**C1.** Derive Eq. (12-22) starting with Eq. (12-21).

**C2.** Derive an equation that is equivalent to Eq. (12-12) for L/(mV) =1, but in terms of liquid mole fractions.

**C3.** Derive an operating equation similar to Eq. (12-40), but draw your balance envelope around the bottom of the column. Show that the result is equivalent to Eq. (12-40).

**C4.** Derive Eq. (10-4), which relates the overall efficiency to the Murphree vapor efficiency, for dilute systems by determining N_{equil} and N_{actual} in Eq. (10-1) from appropriate forms of the Kremser equation.

**C5.** Occasionally it is useful to apply the Kremser equation to systems with a constant relative volatility. Where on the y vs. x diagram for distillation can you do this? Derive the appropriate values ...

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