CHM 5414 Lecture Notes
17 October 1996

God plays dice if not with the Universe then at least with Thermodynamics. But Thermodynamics governs the Universe! So our most probable microstate gives rise to energy state populations and partition functions which we'll show here not only yield thermochemical equilibrium but also speak to pedestrian concepts like heat and work. "Maximization of statistical entropy: it's not just a bright idea; it's the (2nd) Law."

Energy, entropy, work function...piffle. What we as chemists really want is the Gibb's Free Energy so we can predict the (thermodynamic) outcome of chemical reactions. Well, if we weren't so picky, we'd already have it. Our physicist colleagues have been using A, the work function, for this purpose. While G points at equilibrium (G ° = 0) at constant pressure, A ° = 0 points at equilibrium at constant volume. Cold comfort for a gas reaction at atmospheric pressure...which is why we yearn for G. OK...we shall have it.

dE = q_{rev, in} + w_on = TdS - PdV

H = E + PV so that

dH = dE + PdV + VdP = TdS - PdV + PdV + VdP = TdS + VdP
measures reversible q transfer when dP=0, a great convenience.

and G = H - TS so that

dG = dH - TdS - SdT = TdS + Vdp - TdS - SdT = VdP - SdT
has fixed P and T as its experimental handles, a great convenience.

We'll return to these total differentials (State Functions) in a moment, but right now it's the function definitions themselves which interest.

Since the physicists want to control equilibrium at fixed T and V, they would naturally prefer

A = E - TS so that

dA = dE - TdS - SdT = TdS - PdV - TdS - SdT = - SdT - PdV.

But if A = E - TS and E = H - PV and H = G + TS then simple substitution shows us

A = G - PV or G = A + PV.

That's what I really wanted all along because I have simple statistical mechanical expressions for both A and P. Indeed, harkening back a lecture,

A = - kT ln z^N = - nRT ln z

and P = nRT/V ... well, of course that's just the ideal gas equation which evolved from that "independent particle-in-the-box" z_trans. If we turned on non-idealities, P would've had a different form. Let's let sleeping dogs lie on this one, and accept PV=nRT.

So that G = A + PV becomes

G = - nRT ln z + nRT = - nRT [ ln(z) - 1 ] = - nRT [ ln(z) - ln(e) ]

or

G = - nRT ln (z/e) = - RT ln (z/e)ⁿ

which is, mercifully, as simple as the expression for A; indeed it differs from it only by the denominator e, the natural number, 2.7182818... (aren't calculators a blessing?).

One of the satisfying features to point out is the continued dependence of these extensive variables (additive) like G upon the logarithm of the (multiplicative) partition functions. Even the power ⁿ implies multiplicative partition functions due to moles of material becoming additive components of G.

Thus when we relate the Gibbs Free Energy of the many molecules in a reaction:

a A + b B c C + d D,

the stoichiometric coefficients, = a,b,c,d replace the power ⁿ as moles of material in the reaction as written. Thus

G° = _prod G°_prod - _react G°_react
or
G° = - RT [ _prod ln(z/e)_prod - _react ln(z/e)_react ]
or
G° = - RT [ ln(z/e)_prod - ln(z/e)_react ]
or
G° = - RT ln [ (z/e)_prod / (z/e)_react ]

All of this is recalls the defining moment of 1st semester PChem when it was revealed that G° = - RT ln K_eq, where K_eq is the equilibrium constant of the reaction. Thus, we can use with confidence the intuitive expression:

K_eq = (z/e)_prod / (z/e)_react

It's intuitive because (gee, if it's intuitive, I shouldn't have to say why!) it tells us that compounds with more states available to them are going to shift equilibrium their way. Why are we not surprised?

Only a couple of caveats to toss out here: the first is to remind you that although z is a product of z_trans z_rot z_vib z_el, there's still only one factor of e in that z/e expression...rather like there's only one factor of 1/N! in the indistinguishable Z = z^N/N!.

The second caveat has to do with z's all being measured from the same origin. That leads to a factor of exp(-_o/kT) appearing before every z _vib. But since each z _vib appears with its own stoichiometric coefficient as a power, z_vib, the zeropoint correction factor becomes
exp( - _o / kT)
and the z product and quotients in K_eq gather those terms into
exp( - D_o / kT )
where
D_o = _o(products) - _o(reactants)
(assuming we measure _o as molecular zeropoint energies. If we used molar zeropoint energies instead, the kT becomes RT. (Try not to be off by a factor of N_Av ; it tends to get noticed.)

Seque to Traditional Thermo

So there's Good News and Bad News.

The Good News is that we're moving now to things less jarringly unfamiliar. The Bad News is that we're moving to topics too boringly familiar. Win a few; lose a few.

The thrust of this course (somewhat tardy to mention it here, no?) is to leave you as comfortable making intuitive conclusions about the world of chemistry using both your statistical mechanical understanding of the role of availability of states (when that's informative) and your knowledge of the macroscopic relationships of thermodynamical variables and their applications. In other words, we want you to be "switch-hitters," capable of analysing a problem from both the microscopic and the macroscopic view, emphasizing the one most instructive to the problem at hand.

So having explored the one, we now turn to the other, unafraid to have either view inform the discussion of the other! For example, we'll speak of heating as the population of higher energy levels and of work as the retention of populations in levels which separate from one (another along the energy axis). In either case, the average energy rises, but they're due to fundamentally different causes, most easily seen in microscopic arguments.

But we really "see" them through our observation of thermometers and physical compressions of heat/work experiments. Being a theoretical course, we take on faith those experiments and draw on the collective wisdom of their collectors.

So we "know" we can raise the energy of a system by flooding heat into it, +q_{rev., in}, and by working on it, +w_on. It may come of something of a surprise that work doesn't always have the same "sense;" that is, collapse doesn't always increase the energy as it does in a gaseous system. For example,

Gravitational work = _h1^h2 mg dh = mg h
is an expansive work; we increase the energy of systems (rockets, say) by "expanding" them away from the gravitational center. This work is the result of one-dimensional motion in opposition to a one-dimensional force, mass x gravitational acceleration. (We're trusting that h isn't so large that g starts falling off...as distance squared from Earth's center.)

The same thing is observed in two-dimensional work such as that done stretching a balloon or in filling a glass of water slightly more than its capacity. In the latter case, one we'll explore later in the course, water's surface tension, , acts to increase the energy of surface under expansion,

Surface work = _a1^a2 da = a

In contrast, three-dimensional expansion against external pressure is the way to decrease the energy of a gaseous system. To increase the system energy, one compresses it, contracting the volume:

Pressure/Volume work = _V1^V2 - P dV

Also, in contrast to the near constancy of mg and , Pressure may well vary during the compression (or expansion). For example, the familiar case of isothermal expansion of an ideal gas requires substitution of P by nRT/V where the constants nRT come out of the integral, rendering

Pressure/Volume work = nRT _V1^V2 - (1/V) dV = - nRT _V1^V2 d ln(V) = nRT ln (V1/V2)

(We'll discover that there are "ideal surface films" as well as ideal gases.)

We've fixated on the work half of the energy change expression but only identified a paltry 3 kinds of work; there are as many kinds as there are forces against or along which to contend. But the other half, heat, we've visited earlier in this course. Then we identified reversible heat, at least, with our fundamental thermodynamic variable, entropy, but for a temperature scaling,

dS = q_reversible / T

which allows us to complete the fundamental expression of thermodynamics as

dE = TdS - PdV.

In PChem, we recall that while neither heat, TdS, nor work, -PdV, were State Variables, T, S, P, V, and E were! These individual quantities were completely characterized by the state of a thermodynamic system but not dependent at all on the path to that system from any other (including itself). However, different paths between two states of the system (or away and back to the same one) would involve varying heat transfers and work done on the system. (Were work and heat transfers independent of path, cyclic engines could generate no overall work.)

Mathematicians would speak of dE as a perfect differential rather than as a state function, but the property is the same. They would remind us that any perfect differential like dE can be evaluated (in the limit of infinitesimal changes) as

dE = (E/S)_VdS + (E/V)_SdV

where the crux of the differential's "perfection" lay in the Cauchy-Riemann conditions (which we chemists call "Maxwell's Relations") which require that

[(E/S)_V / V]_S = ²E/SV = [(E/V)_S / S ]_V

Well, that isn't the way Maxwell would have you read it; instead from the perfect differential expression, the partial differentials are easily identified as

(E/S)_V = T
and
(E/V)_S = - P

So the Cauchy-Riemann condition above becomes the first Maxwell Relation as

(T/V)_S = - (P/S)_V

admittedly not terrifically illuminating. But wait; there's more! Indeed, there is one Maxwell relation for each perfect differential we can write: dE, dH, dG, and dA so far. Some of them are more useful than others, but all are easily found...and inverted for that matter!

You should be able to show with no difficulty from dG=VdP-SdT that the isothermal sensitivity of entropy to pressure change can be found from the equation of state (the gas law, if the system is a gas) as

(S/P)_T = - (V/T)_P

not a result divined as easily any other way.

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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