Unformatted text preview: 85 3.3. IDEAL MIXTURES OF IDEAL GASES
Mass conservation gives
m2 = m1 = mA + mB . (3.123) − 1W2 , (3.124) One also has the ﬁrst law
U2 − U1 U2 − U1
U2 m2 u 2
(mA + mB )u2
0
0
T2
T2 = 1Q2 = 0,
= U1 , (3.125)
(3.126) = m A u A1 + m B u B 1 ,
= m A u A1 + m B u B 1 ,
= mA (uA1 − u2 ) + mB (uB 1 − u2 ), = mA cvA (T1 − T2 ) + mB cvB (T1 − T2 ),
mA cvA T1 + mB cvB T1
,
=
mA cvA + mB cvB
= T1 . (3.127)
(3.128)
(3.129)
(3.130)
(3.131)
(3.132) The ﬁnal pressure by Dalton’s law then is
P2 =
=
=
= =
= PA2 + PB 2 ,
mB RB T2
mA RA T2
+
,
V2
V2
mA RA T1
mB RB T1
+
,
V2
V2
(mA RA + mB RB ) T1
,
V2
substitute for V2 from Eq. (3.122)
(mA RA + mB RB ) T1
,
T
(mA RA + mB RB ) P1
1
P1 . (3.133)
(3.134)
(3.135)
(3.136)
(3.137)
(3.138)
(3.139) So the initial and ﬁnal temperatures and pressures are identical.
Now the entropy change of gas A is
sA2 − sA1 =
= cP A ln = PA2
T A2
− RA ln
,
T A1
PA1
T2
yA2 P2
− RA ln
,
T1
yA1 P1
yA2 P1
T1
,
−RA ln
T1
yA1 P1 cP A ln cP A ln (3.140)
(3.141)
(3.142) =0 =
= yA2 P1
(1)P1
−RA ln yA2 . −RA ln , (3.143)
(3.144) Likewise
sB 2 − sB 1 = −RB ln yB 2 . (3.145) CC BYNCND. 18 November 2011, J. M. Powers. 86 CHAPTER 3. GAS MIXTURES
So the change in entropy of the mixture is
∆S = mA (sA2 − sA1 ) + mB (sB 2 − sB 1 ), = −mA RA ln yA2 − mB RB ln yB 2 , = − (nA MA ) (3.146) R
MA (3.147) ln yA2 − (nB MB ) =m A = ≥ −R nA ln nA
nA + nB (3.149)
nB
nA + nB +nB ln ≤0 ≤0 0. We can also scale Eq. (3.149) by Rn to get =∆ s ∆s
R (3.148) =RB −R(nA ln yA2 + nB ln yB 2 ), 1 ∆S
Rn ln yB 2 , =m B
=RA = R
MB nA nB = − n ln yA2 + n ln yB 2 ,
=yA2 , (3.150) (3.151) (3.152) =y B 2 = − (yA2 ln yA2 + yB 2 ln yB 2 ) , y
y
= − (ln yAA2 + ln yBB2 ) ,
2
2
yA2 yB 2
= − ln (yA2 yB 2 ) . (3.153)
(3.154)
(3.155) For an N component mixture, mixed in the same fashion such that P and T are constant,
this extends to
N ∆S = −R
= −R nk ln yk ,
k =1
N nk ln
k =1 (3.156)
nk
N
i=1 ni ≥ 0, (3.157) ≤0
N = −R k =1
N = −Rn
= −R nk
n ln yk ,
n (3.158) nk
ln yk ,
n (3.159) k =1
N m
M yk ln yk ,
k =1 CC BYNCND. 18 November 2011, J. M. Powers. (3.160) ...
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 ParkSou
 Dynamics, Trigraph, yk ln yk, ln yA2

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