PS5Solutions - Chemistry 120B SP11 Problem Set 5 Solutions...

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Chemistry 120B SP11 Problem Set 5 Solutions Total: 50 points 1. (20 points) See MQS Solutions Manual. 2. (i) (5 points) W ( V 1 , V 2 ) = V 1 N 1 × V 2 N 2 × (stuff independent of V ) Neglecting volume-independent terms, S = k B ln W = k B ln ( V 1 N 1 V 2 N 2 ) = k B ( ln V 1 N 1 + ln V 2 N 2 ) = k B ( N 1 ln V 1 + N 2 ln V 2 ) (ii) (5 points) W ( V 1 , V 2 ) = ( V 1 + V 2 ) ( N 1 + N 2 ) S = k B ln W = k B ln h ( V 1 + V 2 ) ( N 1 + N 2 ) i = k B ( N 1 + N 2 ) ln ( V 1 + V 2 ) = k B N 1 ln ( V 1 + V 2 ) + k B N 2 ln ( V 1 + V 2 ) (iii) (5 points) Δ S mix = S final - S initial = k B N 1 ln ( V 1 + V 2 ) + k B N 2 ln ( V 1 + V 2 ) - k B ( N 1 ln V 1 + N 2 ln V 2 ) = k B N 1 [ln( V 1 + V 2 ) - ln V 1 ] + N 2 [ln( V 1 + V 2 ) - ln V 2 ] = - k B N 1 ln V 1 V 1 + V 2 + N 2 ln V 2 V 1 + V 2 Δ S mix N 1 + N 2 = - k B " N 1 N 1 + N 2 ln V 1 V 1 + V 2 + N 2 N 1 + N 2 ln V 2 V 1 + V 2 # = - k B " x 1 ln V 1 V 1 + V 2 + x 2 ln V 2 V 1 + V 2 # Since the initial densities are equal, V 1 / ( V 1 + V 2 ) = N 1 / ( N 1 + N 2 ) = x 1 . Δ S mix N 1 + N 2 = - k B ( x 1 ln x 1 + x 2 ln x 2 ) 3. (i) (5 points) Think of the reactive cavity as the system and the special material as the surroundings.
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