Thermodynamics filled in class notes_Part_40

Thermodynamics filled in class notes_Part_40 - 87 3.3....

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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 semi-definite. 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. 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 non-reacting 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 BY-NC-ND. 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 BY-NC-ND. 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 Park-sou during the Fall '11 term at FSU.

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