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Unformatted text preview: 87 3.3. IDEAL MIXTURES OF IDEAL GASES
N = −Rm y
ln ykk , (3.161) k =1 y
y
y
= −Rm (ln y1 1 + ln y2 2 + · · · + ln yNN ) ,
yy
y
= −Rm ln (y1 1 y2 2 . . . yNN ) , (3.162)
(3.163) N = −Rm ln y
yk k , (3.164) k =1 Dividing by m to recover an intensive property and R to recover a dimensionless property,
we get
∆s
∆s
= − ln
=
R
R N
y
yk k . (3.165) k =1 Note that there is a fundamental dependency of the mixing entropy on the mole fractions.
Since 0 ≤ yk ≤ 1, the product is guaranteed to be between 0 and 1. The natural logarithm
of such a number is negative, and thus the entropy change for the mixture is guaranteed
positive semideﬁnite. Note also that for the entropy of mixing, Truesdell’s third principle
is not enforced.
Now if one mole of pure N2 is mixed with one mole of pure O2 , one certainly expects
the resulting homogeneous mixture to have a higher entropy than the two pure components.
But what if one mole of pure N2 is mixed with another mole of pure N2 . Then we would
expect no increase in entropy. However, if we had the unusual ability to distinguish N2
molecules whose origin was from each respective original chamber, then indeed there would
be an entropy of mixing. Increases in entropy thus do correspond to increases in disorder.
3.3.1.3 Mixtures of constant mass fraction If the mass fractions, and thus the mole fractions, remain constant during a process, the
equations simplify. This is often the case for common nonreacting mixtures. Air at moderate
values of temperature and pressure behaves this way. In this case, all of Truesdell’s principles
can be enforced. For a CPIG, one would have
u2 − u1 = cA cvA (T2 − T1 ) + cB cvB (T2 − T1 ),
= cv (T2 − T1 ). (3.166)
(3.167) cv ≡ cA cvA + cB cvB . (3.168) h2 − h1 = cA cP A (T2 − T1 ) + cB cP B (T2 − T1 ),
= cP (T2 − T1 ). (3.169)
(3.170) where
Similarly for enthalpy CC BYNCND. 18 November 2011, J. M. Powers. 88 CHAPTER 3. GAS MIXTURES where
cP ≡ cA cP A + cB cP B . (3.171) For the entropy
s 2 − s 1 = cA (s A 2 − s A 1 ) + cB (s B 2 − s B 1 ),
(3.172)
T2
y A P2
T2
y B P2
= cA cP A ln
− RA ln
+ cB cP B ln
− RB ln
,
T1
y A P1
T1
y B P1
P2
T2
P2
T2
− RA ln
+ cB cP B ln
− RB ln
,(3.173)
= cA cP A ln
T1
P1
T1
P1
T2
P2
= cP ln
− R ln
.
(3.174)
T1
P1
The mixture behaves as a pure substance when the appropriate mixture properties are deﬁned. One can also take
cP
k= .
(3.175)
cv 3.3.2 Summary of properties for the Dalton mixture model Listed here is a summary of mixture properties for an N component mixture of ideal gases
on a mass basis:
N M= yi Mi , (3.176) ρi , (3.177) i=1
N ρ=
i=1 1 v= N
1
i=1 vi 1
=,
ρ (3.178) N u= ci u i , (3.179) ci hi , (3.180) i=1
N h=
i=1 =R R= R
=
M N N ci Ri =
i=1 yi Mi Ri = N
i=1 yj Mj N yi ,
i=1
=1 j =1
=M CC BYNCND. 18 November 2011, J. M. Powers. R
M (3.181) ...
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Parksou during the Fall '11 term at FSU.
 Fall '11
 ParkSou
 Dynamics

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