Question

# For an ideal solution, we discussed in class that the chemical potential is given by: μ=μ_0+RT...

For an ideal solution, we discussed in class that the chemical potential is given by: μ=μ_0+RT "ln" (c/c_0 ) where the units of µ written in this form are Joules/mole. The concentration c_0 is a reference concentration, which we will take to be 1 Molar for this problem. Suppose solution 1 with solute concentration of 1 mM (i.e., 1 x 10-3 Molar) is in contact with solution 2 with solute concentration of 0.1 mM (i.e., 1 x 10-4 Molar). If a differentially small amount of solute, dn, moves from the higher concentration to the lower concentration, the differential change in free energy is calculated as follows: dG = [-((∂G_1)/dn)_(P,T) dn + ((∂G_2)/dn)_(P,T) dn] where G_1 is the free energy of solution 1 (the higher concentration solution), G_2 is the free energy of solution 2 (the lower concentration solution), and the relative signs indicate that the number of moles of solute in solution 1 is decreased by dn, while the number of moles of solute in solution 2 is increased by dn. Recalling the definition of chemical potential, we can write: dG = [-μ_1 dn + μ_2 dn] = [μ_2-μ_1 ] dn

2a. Show that for the case described above dG = [RT "ln" (c_2/c_1 )] dn and insert the values of the high (c_1) and low (c_2) concentrations to confirm that, indeed, dG < 0 for transfer of solute from a high concentration solution to a low concentration solution. This result confirms that such transfer will be a spontaneous process if the solutions are placed in contact with one another.

2b. Without worrying about how it might be done, at T = 298 K, by what amount (in kJ/mole) would the chemical potential of the solute in the low concentration solution need to be increased in order for the system to be at equilibrium, even though the two solutions remain at their different concentrations of c_1=1 "mM" and c_2=0.1 "mM" ? (Note: In the case where the solute is an ion, and therefore charged, the most common way of changing the chemical potential of one solution versus the other is to apply a different electrical potential (in Volts) to one solution versus the other. Depending on the signs of the charge on the ion and of the electrical potential, the free energy of a charged solute can be either increased or decreased. This situation of maintaining different ionic concentrations in equilibrium is common in electrochemistry and in living cells.)