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I'm satisfied with the explanation about Raoult's law that the chance of vaporizing will depend on the surface exposed to the atmosphere and solute will decrease such chance by taking portion of the surface at rate of $X_{solute}$.

However, incase of elevation and depression of boiling point, I can't find any explanation why it should depends on the molality. There's a molarity, molality, mole fraction, etc.. why molality among them? I know that since it's microscopic, we can't be sure about anything since we cannot observe. But still I'm looking for some satisfactory explanation.

And below is the example of boiling-point-elevation equation that I made by looking at the graph of tripoint and considering the fact that boiling point elevation occurs due to the vapor pressure change. Why can't we use such method? it seems accurate to me though..

Let $$P(t):T \to P, when$$ $$t_{tri}<t, t \in T (temperature), t_{tri} \in T (temperature), p\in P(pressure)$$, $$P^{-1}(p): P \to T $$ $$p_{tri}<p, t \in T (temperature), p_{tri} \in P (pressure), p\in P(pressure)$$ $p_{tri}, t_{tri}$ are pressure and temprature at tripoint $P, P^{-1}$ are function for solvent before dissolve.

According to the Raoult's law, we can calculate vapor pressure of solution's solvent $P_{2}(t)$ as such $$P_{2}(t)=P(t)*X_{solvent}, X_{solvent}=mol_{solvent}/(mol_{solvent}+mol_{solute})$$ Let's say that we want to know the elevation of boiling point at pressure $P(t_0)$ when $t_0$ is former boiling point. According to the Raoult's law stated above, we can say that boiling point elevation $\Delta t=P^{-1}(\frac {P(t_{0})}{X_{solvent}})-t_0$. (I'd tried to write down the specific process. But I couldn't do it well on words. Still you can see it by looking at the graph of any tri point.)

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Molality is a useful unit in colligative properties because it is independent of temperature, it is not affected by volume changes upon mixing and it will allow you to calculate mole fractions. Keep in mind that colligative properties only depend on the number of particles. Fortunately, mass is additive but volume is not.

Molarity is way more complex, where 2+2 is not equal to 4. For example mix 50 mL water+50 mL ethanol. The volume is quite less than 100 mL but if you mix 50 g of water with 50 g of ethanol, it will always be 100 g.

I am not sure why set theory and mapping notation was needed here?

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It is much easier to work with molality than molarity when measuring freezing point depression. Imagine. You weigh $10$ g, or $20$ g ice (or any other amount); you add $1$ g salt, $2$ g salt, or any other amount of any sort of solute. You drop a thermometer in the mixture, and you measure the fusion temperature. It is done quickly, and the scientist does not mind about the molarity of the obtained solution.

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