Treatment of Chemical Equilibrium without Using Thermodynamics or Statistical Mechanics P. G. Nelson The University, Hull HUB 7RX, England
Most textbooks of advanced physical chemistry treat the subject of chemical equilibrium in the same way. Two expositions are given: one based on classical thermodynamics, showing how equilibrium constants can be calculated from thermal data; and the other based on statistical mechanics, in which equilibrium constants are derived from spectroscopic data. This approach has much to commend it. In the first place, it introduces students to thermodvnamics. one of the "ereat branches of natural science", as iewis and Randall (17 described it, standing out "by reason of the variety of far reaching deductions drawn from a small number of primary postulates". Secondly, it also introduces them to statistical mechanics, a branch of science no less impressive for its economy and power. Thirdly, i t keeps the two approaches separate, so that the power of the one tomake predictions by considering phenomena more or less exclusively a t the bulk level, and the power of the other to make predictions by working more or less exclusively a t the molecular level, are brought out as clearly as they can be. The conventional approaches to chemical equilibrium do, however, have their drawbacks. If i t is a strength of the thermodynamic approach that i t treats phenomena more or less exclusively a t the bulk level, it is also a weakness. I t means that the student of chemistry finds him- or herself no longer thinking in terms of atoms and molecules hut of concepts that are less familiar to him or her, like heat engines and thermal efficiency (or the more abstract ideas of a treatment stripped of these). Furthermore, thermodynamics is difficult. The conceptual problems raised by the second law are considerable, and the logical trail from the three laws to the equilibrium constant is a long one. I t may all he very good for the souls of the strong, hut for the majority i t is a burden they find heavy to bear. The statistical mechanical approach is no less troublesome. It, too, brings in ideas that are strange to the student-not least the idea of an ensemble. I t is also difficult; the logical path from axioms to conclusions is again a long one. A further problem, not always mentioned in textbooks, is that the statistical mechanical treatment of thermodynamic systems raises some very difficult questions to which i t is not yet possible to give completely convincing answers ( 2 ) .In the words of de Broglie (3, written in 1963, hut still applicable), "Although nobody is in doubt today of the validity of the remarkable interpretation of Thermodynamics with which Statistical Mechanics, following the efforts of Boltzmann and Gibbs, has recently provided us, it still remains extremely difficult t o give a completely accurate justification for it!' Alternative Appmach An alternative approach to the treatment of chemical equilibrium is t o take as the starting point Boltzmann's distribution law. Such an approach was discussed a t a meeting of the Low-Temperature Group of the Physical Society
'
It Is assumed for simplicity that molecules can have zero translational energy, so that Ar, is independent of volume.
852
Journal of Chemical Education
of Great Britain in 1946 ( 4 ) and taken up by Guggenheim in a booklet first puhlished in 1955 (5).A more complete treatment has recently been developed by the present author (@, a sketch of which is given below. Consider the equilibrium (1) PQ(d =Pfd + Q(d If the mixture of gases behaves ideally, the equilibrium constant for this reaction is given by
K
= (PPIP%)@QIP%)I(PPQ~P%) = PPP~IPPQP*
(2)
where p x (X = P , Q, PQ) is the partial pressure of X and p e = 1 atm or lo5 Pa. Application of Boltzmann's distribution law t o this equilibrium gives
where is the number of molecules of type X present a t equilibrium, qx is the partition function of X, A q is the difference in energy between P + Q and PQ when each species is in its lowest energy state,' k is the Boltzmann constant, and T the absolute temperature. The partition functions are defined by
where ri is the energy of the species concerned in its ith energy state. From eq 3 the equilibrium constant of reaction 1is given by
K = (q*m , * lL*~ Q ) ~-A'dkT ~ ~ m%. . P
(5)
qEx
where is the partition function for one mole of pure X at 1atm or lo5 Pa, and L is the Avogadro constant. Equation 5 can be used as i t stands for the evaluation of K from spectroscopic data (7). Alternatively, the quantities in it may be derived from thermal data as follows. From Boltzmann's law the internal energy of a pure ideal gas is given by
where N is the number of molecules. From eq 4 this is identical t o The corresponding equation for the enthalpy is Integration of this by parts gives Nk in [q(T")lq(T")l= -I[H(T) - H(O)IIT- S(T)IIy
(9)
withp constant. Here S is defined by (as), = (aH),IT. The quantity corresponding to (8H)dT when eq 7 is integrated is (8U)vIT. These can be brought together by making the general definition of S the familiar thermodynamic one
where 66, is the heat absorbed by a system in an infinitesimal reversible change. Equation 9 enables values of q to be obtained from thermal data once one value is known a t the pressure of interest. Although there are no obvious values of q for gases, the corresponding quantity for a crystal has a predictable value at T = 0. This quantity is q = N@
(crystal)
(11)
where Q is the partition function for the crystal as a whole. For a perfect crystal a t T = 0, Q = 1and q = 1.By applying Boltzmann's law to the appropriate phase equilibria, and using eq 9 for each phase, q(T) for the gas (g) can be related to q(0) for the crystal (c) to give
Nk In lq(g,nlNl
= -[(H(g,n
- H(g,O)lIT+ S(g,n
(12)
(witbp constant) where
Since H(g,O) = U(g,O) = N E Osubstitution ,~ of eq 12 into eq 5 gives the familiar equation -RT In K = e ( T ) - TASz(:O = AG:(TJ
= -dnB (nx = amount of X) this is equivalent to (~G),,,T
