"Liquids A and B, which have identical molar volumes, are mixed together in equal volumes to form a 75 mL mixture. The liquids are partially miscible and the mixture phase separates into an upper phase with a composition xA=0.45 and a lower phase with a composition xA=0.82. How many milliliters of each phase forms? What will happen if another 25 mL of liquid B were added to the mixture?"

I know that I need to use the lever rule, and the V(total)=V(a)n(a)+V(b)n(b)
I am 100% convinced I am over thinking this, but are the n(a) the same as n(alpha) in the lever rule?

Yes, in this context, the n(a) represents the number of moles of component A in the mixture, and n(b) represents the number of moles of component B in the mixture. These values are used in the lever rule to determine the proportions of the phases formed during phase separation.

To apply the lever rule, you need to calculate the number of moles of A and B in each phase. Given that the molar volumes of liquids A and B are identical, and they are mixed in equal volumes, the total volume of the mixture is 75 mL. Therefore, each phase would have a volume of 37.5 mL.

To find the number of moles of A in each phase, you can use the composition xA and the total number of moles of A in the mixture. Let's assume that the total number of moles of A in the mixture is n(A_total). Then, the number of moles of A in the upper phase (n(A_upper)) can be calculated as:

n(A_upper) = xA_uppear * n(A_total),

where xA_upper is the composition of A in the upper phase (0.45).

Similarly, the number of moles of A in the lower phase (n(A_lower)) can be calculated as:

n(A_lower) = xA_lower * n(A_total),

where xA_lower is the composition of A in the lower phase (0.82).

Once you have the number of moles of A in each phase, you can calculate the number of moles of B in each phase using the molar volumes and the total number of moles of B in the mixture.

Finally, to find the volumes of each phase, you can use the formula:

V(phase) = (n(phase) * molar volume) / (n(A_total) * molar volume),

where V(phase) is the volume of the phase, n(phase) is the number of moles of A or B in the phase, and molar volume is the molar volume of the liquids A and B.

If another 25 mL of liquid B were added to the mixture, the total volume of the mixture would increase to 100 mL (75 mL + 25 mL). The exact behavior and phase separation of the system would depend on the specific properties and miscibility of liquids A and B. Without further information, it's difficult to determine the outcome. However, in general, adding more of one component can potentially result in changes in phase separation and properties of the mixture.