Comment On "On Euler's Theorem for Homogeneous Functions and Proofs Thereof" Michael A. Adewumi Department of Gas Engineering, lnstitute of Gas Technologyllllinois Institute of Technology, Chicago, IL 60616
R. J. Tykodil presented an excellent discussion of Euler's theorem as anolied to thermodvnamic functions. He also gave a suggest'ed procedure of teaching this topic to students of thermodynamics; step 3 in his procedure involves revising Taylor's series expansion and L'Hopital's rule with the students. However, many students (even graduate students) are confused by L'Hopital's rule since it requires taking the derivative of the numerator and denominator separately. We therefore suggest a more straightforward way of resolving the term:
knowledge of elementary algebra. Consider the geometric series
.,
a, or, or2,ar3,,, ar"-'
(n term)
whose sum, S, is given by S=o
+ ar + or2 + . _ _ +a?"-'
S-rS=(l-r)S=n+ar+
...+ n r " - ' - o r - a r z -
Hence in eq 38 of the article. A well-known (and easy-to-remember) expression in elementary algebra is A" - 1 = (A
- l)(A"-' + A"-* + ... + A + 1)
(I-r)S=a-ar"=a(l-r")
and
(1)
Hence,
Thus, using eq 3, setting a = 1and h = r gives
and limit r-I
1 (---) liilt (A"-' + A-1 A" -
=
A"-%
+ ...+ A + U
+ + .. [ ( n - 1)times] + 1 = n
limit (E) =1 1 1-1 A-1
Furthermore, the expansion can easily hederived from the
610
Journal of Chemical Education
(2)
Multiplying eq 2 through hy r and substracting the resulting equations from eq 2 gives
' Tykodi, R. J. J. Chem. Educ. 1982,59,557.
ar"
