JOHN J. ALEXANDER University of Cincinnati Cincinnati, 45221
tem is left to itself for as long a time as necessary for "equilibrium" (uniform temperature throughout) to he attained. Assuming T, > T, then i t must of course follow that T, > TI > T, where TI is the final temperature.
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(a) Show that Tf = (T,, 8T,)I(1 8 ) where 8 = mln ( b ) Show that the total entropy change is
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AS = n(l8) In (TIIT.)
where the heat capacity for water has been taken to he that for constant pressure, C,, and is assumed constant at approximately 1.00 cal deg-' mol-'. (c) Setting n = m = 1, show that AS > 0. (d) Show that for any valuesof n, rn, T,,,, and T, it must followthat AS b 0. (e) Suggest some physical reasons for the positivity of AS Answers to the Mathematical Parts (a) Then moles of water at T, will absorb n(18)(Ti - T,) calories from them moles at T,, the latter of course loslng rn(18)(TIT,) calories. Thus
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n(18)(Tf- T,) rn(18)(Tf - T,) = 0 sinceenrrgy isrwwerved i n the isolstrd cgztem.Thm.sincrfl= rn,,,. 7 j ir found h) d v i n g the equation. tbl l'hc entropy chnn~efw the ,I moles orwnally at T, is. by a rrversible path AS, =
n-p T ST'dTlT .,. C ~ ~ T I n(18) =
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7-,.
= n(18) In (TilT,,).
Similarly, AS, = rn(18) In (TIIT,). The total entropy change AS, since entropy is additive, is then AS = AS, AS,. (c) When n = rn = 1,8= 1 and Tf = 1/2(T, T,,) so that, since rn may now he replaced by n
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Entropy Calculations
... .. NOWAS>O~~(T,+T.)~>~T,T,,O~T,~+~T,T~+T,~> 4T,T..orTm2-2T,T.+T.2>0.ButT,2-2T,T,,+T,Z = (Tside is elearlvrecomizedas . ,,.- T.12: ... . the rieht-hand " positive, even if T. > T,,,
(d) In general F r a n k R. Meeks Uniuersity of Cincinnati Cincinnati, Ohio 45221 T h e positivity of AS, and incidentally the nonconsewative character of entropy, for an irreversible thermodynamic process can he illustrated by calculations for a change familiar from direct experience to everyone. The question is designed to he aiven t o juniors or seniors midway in a chemical thermodynamics course' or after some mastery of entropy calculations in a more general course in physical chemistry. Question Suppose n moles of water a t temperature T, and m moles of water a t T,,, are placed together in a vacuum bottle and quickly isolated, thermally, from the surroundings. T h e sys-
Meeks, F. R., "Classical and StatisticalThermodynamics," AUyn &Bacon, Boston, forthcoming.
where X = T,/T.
b 1.Then
ASICp = (rn + n) In
- ->> t
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(-)1 + 8 X
- rn In X
When X 1, ASICp 0. When X >> 1, three cases arise: (i) 8 1;(ii) 0 and (iii) 8 OandsoAS>O.
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Exam Question Exchange offers teachers an opportunity to share prize exam questions with others. Guidelines for preparing exam questions for submission were outlined on page 608 of the October 1977 issue. All questions submitted become the property of the Journal of Chemical Education and will not be returned. Questions should be submitted to the column editor. Volume 55, Number 10. October 1978 I 653
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(rn n) In (0x1 - rn In X = Case (ii). When 0 >> 1, ASICp rnlnB+mlnX+nln(8X)-rnlnX=rnlnR+nln(8X) >OandsoAS>O. Case (iii). When 0 I ) , we note that AS/Cp (rn+n)In(1+8X)-mlnXsinceln(1+0)-ln(1)= 0.Further, R
