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Thermodynamics filled in class notes_Part_42

# Thermodynamics filled in class notes_Part_42 - 91 3.3 IDEAL...

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Unformatted text preview: 91 3.3. IDEAL MIXTURES OF IDEAL GASES Example 3.5 Find the expression for mixture entropy of the ideal gas. si = ci si = ci si = N s= i=1 ˆ cP i (T ) ˆ Pi , dT − Ri ln ˆ Po T 298 T ˆ cP i (T ) ˆ Pi ci so ,i + ci , dT − ci Ri ln 298 ˆ Po T 298 N N N T ˆ cP i (T ) ˆ o ci ci s298,i + ci Ri ln dT − ˆ T 298 i=1 i=1 i=1 T so ,i + 298 N ci so ,i + 298 = i=1 = so + 298 T 298 N T 298 i=1 ˆ ci cP i (T ) ˆ dT − ˆ T N ˆ cP (T ) ˆ dT − ˆ T i=1 (3.211) Pi Po ci Ri ln i=1 , (3.212) Pi Po N Pi Po ci Ri ln (3.210) , (3.213) . (3.214) All except the last term are natural extensions of the property for a single material. Consider now the last term involving pressure ratios. N − ci Ri ln i=1 Pi Po N = − i=1 N = −R ci i=1 Ri ln R N = = −R Pi Po ci Ri ln i=1 N −R i=1 −R −R i=1 i=1 = = −R ln −R ln i=1 Po P P N (3.216) P P − ln Po Po , (3.217) (3.218) P P + ln − ln , Po Po , (3.219) P P + ln Po Po , (3.220) yi + ln + ln PN Po i=1 P Po , yi (Pi ) i=1 (3.221) , (3.222) N 1 yi Po P + ln P Po yi Pi Po i=1 i=1 , − ln Pi Po Po PN (3.215) P P − ln Po Po yi Pi Po ln + ln , + ln Pi Po yi ln −R ln Pi Po =y i N = Pi Po ln ln N j =1 cj /Mj N = P P − ln Po Po + ln ci /Mi N = Pi Po R/Mi N j =1 cj R/Mj ci P P − R ln Po Po + R ln yi + ln P Po , (3.223) CC BY-NC-ND. 18 November 2011, J. M. Powers. 92 CHAPTER 3. GAS MIXTURES N = −R ln i=1 N = −R ln yi Pi P + ln yi P P i=1 yi N y yi i = −R ln P Po + ln i=1 + ln P Po , P Po (3.224) , (3.225) . (3.226) So the mixture entropy becomes s= = so + 298 so + 298 T 298 ˆ cP (T ) ˆ dT − R ln ˆ T T 298 N y yi i + ln i=1 ˆ P cP (T ) ˆ dT − R ln − R ln ˆ Po T classical entropy of a single body P Po , (3.227) N y yi i . (3.228) i=1 non−Truesdellian The extra entropy is not found in the theory for a single material, and in fact is not in the form suggested by Truesdell’s postulates. While it is in fact possible to redeﬁne the constituent entropy deﬁnition in such a fashion that the mixture entropy in fact takes on the classical form of a single material via the T (ˆ Pi ˆ deﬁnition si = so ,i + 298 cP iˆ T ) dT − Ri ln Po + Ri ln yi , this has the disadvantage of predicting 298 T no entropy change after mixing two pure substances. Such a theory would suggest that this obviously irreversible process is in fact reversible. On a molar basis, one has the equivalents N ρ= ρi = i=1 1 v= N 1 i=1 v i ρ n = , V M = V 1 = = vM, n ρ (3.229) (3.230) N u= yi ui = uM, (3.231) yi hi = hM, (3.232) i=1 N h= i=1 N cv = yi cvi = cv M, (3.233) i=1 cv = cP − R, (3.234) N yi cP i = cP M, cP = i=1 CC BY-NC-ND. 18 November 2011, J. M. Powers. (3.235) ...
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