Question

2. *D1* Salt water is plentiful on earth; natural supplies of fresh water, on the other hand, are dwindling while mankind's demands grow. One appealing solution to this imbalance is to remove the salt from ocean water by some means. Here you will investigate the energy input that is necessary to accomplish this task. Special membranes can be manufactured that allow water, but not ions, to pass through. By pushing such a membrane through salt solution, one can separate salt and water, as depicted below: semipermeable membrane /レ (a) Explain, in terms of the number of microstates, why work is required for this process to be carried out at fixed temperature and total volume of the solution. (1 pt) (b) Imagine that the solutes, whose initial concentration is co (i.e., the number of solutes is N -coV, where V is the initial volume), do not interact with one another, so that the energy of the system does not change throughout the process. (In other words, the solutes can be treated like an ideal gas at temperature T, which is fixed by the solvent.) Calculate the partition function Qf for the spatial arrangement of N solute molecules when a fraction f of the solution has been purified. Write your answer in terms of N, f, and the total volume V. You can ignore terms in Qf that do not depend on V, since these do not matter for the rest of the problem. (2 pts)

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