The Legendre Transformation and Spreadsheets Carl W. David University of Connecticut, U-60 Rm. 161,215 Glenbrook Road, Storrs, CT 06269-3060 Having failed in two attempts1.' to explain Legendre transformations and having students read the description in Berry, Rice and Ross3 and the one in Calen4 without obtaining undrrst;lnding, it secmed reasonable to convert the act oflearning thc Lcgendre transformation into a kind of laboratorv exeieise in which the students were forced to grapple witk the problem each in his or her own way. This then becomes a n experience, a "hands-on learning" experience, which appears to actually be more effective than any reading, talking, andlor listening. Consider the function (1) E(VJ= 14 V2 which can be changed to any other well-behaved function as one sees fit. We make a small table of values of V, E(V) and the derivative of E(V) with respect to V, that is, aEIaV, a s seen i n Table 1.
Table 2.First Expansion of the Spreadsheet V
E( '4
intercept ( H )
.3E
av
p
Table 3. Final Expansion of the Spreadsheet V
E( '4
intercept (HJ
aE
-
av
ap
Table 1.The Initial Spreadsheet
We wish to perform a Legendre transformation of E(V), using the partial of E with respect to V as a "variable". In Figure 1we show the function E(V), versus V a s well a s the slope of E(V) a t two different values of V. The intercepts of these straight lines (slopes) are also shown. We defme a new variable,^, through the following: ,
,intercept = -56
that is, p = -28V and V = -p/28. The Legendre transformation requires us to express the energy (E) in terms of the straight line construction (Fig. 11, that is, i n the form: E = slope x V + intercept (3)
V
Figure 1. E ( q versus V, with slopes and intercepts
that is, 14v2=(+28VJv+~
(4)
where the intercept is traditionally called "H".Notice t h a t since H = E + p V also, we have: H = 1 4 ~-' (28 V)*V (5) H = -14V2 - 2 8 ~ = ' -14V2 (6) which agrees with our prior result. This is expressing11a s a function of V, hut we want it as a function ofp, so we have:
'David, C. W. Int. J. Math Sci. Technoi.1986,17,201-204. 'David, C. W. J. Chem. Educ.1988.65,876877. %erry, R. S.;Rice, S. A.; Ross, J. PhysicalChemistry; Wiley: New York, 1980: p 648. "Callen, H. 0. Thermodynamics andan Introduction to Thermostatics, 2nd ed.; Wiley: New York, 1985; p 137.
Figure 2. Plot of H(p) versus p Volume 68 Number 11 November 1991
893
H = -14[$J
JH H@)=-p + E ap
(7)
which gives H = - (p2)/56,as can be seen in Table 2. From this second spreadsheet we can obtain a plot ofH@) versus p by plotting column 4 against column 5. This is shown in Figure 2. We can do a similarly constructed Legendre transformation on H(p) as we did on E(W. We calculate dHl$ and attempt to representHin a straight line fashionin the form, (8) H@)= slope x p + intercept
which is:
(see Table 3). We now have:
which recovers E(W.
894
Journal of Chemical Education
(9)
2
- 56 L=%p+~ 2,
A)
L--= 56 L=E
(56
(10)
2
-' - '28w2 = 1 4 ~ =E(V) 56
(11) (121
