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Thermodynamics filled in class notes_Part_39

# Thermodynamics filled in class notes_Part_39 - 85 3.3 IDEAL...

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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 BY-NC-ND. 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 BY-NC-ND. 18 November 2011, J. M. Powers. (3.160) ...
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