CHM 5414 Lecture Notes
31 October 1996

Even without the hint of the shovel and bucket, you would not mistake that sand pile as the spontaneous work of an ocean wave upon this beach. Waves dissolve sand castles; they don't build them. This artistic endeavor is the spontaneous work of a child. The beach has suffered a drastic loss of entropy by being order so; the child and indeed the rest of the Universe has supplied that entropy and more to so reduce the W (number of microstates) of sand.

Unfinished Business

While we've dwelt at length on the utility of Maxwell's Relations, they offer us only the four cross-derivative expressions involving several dependent, independent, and fixed variables. Sometimes we need alternative relations not immediately apparent from Maxwell's. Then the following three rules:

The obvious one is the Inverter:(X/Y)_Z = 1 / (Y/X)_Z but the others are more subtle
like (X/Y)_Z = - (X/Z)_Y (Z/Y)_X sort of chain-rule-like
or (F/X)_Z = (F/X)_Y + (F/Y)_X (Y/X)_Z a bit of left field.

May they bring you happiness and good fortune. :-)

The other missing feature from last lecture is the (again obvious) temperature dependence of entropy, expressed by a macroscopic change in that state function, S, as the temperature integral of its temperature derivative,

(S/T)_P = (1/T) (H/T)_P = C_P / T or
S = _T1^T₂ (C_P / T) dT which, if C_P reasonably T-independent (over T) means
S = C_PT1^T₂ (1/T) dT = C_P ln(T₂/T₁)
Just as integration over large T ranges may span phase changes in a H calculation, necessitating breaking the ranges and interposing H_transition, the same thing will happen to S:
S=_T1^T_trans C_{P,phase A}(1/T)dT + H_trans/T_trans + _Ttrans^T₂ C_{P,phase B}(1/T)dT

Finally, since C_P evaluations are dicey at low temperature, it's valuable to know that all heat capacities fall off toward T=0°K as aT ³ (due to phonon frequency distribution calculations), where a bears no relation to the empirical constant in the usual fitted heat capacity T dependence. This a can be fixed by setting aT ³ equal to the C_P measured at the lowest (reliable) T.

This means that S can be evaluated from S(0°K)=0.0 J/mol°K on up as far as C_P data is available (including H_transitions where appropriate). That S(0°K)=0.0 is an article of thermodynamic faith which stat mechers can take without flinching. At 0°K, in a perfect crystal, all the atoms are in their same ground state, making W=1. So it's unsurprising that S = k ln(W) = k ln(1) = 0.0

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Come to think of it, in Stat Mech, we were satisfied that systems always evolved toward total W becoming maximal (subject to energy constraints). So for systems in (thermal) contact, W = W₁W₂ and S = S₁+S₂ (law of the logarithm of products), and heat flowing, cooling the hotter while warming the colder, it does so in such a way that W grows until T₁=T₂. So S₁ may fall, but S₂ rises further.

T₁=T₂ stops the process since S=q_rev/T; so at equal T, q_rev exchange causes equal and opposite changes in the two S_i. Thus S_total=0.

But this maximization of global W tells us S_universe increases in spontaneous processes until further change cannot increase W, whereupon S_universe is maximal (subject to energy constraints).
S_universe > 0 is thus a reliable bellweather of spontaneity...as reliable as S=0 is of equilibrium.

In aerial dogfights, pilots are often defeated by unseen enemies as they fixate upon a target before them; it's called target fixation. (I sometimes fear that it operates in Thermo tests as well!) The point here is that we chemists fixate on the properties of our system not of the Universe! So we need a system parameter that is as reliable a spontaneity bellweather.

That's Free Energy, G! ...Why?

Because G_sys = + H_sys - T_sysS_sys.

No apparent "Universe" there unless we recognize that heat leaving the System must enter the Surroundings (which constitute the rest of the Universe). So if we didn't mind mixing our metaphors (shouldn't there be paraphors as well? Orthophors sound horrible),

G_sys = - H_surr - T_sysS_sys which at constant T means

G_sys / T = - H_surr / T - S_sys = - S_surr - S_sys = - S_universe < 0
points toward spontaneity. Ahhh...G is the system variable we need to ensure universal entropy maximization.

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Life would be peaches and cream if H<0 and S>0 since that would always ensure G<0. Unfortunately, H<0 implies a change toward more tightly-bound products with fewer motional degrees of freedom and a consequent S<0 not >0. Pity. Because if the absolute magnitude of H is small, that means that for T>H/S, entropy wins, and the process isn't spontaneous.

Conversely, for the reverse process, H>0 while S>0, entropy wins again for T>H/S, and the process is spontaneous.

All this warns us that reactions can be driven left and right with T if, as is usual, H and S are of the same sign. In other words, LeChatlier was right when he said that exothermic reactions become less spontaneous at higher T; the system reacts to absorb the environmental change.

dG = VdP - SdT

tells us how this bellweather (definition: a bellweather is an indicator of change) varies with its preferred variables. They're preferred, of course, because if they're fixed, dP=dT=0, G is unchanged? Well...almost. We have to specify that no chemistry occurs!

Under conditions of no chemistry, dn_i, now the number of molecules of type i, are all zero as well. Being an extensive variable, G must vary with the number of molecules. The molar free energy by which G scales is called the chemical potential and is given the Greek symbol mu, , and

dG = _i [ V_idP - S_idT + _idn_i ]
Indeed, this is what STANJAN is playing with to arrive at equilibria, {n_i}_eq.

Let's take the easiest case of change but no chemistry: phase change of a single component...fusion or vaporization, for example. If phase A and B are in equilibrium at some P and T, then G_A=G_B or G = 0. If we tweak the P and T so that they're different but equilibrium is re-established, then the differentials for G change in each phase must again be equal, dG_A=dG_B, or dG = 0.

Given G's sensitivities to dP and dT, that means
VdP - SdT = 0 or
(dT/dp)_dG=0 = V/S = V/(H/T) (Clapeyron) which means
(1/T)dT = (V/H) dP or integrating
ln(T₂/T₁) = (V/H) (P₂-P₁) usually written
T₂ = T₁ exp[ V P / H ]
which permits determination of the new transition temperature given a change in pressure (usually away from 1 atm).

In the special case of vaporization, V = V_{ideal gas} with little loss of generality (molar V_liquid is negligible). Thus V = RT/P, which makes the Clapeyron equation a tiny bit harder but a whole lot more useful:

(1/P)dP = ( H_vap/RT²) dT integrating to
ln(P₂/P₁) = - (H_vap/R) [ (1/T₂) - (1/T₁) ] (Clausius-Clapeyron) or
P₂ = P₁ exp[ - (H_vap/R) (1/T) ]
chronicling the change in vapor pressure with temperature.

Note that a plot of ln(P) vs (1/T) has a slope -H_vap/R.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
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Richardson, TX  75083-0688

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