic energy.
This is NOT the same thing as trying to utilize the heat of splashing as
an energy source for a heat engine to raise the water. In other words, using
the energy before it becomes heat is much more efficient than trying to use 
it after it becomes heat.

If a water wheel is connected by shafts, belts, pulleys, etc. to a pump, the
pump can raise water from the downstream side of the water wheel to an
elevation even higher than that of the upstream reservoir. SOME of the water
would spontaneously raise itself to an elevation even higher than original,
but the rest of it would end up below the water wheel on the downstream side.

While it is not possible for ALL of the water to raise itself to an elevation
higher than its initial elevation, it is possible for SOME of the water to
spontaneously raise itself to an elevation higher than initial. 

As with any other irreversible change, there will be an increase in over-all
entropy. This means that the entropy increase of the water going over
the wheel is greater than the entropy decrease of water pumped up to the
higher elevation. 

This will be confirmed mathematically in the following paragraphs. As before,
in order to avoid problems with special characters not reproducing properly,
& will be substituted for the Greek letter gamma, representing unit weight
in pounds per cubic foot; # will be substituted for the Greek letter delta,
representing an increase in the value of a parameter.
       
       Let: & = unit weight of water, lbs./cubic foot
            h = height of reservoir above downstream side, feet
           &h = potential energy of water in reservoir
           #h = additional height above reservoir (to which water is pumped)
       h + #h = height to which water is pumped, feet above downstream side
            w = work of pumping water to higher elevation
            q = heat loss due to pump friction and downstream agitation
            f = fraction of water pumped to elevation h + #h


From the flow equation, energy in = energy out:

      &h = q + w
       q = T#S           (equation 1)
       w = f&(h + #h)
      &h = T#S + f&(h + #h)        
     T#S = &h - f&(h + #h)          (8)

In the case where the water falls freely without turning the water wheel or
operating the pump:

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


  let q'= heat of splashing
      q'= T#S', where #S' is the entropy increase on free fall
    
Since, in free fall, all the potential energy is converted to heat of
splashing:        
    
     &h = T#S'           (9)

Combining equations (8) and (9):

     T#S' - T#S = &h - [&h - f&(h + #h)]
     #S' - #S = f&(h + #h)/T              (10)

Equation (10) shows that #S' is larger than #S, and that the entropy
increase due to pump friction and downstream agitation is "backed up" by
the even larger entropy increase that takes place when water falls freely.

Equation (10) also shows that the lower the value of #S, the more efficient
the pump, and the greater the value of f, the fraction of water pumped.

Creationists assume that a change characterized by a decrease in entropy
can not occur under any circumstances. In fact, spontaneous entropy
decreases can, and do, occur all the time, providing sufficient energy
is available.

The fact that the water wheel and pump are man-built contraptions has no 
bearing on the case: thermodynamics does not concern itself with the detailed
description of a system; it deals only with the relationship between initial
and final states of a given system (in this case, the water wheel and pump).

A favorite argument of creationists is that the laws of thermodynamics would
not allow a junkyard to spontaneously become an airplane. Nevertheless,
there is nothing in thermodynamics that would prohibit the formation of said
airplane by an array of incredibly sophisticated robotic machines that would
require nothing more than the spontaneous flow of electrical energy. There is
also nothing in thermodynamics that would prohibit the manufacture of said
robotic machines from the energy of other machines. All that is really 
essential, as far as thermodynamics is concerned, is a source of sufficient 
energy.

Considering the earth as a system, any change that is accompanied by an 
entropy decrease (and hence going back from higher probability to lower 
probability) is possible as long as sufficient energy is available. The 
ultimate source of most of that energy, is of course, the sun.

The numerical calculation of entropy changes accompanying physical and
chemical changes are very well understood and are the basis of the
mathematical determination of free energy, emf characteristics of voltaic
cells, equilibrium constants, refrigeration cycles, steam turbine operating
parameters, and a host of other parameters. The creationist position would
necessarily discard the entire mathematical framework of thermodynamics and
would provide no basis for the engineering design of turbines, refrigeration
units, industrial pumps, etc. It would do away with the well-developed
mathematical relationships of physical chemistry, including the effect of
temperature and pressure on equilibrium constants and phase changes.

Copyright 1995 Frank Steiger. This document may be reproduced without royalty
for non-profit, non-commercial use.

============
: [1] Note that Gans used the term "isolated system", and used "closed
:     system" to mean (as I recall) simply no exchange of mass.  There
:     is a difference in terminology in the way thermodynamics is taught
:     to physics students (or perhaps "by physics professors") and to
:     chemistry students (or perhaps "by chemistry professors").  The point
:     is, whatever term you use for it, if the relevant system has energy
:     or mass added or removed by any process, the 2nd law doesn't apply
:     to that system (though it may apply to a containing system).  By which
:     I mean that the entropy of that system *can* *decrease*, for example,
:     energy entering a still can separate salt water into fresh water
:     plus brine, which is a decrease of entropy.

: [2] From the Oxford Concise Science Dictionary, "the entropy of a closed
:     system increases with time".  This, and the surrounding slightly more
:     formal definitions make it crystal clear that the entropy that must
:     increase is defined over the system as a whole, and is a property of
:     the system as a whole.  The 2nd law makes no statement whatsoever
:     that a subsystem, or an unclosed system, cannot decrease in entropy.
:     In fact, the statement of the law as delta(U)=T*delta(S)-W shows that
:     the entropy of a system can decrease, as long as delta(U) is nonzero.

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