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(a) Derive the equation relating the molality $(m)$ of a solution to its molarity $(M)$$$m=\frac{M}{d-\frac{M M}{1000}}$$where $d$ is the density of the solution $(\mathrm{g} / \mathrm{mL})$ and $\mathcal{U}$ is the molar mass of the solute (g/mol). (Hint: Start by expressing the solvent in kilograms in terms of the difference between the mass of the solution and the mass of the solute.) (b) Show that, for dilute aqueous solutions, $m$ is approximately equal to $M$.

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a. $\mathrm{m}=\mathrm{M} /(\mathrm{d}-\mathrm{MM} / 1000)$b. $\therefore \mathrm{m} ? \mathrm{M}$

Chemistry 102

Chapter 12

Physical Properties of Solutions

Solutions

University of Central Florida

Drexel University

University of Toronto

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All right. So let's derive this equation that relates the morality to the polarity. So there will be a lot of symbols. So for a start, I'll use am little him for morality. Right. Let's recall that that's the number of malls. Number of moles. I'm going to represent this in divided by the mass off the solvent. So the mass of the solvent, our using w Okay, uh, in the other hand, more clarity will be a capital whim. And is the number of malls divided by divine Okay, uh, so let's to start working out in this side here. So the first step is to express to rewrite w. Okay, so w is the mass off the solvent. Okay, but let's think about the whole solution. So I'm gonna write s for the mass off the solution. Okay? The mass of the solution is the massive the solvent, plus the mass of the soy it. Right. So for the mass of the science, I'm just going to use acts. Okay, so this will be the Salyut, and that's the solvent. So hopefully that makes sense. Massive. The solvent plus a salyut. It's the total which is the mass of the solution so we can re express the morality as the number of malls over. So w I'm gonna say the W is rico s meant minus X. Okay. Like this. Cool. No, let's start replacing things. So instead of number of moles Ah, let's replace that with Ah. So why Why do we want to get rid of number of moles? Because we want express the formula as it was given They're Onley depends on density and more mass again. So for the number of miles, let's write this using the polarity. So the polarity says the number of malls over volume is equal. The capital m so we can ride the end will be Dimler tee times V. Okay. Ah, now for the mass of the solution, that's a great opportunity to use the density of the social. Okay, recall the density is mass over volume. Therefore, we can express mass as volume times density. So that's what I'm gonna do the for the density of the solution. Times feet. Okay, great. So we can get rid of E if we can also find a key factor in X. So let's try that. So X is the mass of the Salyut. So let's write that as number of most times. Moeller mess right? That is valid, recalled the molar mass is the mass off the substance divided by the number of malls. So you can work this out. However, molar mass is normally expressed in kilograms. I'm sorry in grams and we won in kilograms. Right? So we That's how we get the factor. 1000 to convert grams. Two kilograms. Okay. Why is that? Well, because since we're working with morality, morality assumes that this mass here on the bottom, this has to be in kilograms. All right, so now we can take the number of malls right here and do the same replacement that we did up here using the polarity and the volume. Great. So morality will be m V over the minus. Now, I'm gonna replace that with MV Molar mass. Over 100. Now, hopefully, you see that we have a bunch of these. We can actually cancel, then out. So we arrive at the desired former. All right, That's the relationship between polarity, morality and mole aridity. All right, so that's the first part of the question. The second is just to justify why? For water, when it's very diluted, morality is pretty much equal malaria. Gee, Okay, so if we look at this formal, I'm gonna repeat it here. So morality is equals malarkey. Density minus this factor here. The 100. Okay, if this solution is very, very diluted, what is going on here? In the bottom, You're getting up. You have ah, number. Mine is a tiny little number. So when you have something big and that's you can think about calculus like limits when you have something big time minus something small, that is approximately the big Okay, So if we take the and subtract this a small factor, why is it small? Because we're talking about diluted solutions. So am is a small All right, so we can rewrite this as I am over the okay, This just kind of cancels out. Now, what's the density for water? Normally, if that water is pure and the density will be very close to one kilogram for later, Okay, Demings that numerically, this D will be just one. So M equals capital win

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