p:(
Glbbs' Paradox: Two Vlews on the Correction Term
= Rln
To The Editor:
In a recent article Kempl discussed the calculation of the entropy (change) of isothermal mixing of ideal gases. The equation, AS,,,,,
Vf Vf = nARln- n&ln -
VA
+
VB
+ ...
v, + v~
vt v~
v, + v~
vt + v~
v~
(?)"
mo1
Generalizing,
(1)
where Vf is the final volume of the mixture, gives wrong answers when some of the gas samples are composed of same molecules. Kemp developed a correction term that, when subtracted from the right-handside of eq 1, allowed for some of the initial samples having indistinguishablemolecules D, E. This correction term is n a l n u , where n. is the total amount of indistinguishable species. Kemp's correction term is applicable only when the numbers of molecules in each sample are equal (as well as the temperatures and pressures). Kemp's correction term stems from his derivation of ASb, his eqs 7 and 8. A more general correction term is obtained by deriving a different expression for ASb. T o do this we introduce an additional step, step x, which is the sum of steps a and b. Thereafter, ASb = AS, - AS,. Now
v, + v~
m0l + Rln
AS, = n A R h nBRh- ncRln -
where i, j, . . . are indistinguishable species, ni, n,, . . . are their respective amounts, and n, is the summed amount of the indistinguishable species. This expression for ASb then becomes the correction term (c.t.) that is applied in lieu of -n&ln ca. Solutions of the correction term are negative when samples are not all different. Solutions approach zero as the amounts of same samples become very small. When the samples are all different, AS, and AS, are equal, ASb is zero, and the correction term is zero and vanishes, as does route b of Kemp's Figure 1. There are as many quotients in the logarithmic part of the c.t. as there are same samples. When the samples are all different there is one quotient, say nilni, which is unity. Conclusion. The writer finds that the use of the above c.t. in place of Kemp's relations 7 and 8 gives derived relations that give correct answers whatever the initial volumes of the samples. Hence Kemp's relation 12 becomes
vt
AS, = nARln- nBRln- ncRln -
[
+ .. . Rln
(:y -
X
Rln
'n' .. .] mol (1) '
X
(6)
and his relation 21 becomes Subtracting AS, from A&,
+
AS, = -nARlnxA -nBRhuB
+...
[
(3"
R l n
]
(
XRlnL
'X...mol
(7)
Application. Let A, B, C, and D be different gases, and let E be the same as D. Initial temperatures and pressures are = the same. Let the initial volume ratio be VA:VB:VC:VD:VE 1:3:1:2:3. Let nt = 1 mol; n, = 0.5 mol; mole ratio = 0.1:0.3:0.1:0.2:0.3. If the gases were all different, AS. = AS, = 12.5 J K-1. But since D and E are same samples, c.t. = Rln
Since ASg = ASa J K-'.
0.2 0.2 0.3 0.3 [(=) (=) ]m d
+ c.t., then AS, = (12.5 + -2.8)
since n o; V (T,p constant)., Let nD = n~ mol, where nDis the number only (since only numbers can go into the in term), then:
J K-I = 9.7
J. J. MacDonald Bishopbriggs High Schwl Glasgow, Scotland
' Kemp. H. R. J. Chem. Educ. 1985,62,47. Volume 63
Number 8
August 1986
735
