Entropy and Gibbs free energy, ΔG = ΔH - TΔS

S: Δ G/T = Δ H/T - Δ S. Whaddya mean, look like entropy change? Δ S is q/T.

P: Sure, but q is the transfer of thermal energy (that we often, too loosely, call "heat"). So isn’t ΔH really a "q", a thermal energy transfer? Also, ΔG/T has the form of an energy transfer/T just as that ΔH/T does. All ΔSs! Therefore, the Gibbs equation really is

        Now, if chemicals are mixed in the system (at constant T and, normally, constant P) and a reaction occurs, some thermal energy transfer ("heat"), q, takes place in the reaction. How much q? That's easy: q is the change in enthalpy; q = ΔH(system). (The sign of ΔH can be + or - , but we're just talking broadly and generally, so let's start simply with + ΔH.)

However, from the viewpoint of the surroundings, the sign of ΔH changes when thermal energy is transferred from the system and becomes absorbed by the surroundings, i.e., a + ΔH(system) when transferred to the surroundings becomes - ΔH(surroundings) -- (and, of course, a -ΔH(system) when transferred out of the system becomes ΔH(surroundings). As a general equation, simply to express that change of sign, here's (2):

                     -ΔH(surroundings) =ΔH(system)                                       (2)

                    -ΔH/T(surroundings) = ΔH/T(system)                               (3)

Then, since ΔS = q/T, and the only q in the surroundings right now is - ΔH, that means that - ΔH/T(surroundings) = - ΔS(surroundings). Therefore, inserting this result in equat. (3):

                    -ΔS(surroundings) = ΔH/T(system)                                  (4)  

or, changing signs merely so we have a + ΔS to work with in a moment,

Now, replacing DS(surroundings) in equat. (1) with -ΔH/T(system) as just justified by (5):

                   ΔS(universe) = - ΔH/T(system) + ΔS(system)                 (6)

ΔS(universe)?? Just to say that aloud seems like a really big mouthful -- and head-full! But remember that we started out originally by saying that the only reaction happening in the whole universe was the one in our constant T, constant P system. So any DS(universe) would be perfectly measured by what happens only in our system.

Let's see. To put it in the most general terms, the energy change that occurred in the reaction in the system -- and which entropy measures by ΔS = q/T -- has been spread out, some remaining in the system as ΔS and some being transferred to the surroundings and identifiable as the ΔH/T(surroundings). (It can be used for work, but right now we're looking at it either as though it was all dissipated as unavailable thermal energy or as though the work itself had resulted in the same amount of waste energy, i.e., as ΔS(surroundings).

But exactly what was the original energy change that came out of the reaction in our system? We don't know, but let's give it a symbol of ΔG(system). OK, since that was the only event that occurred in the universe, and because energy flow (thermal energy transfer, "heat") divided by T is entropy, ΔG/T(system) is equal to the total entropy change in the entire universe due to this reaction! ΔS(universe) = ΔG/T(system)

                    - ΔG/T(system) = - ΔH/T(system) + ΔS(system)

          Starting with that Equation (1), we have found the breadth of the concept we call entropy. Entropy change, in the classical "macrothermodynamics" we've been looking at, is the ratio-measure, q/T, of the driving force for every spontaneous chemical reaction in the universe. Even when we focus on a little chemical system in our lab on earth by means of the Gibbs equation, we see the driving force -- the tendency of energy formerly bound in reactants to be spread out in the products. From information in other areas of chemistry than thermodynamics, we now know that the ΔH of Gibbs primarily comes from the difference in electronic binding energy between products and reactants. Usually in exothermic reactions, the products have stronger bonds and so some of the greater binding energy in the less strongly bound reactants is released as heat. From the energetics of molecules (again, outside of macrothermodynamics) we know that this increased thermal energy is due to molecules moving more rapidly and colliding more forcefully with one another. When the resultant DH(system) gets to the surroundings, it can be used to boil water or run a steam engine or charge a battery. But if it isn’t used for doing any work, it is simply dissipated in the surroundings as DH/T(surroundings), i.e., as ΔS(surroundings). It becomes entropic "waste heat", no longer capable of causing change because it is at the same temperature as the surrondings (even though it may have raised the terperature of the surroundings and infinitesimal amount.)

        The Gibbs equation is only concerned with macrothermodynamics, with what we can measure in the lab rather than what is happening down there inside the molecules during chemical reactions. What IS entropy change from the viewpoint of a molecule? Boltzmann and "microthermodynamics" deal with that. (You've heard a lot about it in other sections of this Web site.) To begin with, depending on their temperature and solid/liquid/gas state, all molecules need a certain amount of energy to move and rotate and vibrate internally, in addition to the large amount of electronic energy in their bonds that holds them together. S(298^oK, formation) in your textbook tables -- that's really DS(from absolute zero, formation) because entropies are assumed to be 0 there -- measures all those kinds of energy "in a bundle". So TΔS in the Gibbs equation is the amount of energy a mol of product molecules needs to exist at a temperature T compared to one of reactant molecules. (Give them more energy and they scoot around faster because the added energy goes into their translational/rotational/ vibrational modes. Heat them up to a high enough temperature and you even boost them to an excited electronic state where bond breaking can occur.

        The confusion -- about "If it's enthalpy, how can that be entropy???" -- comes in because the BIG equation (1) is actually the ultimate scenario for energy transfer in any chemical change. (Or "physical change" too, but let's skip that in concentrating on Gibbs.) We may be able to sneak some work out of the ΔH when or as that energy transfer from system to surroundings occurs and that's fine. Perhaps we can use it for human purposes. Good. Go ahead and calculate how much w.

        So you now have a better picture of fundamentals and of the Gibbs equation. There is no "conflict" betweenΔS. They're really just different aspects of entropy change: ΔH will become entropy in the surroundings if it does no work, and TΔS just represents what little energy is left over to become dissipated (as increased entropy in the surroundings) after the ΔS(formation) of the product molecules has been taken care of.

S: Ho ho ho! Now I see better what that "crossover T" point in the problems I worked really means. At low temperatures many reactions, say the old one of water being broken into hydrogen and oxygen gas, just don't go to any appreciable extent. Why not? Well, you simply don't have enough energy around in the system to put it into any new hydrogen and oxygen molecules so they can exist with their own individual allotments of ΔS even in their minimal quantum levels. However, as you raise the temperature more and more from outside the system -- and that means more and more intense energy, not just more energy -- there comes a temp point where you have enough flowing into the system to give the H₂ and O₂ molecules adequate energy for them to exist. Then any temperature above that is more than they need: they show it by flying around faster and hitting each other harder and harder.

P: Hooray. Ya got it.