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Let's say I want to lower the freezing point of water by 5 degrees Celsius. I know that for a fixed volume of water, I will need $x$ moles of solute to do so.

Because freezing point depression is a colligative property, does that mean I can use any solute to lower the freezing point of water as long as the molality of the solution is maintained, or are there other properties of the solute that need to be considered?

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The key is in the terminology- the freeze point suppressant must be a solute. Oil won't suppress the freeze point of water because it doesn't dissolve. A small amount of sodium chloride or ethanol will suppress the freeze point of water to -5°C.

The first exception is when the solute stops dissolving. This is called the eutectic point. For example, you can only dissolve 23.3% NaCl in water (by mass). That mixture will freeze at -21.1°C. After that, adding more salt won't reduce the freeze point further.

The second exception is with large amounts. If you mix water and ethanol in a 50/50 mol ratio, which is the solvent and which the solute anymore? You can calculate the freezing point suppression of each with respect to the other and they won't match. Liquid/liquid mixtures also have eutectic points at the minimum freezing point, which is less than the freezing point of either component.

To get the actual freezing point for non-dilute systems, try searching for "solid liquid equilibria", "eutectic point", or "binary phase diagram".

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