Lecture Notes from CHM 1341
18 July 1996

Thermochemistry

Entropy and Free Energy

ENTROPY

Eggs don't unscramble. Sand castles do not arise spontaneously from pounding surf. When we see such things on film, we know that the film is running backwards.

This means that we have a finely developed sense of the direction of time's arrow. It points toward increased randomness and away from increased order.

But seemingly not always...a (highly disordered) gas can solidify to a (highly ordered) crystalline state if we but reduce its temperature. However, that's not ALL we've done. We had to run a refrigerator to cool the gas sufficiently.

The refrigerator consumed a concentrated (ordered) power source and generated waste (disordered) heat out its coils. To anyone who has stood next to a refrigerator on a hot day, it comes as no surprise that the disorder from the coils is significant.

Indeed, the refrigerator has created more disorder in the air of the room than there would have been simply leaving the original gas alone! So our instincts about "time's arrow" are indeed correct.

And quantifiable.

The measure of this instinct is called entropy, and its original definition, not surprisingly, involved heat flow...since heating a system makes it less orderly.

Last century's definition of entropy change was dS = q_rev / T.
The rev is an abbreviation for reversible and implies that the heat transfers should be so slow that we don't "shock" the system from equilibrium. It wouldn't do to have the heat dumped in such that only PART of the system got hot.

And the very definition starts to explain the way the world works. For example, take 2 blocks of metal, one hot and the other cold. Put them into contact and heat flows from the 1st to the 2nd until they're at the same T. Why?

If our dS expression above is valid, it says that dS for the hot block is NEGATIVE as it cools because heat is leaving; so q is negative. That corresponds to becoming more ordered.

Similarly, dS for the cold block increases as q is pumped INTO it. It becomes more disordered faster than the other is becoming ordered because it has a smaller T in its denominator.

So here it is...the 2nd Law of Thermodynamics...this equilibration towards an intermediate T occurs because greater disorder than order is created...the SUM of the dS is positive. The disorder of the universe always increases spontaneously!

Or "the universe is running down." It's not running out of energy (that would be a violation of the 1st Law) but rather the energy is becoming more diffuse and therefore less available to perform work...like the work of evolving sentient beings.

After all, except on OU weekend, perhaps, we're highly ordered collections of molecules which represent significantly less entropy than our decomposed remains, let us say. But we keep ourselves that way at the expense of our environment.

We decompose plant and animal tissues, increasing their entropy to keep ours low. The plants turn sunlight (a rich power source) into waste heat (disordered remains of sunlight) to constitute those tissues.

So we prosper by driving up the overall entropy of the universe quicker than the species or processes with which we compete! And that process "wins" which can most efficiently degrade an energy gradient. (Prigogine's Principle)

It's only a little disconcerting to discover that we're in ascendance because we rape the environment better than our competitors. It's kind of kin to discover our eyes face front...like every other predator's eyes do. (Pleasant dreams.)

A prey animal's eyes face sideways the better to see us predators coming. See? Chemistry fuels moral dilemmas of all sorts! Everything's philosophy.

Boltzmann gave us the most powerful way to think about entropy. He showed that it really just measures the number of ways a microscopic system can be found consistent with its macroscopic properties.

In other words, entropy is a measure of likelihood since the more possibilities for rearranging a system, the more systems will be found in that collection of rearrangments.

The trivial case is gas molecules in a box. What's to prevent all of them from spontaneously moving to one side? Nothing. It is merely phenomenonly UNlikely. Indeed, it's easy to count the ways.

The chance that one molecule is in the left half of the box is 1/2. The chance that the next one joins it is 1/2. The combined probability of finding them both there is (1/2)(1/2)=(1/4). What happens with Avogadro's Number of molecules?

                                                        NA  
The likelihood of all NA being in the left half is (1/2)   

    NA                          1.8x1023
or 2   to 1 against!  (That's 10        a number with (1/3)NA digits!)

In other words, the statistics of large numbers tells you such entropic violations cannot occur in the lifetime of the universe and far beyond!

Boltzmann was so (justifiably) proud of this gift to understanding that he had its expression engraved on his tombstone as his only epitaph: S = k lnW. W is the microstate count (example above) and k is (no surprise) Boltzmann's Constant.

So, unlike internal energy or enthalpy, S is easy to describe as value rather than as a difference. Indeed, the 3rd Law of Thermodyamics sets the origin for entropy: S is always zero at 0 degrees Kelvin.

At absolute zero temperature, there can be NO disorder. Everything's locked up in crystalline solids.

Well, almost everything. Imperfect crystals will have residual disorder; so the 3rd Law applies to making perfectly ordered solids at zero Kelvin.

But while you can calculate, tabulate, ruminate upon absolute entropies, it is still dS entropy changes which will be of most interest. They will point the way to spontaneous reaction...but only if we know the universe's dS not just the system's.

And we do!

While the tables (like 9.3 or C7 in the appendix) list system entropies, the surrounding's (rest of the universe's) entropy is easy to come by. The surroundings simply react to heat flow to/from the system!

So  dSUNIVERSE = dSSURROUNDINGS + dSSYSTEM  > 0 

                  - dHSYSTEM
               =  ----------- + dSSYSTEM    > 0
                      T

   TdSUNIVERSE =  - dHSYSTEM + TdSSYSTEM    > 0

 - TdSUNIVERSE = dH - TdS  < 0  (where the missing "SYSTEM" is understood)

This development points to a new state function called Free Energy and defined as the lefthand side of the last equation and symbolized by the dG given in the tables. Since it is (T scaled) NEGATIVE univeral dS, it must decrease spontaneously.

dG° = dH° - TdS° < 0 (for spontaneous reaction) where we've returned the ° to signify standard conditions, 1 atm and 298 K.

We can now use dG_f° just as we would dH_f° to determine spontaneous reaction.

And more! For the -TdS component quantifies the "wasted" energy in any reaction, leaving dG to tell one the amount of reaction enthalpy which can be harnassed to do useful work.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688

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