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Prove the statement that when the concentration of a weak acid HA decreases by a factor of $10,$ its percent ionization increases by a factor of $\sqrt{10}$. State any assumptions.

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$$\sqrt{\frac{k_{a}}{[H A].}}$$

Chemistry 102

Chapter 15

Acids and Bases

Liquids

University of Maryland - University College

University of Kentucky

Lectures

03:07

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, a liquid is one of the four fundamental states of matter (the others being solid, gas and plasma). A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena.

04:38

A liquid is a state of matter in which a substance changes its shape easily and takes the form of its container, and in which the substance retains a constant volume independent of pressure. As a result of this, a liquid does not maintain a definite shape, and its volume is variable. The characteristic properties of a liquid are surface tension, viscosity, and capillarity. The liquid state has a definite volume, but it also has a definite surface. The volume is uniform throughout the whole of the liquid. Solids have a fixed shape and a definite volume, but they do not have a definite surface. The volume of a solid does not vary, but the volume of a liquid may vary.

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Show by calculation that w…

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we are going to do a mathematical proof. Yes, a mathematical proof in chemistry. Not my favorite, but we're gonna prove this statement. The statement is when the concentration of a weak acid a so n h a look when the concentration of a weak acid h a decreases by a factor of 10. But it's percent ionization increases by a factor of the square root of 10. And then we'll have to state any assumptions were doing this proof. So again, the concentration of a J goes down by a factor of 10% of the ionization increases by a factor of 10. Okay, let's begin our proof. I'll be getting on this page. I get on one page if I can. Okay, so let's right. First, our equation for the hydraulics is of weak acids. So I've got a J A. When leave states off this plus water because there's a side. Raul Icis and I will have the Hydro Nehemiah and HBO plus and my and I there's that's now at equilibrium when were at equilibrium. We know that um, r h 30 plus is going to eat well, our and I on and we know that our K expression for this will be H 30 plus times a minus, which will be eat the same as saying H 30 plus squared. That's equivalent. And I have a my denominator there. Okay, this is a weak acid. So what assumption is, and this should be I should have written H Yeah. Cut H A is going to be approximately equal to the original given concentration of the asset because so little of its gonna lionize. So we're gonna make that assumption. And I think that our text is gonna refer to that as h A o for original okay, or a time zero at time zero. So I know that. So now, um, let me rewrite this equate. I'm just erased this part, so h a at time zero. So now I'm going. Teoh saw this equation for H 30 plus. Let's solve this equation for H trio Plus, So the concentration of ATRIOs plus it's gonna equal the square root off Mike a Times night, this concentration. In order to do that, I just multiplied each side by a J. So I got k a times a J zero equals h trio squared, and then they took the square root of that whole thing. Okay, so there's this now, by definition, what is ionization? What is percent ionization? It's my aged Rio plus concentration divided by my this concentration times 100. So my h 30 concentration. I'm gonna substitute for this. My age To your concentration. Right here is K a square root of K times h A time zero divided by h A time zero. And that would be times 100. Okay, so, um, we can reduce this one too. Okay. Over H A zero. So this is our percent. Our equation for percent ionization, This is percent ionization. So that says if we increase if we increase our h no decrease if we decrease this factor, If this factor goes down by 10 the ionization will increase. So if that h a goes down by a factor of 10 what will happen? The ionization will increase by the square root of 10. Because if I put a 10 right here, if I increase that by 10 if this turns into a 10 let me just erase that. Put up 10 in there. If this goes up by 10. Then my, um the percent ionization is gonna decrease because it's in the denominator by the square root of 10. Okay? And that is in accord with the shot ladies principle because shot liaise. Principle tells us that ionization percent ionization should increase as we dilute as we become less and less concentrated. That's it.

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