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# Introductory Chemical Engineering Thermodynamics

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6.2 (a) General: q q = N i i where q ¶q ¶ i i T P N N j i = a f , , and d dN N d i i i i q q q = + (1) However, we also have that d T dT V dV N dN V N T N i T V N i j i q ¶q ¶q ¶q = F H G I K J + F H G I K J + F H G I K J , , , , (2) Subtracting (2) from (1) yields 0 = - F H G I K J - F H G I K J + - F H G I K J L N M M O Q P P + ¶q ¶q q ¶q q T dT V dV N dN N d V N T N i i T V N i i i j i , , , , At constant T and V 0 = - F H G I K J L N M M O Q P P + q ¶q q i i T V N i i i N dN N d , , (general equation) For q = A , q i i A = and ¶q N A N G i T V N i T V N i j i j i F H G I K J = F H G I K J = , , , , . Thus, q ¶q i i T V N i i i N A G PV j i - F H G I K J = - = - , , and N dA P VdN i i T V i i T V = , , specific equation for q = A (b) Following the analysis above, we also get

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Solutions to Chemical and Engineering Thermodynamics, 3e 0 = - F H G I K J - F H G I K J + - F H G I K J L N M M O Q P P + ¶q ¶q q ¶q q U dU V dV N dN N d V N U N i i U V N i i i j i , , , , and, at constant U and V 0 = - F H G I K J L N M M O Q P P + q ¶q q i i U V N i i i N dN N d j i , , Now, choosing q = S , and using that S N G T i U V N i j i F H G I K J = - , , , which is easily derived, yields - = T N dS H dN i i U V i i U V , , (c) Following a similar analysis to those above, we obtain 0 = - F H G I K J - F H G I K J + - F H G I K J L N M M O Q P P + ¶q ¶q q ¶q q S dS V dV N dN N d V N S N i i S V N i i i j i , , , , which, at constant V and S , reduces to 0 = - F H G I K J L N M M O Q P P + q ¶q q i i S V N i i i N dN N d j i , , Finally, using q = U , and U N G i S V N i j i a f , , = yields N dU PV TS dN i i S V i i i S V = - + , , l q 6.3 (a) At constant U and V , S = maximum at equilibrium S S S N S N S i i i C i i i C = + = + = = I II I I II II 1 1 but dS S U dU S V dV S N dN S U dU S V dV S N dN V N U N i U V N i V N U N i U V N i j i j i = = F H G I K J + F H G I K J + F H G I K J + F H G I K J + F H G I K J + F H G I K J 0 I I I I I I I I I II II II II II II II II II , , , , , , , , Since U U U = + = I II constant, dU dU II I = -