On the Legendre Transformation and the Sackur-Tetrode Equation Carl W. David University of Connecticut, Storrs, CT 06268 I t is desirable t o make the manipulations of thermodynamics comprehensible. Specifically, the apparent conflict between the statement for ideal gases that 3 E=NkT 2
(1)
I n t e r c e p t
(where E is the energy, T is the absolute temperature, and N is the number of molecules or atoms in the system) and
(HI
-
>
should cause a modest amount of perplexity for beginning students. Where is the entropy (S) on the right-hand side of eq I? If the student concedes that the energy of an ideal gas does not depend on the volume, the student must still wonder how to implement the equation dE = TdS - pdV = TdS
V)
w
(3)
for the ideal gas. Even if the student is willing t o say that E truly is a function of T through S, since does the student then appreciate that in going from E to A (the Helmholtz free energy), (s)he has changed independent variable from the pair (S, V) to the pair (T,V). Itis true that sometextslshow how the Legendre transformation works in the abstract. formal sense. It is also true that appropriate examples of the Legendre trannforrnation' exist. Nevertheless. the best example of how the Leaendre transformation works, and how it relates the seemingiy conflicting statements above, remains the Sackur-Tetrode equation3, which, unfortunately, is not included in many elementary physical chemistry courses. Even if its derivation is omitted, the structure of the Sackur-Tetrode equation is illuminating to students and is worth introducing. The Legendre Transformation
In the case of the Legendre transformation, a picture really is worth a thousand words. The reason for this is that the relationship between a function and its variable, on one hand, and the function's derivative and the intercept of that derivative, on the other, are intimately related. That relation, shown in the figure, is as easy t o understand as the beginner's development of the form for the enthalpy (H): dH=dE+d@V) = dE+pdV+ Vdp = TdS - pdV + pdV + Vdp = TdS Vdp
+
(5)
If the intercept of the slope is H, and the slope itself is m, then the equation of that line is just
'
Beriy, R. S.; Rice. S. A,; Ross. J. PhysicalChernistrxWiley: New York, 1980; p 648. David, C. ht. J. Math. Sci. Technol. 1986, 17, 201-204. Desloge. E. A. Statistical Mechanics: Holt. Rinehan 8. Winston: New York, 1966: p 261. 876
Journal of Chemical Education
The constant temperature Legendre transformation of E(S. Y) into H(S, p) showing the slope (m) of E(S. W versus Vand the intercept of Uw tangent to the E(S, W versus Vcurve (M.
or, written another way, we have:
which is, obviously, The advantage of explicitly descrihing the transformation is in understanding that H = H ( S , p ) is true, vis-a-vis E = E(S, V). Even more helpful in understanding these transformations is an example. The Sackur-Tetrode Equation
One form of the Sackur-Tetrode equation for an ideal monatomic gas is:
i.e., S = S(E, V) and assorted atomic constants. For clarity, we rewrite this equation in the form S=N~~I(E~"V)+O+~
which allows us to solve for E(S, V) explicitly, i.e.,
(10)
which, surely, presents E as E(S, V)! This form begs us to take the partial derivatives of E with respect to V (at constant S), or with respect to S a t constant V. We obtain:
which is a well known result for ideal gases. Continuing along the same vein we have
E = E(S, I.? and E = (312)NkT are compatible statements. In the set of variables IS, T, p, and V] the form of the partial derivatives is simplest if one chooses them appropri-
ately. E = E(T, V) is an inappropriate choice, since the partial of E with respect to T does not lead to another variahle from the set. Choosing instead E = E(S, V )guarantees that the partial of E with S gives another element of the set, namely T. The variable pairs S a n d T, on the one hand, and p and V , on the other hand, are called cannonically conjugate variahles4. Posing thermodynamic equations in properly chosen pairs (one from each pair) results in equations whose information content is maximum and whose appropriate partial derivatives are themselves the canonically conjugate opposites.
Callen. H. B. Thermodynamics; Wiley: New York. 1961.
Volume 65
Number 10
October 1968
877
