2015 AP Chemistry free response 1d

Metal air cells need to be lightweight for many applications in order to transfer more electrons with a smaller mass. Sodium and calcium are investigated as potential anodes. A 1.0 gram anode of which of these metals would transfer more electrons, assuming that the anode is totally consumed during the lifetime of a cell? Justify your answer with calculations.

So, let's go to the top where they showed the reactions, where they showed the reactions involving sodium and calcium. Actually, let me just copy and paste this so it's easier for me to reference near the bottom where we're actually doing the question. So, let me just copy this and then let's paste it down here so that I can reference it. Okay, I'll even make it a little bit smaller just so I don't take up too much real estate.

The reactions that we care about are actually going to go in the reverse direction of these half reactions. Remember, when we looked at our power source right over here, this first half reaction is what produces the hydroxide ions. The hydroxide ions migrate their way to the left-hand side, and then they are involved in oxidizing whatever metal you have here. If this was zinc, the hydroxide combines with the zinc; they are going to oxidize the zinc, and those extra electrons that are released in that reaction are the ones that are going to travel across the wire and provide the current.

But the hydroxide could react with sodium if this metal is sodium; it could react with calcium if this metal is calcium. So, the reactions we care about are going from this side, the right side, to the left side. Let me just rewrite that again, and actually, just the act of rewriting it will help. Well, my pen is having trouble actually scrolling down, but just the act of rewriting it will help with digesting what's going on.

The sodium reaction for every two molecules of sodium solid is actually forming our anode plus two molecules of hydroxide dissolved in water aqueous solution. We are going to yield sodium oxide solid version of that plus water in the liquid state and plus two electrons. This is really important because we want to think about one gram of each of these metals—how many electrons would they yield? How many electrons are they going to be able to transfer, potentially through that wire?

So, that's the first relationship up there. Let me keep the sodium there and then let me do the calcium over here. Remember, we want to go from the right to the left, so let me rewrite it that way: calcium solid state plus two hydroxides dissolved in water is going to yield calcium oxide solid plus liquid water plus two electrons.

So, once again, I just rewrote each of these in the opposite direction. If I were to write the voltage, I would flip the sign now because I have written them in the other direction. But they are not asking us about the voltage; they want us to know what a 1 gram anode of which of these metals would transfer more electrons.

If we think about sodium, the case with sodium—we have 1.0 grams of sodium. Well, let's think about how many moles of sodium that is. Well, let's see, we can look up sodium on our periodic table of elements. This is actually the one that they give you when you take the AP test, and you see sodium right over here. We could read its atomic weight, sometimes now called relative atomic mass, and that's the weighted average of the atomic masses of the isotopes that you would typically find of sodium on Earth. If you just took a random sample of sodium and averaged the atomic masses, it would be 22.99.

So, that tells us that a mole of sodium is going to weigh—it’s going to have a mass of 22.99 grams. You could say a mole is going to have a mass of 22.99 grams. So, I could write one mole of sodium is going to have a mass of 22.99 grams. Of course, this is still talking about sodium, and I made sure that I have it in terms of moles per gram. Because when you do our dimensional analysis, grams divided by grams or grams of sodium divided by grams of sodium are going to cancel out, and I’m going to be left with moles of sodium.

So, this part of the expression is going to give me how many moles of sodium we are dealing with if I have one gram of sodium. But we need to think about how many electrons we are going to be able to transfer, and that's where we actually have to look at this reaction. So, for every two moles of sodium, we are going to transfer two moles of electrons. Or, another way you can think about it is, for every mole here, you have a two to two relationship, or you could say for every mole of sodium, we are going to transfer one mole of electrons.

So, let’s do that; we’re going to transfer one mole—let me write it this way: one mole of electrons per mole of sodium. If you have two moles of sodium, you’re going to get two moles of electrons. If you have one mole of sodium, you're going to have one mole of electrons.

What is this going to give us? Well, if you just multiply out—actually, let’s look at the units first. Grams of sodium cancel out with grams of sodium, moles of sodium cancel out with moles of sodium, and so you're left with just moles of electrons. It’s going to be 1.0 times 1 divided by 22.9 times 1.

So, it’s 1.0 divided by 22.9, and let’s see; I’m going to have two significant digits right over here, two significant figures. So let me get the calculator out, and it is—let’s see, so it’s just one. I could just say one divided by 22.9 is equal to—it’s approximately 0.0434. So I’ll say approximately 0.044 moles of electrons.

Remember, moles, Avogadro’s number—this is still a lot of electrons. The mole is 6.022 times 10 to the—you know roughly that times 10 to the 23rd, so it’s a very large number of electrons that we are going to potentially transfer over here if we have one gram of sodium.

Now, let’s do the same thing with calcium. If we have calcium, it’s going to be the same exercise here. We have one gram of calcium times one mole of calcium has what mass? One mole of calcium has a mass of how many grams? So let’s go back to the periodic table, and there you have calcium right over here; the atomic weight is 40.08, or often times now the more modern name is relative atomic mass—the weighted average of the typical isotopes: 40.08.

So, I can write 40.08 right over here. In our reaction, for every mole of calcium, how many moles of electrons are we going to generate? Well, we have in this reaction one: for every one calcium, we have two electrons. So for every mole of calcium, we're going to have two moles of electrons.

We could say times two moles of electrons for every mole of calcium that we have as a reactant over here. This is going to give us—this is going to be approximately—see, our units grams of calcium cancel with grams of calcium, moles of calcium cancel with moles of calcium, and so we're going to have 1 times 1 divided by 40.08 times 2.

So, it’s going to be 2 divided by 40.08, which is—so 2 divided by 40. We’re gonna have two significant figures right over here, because this two, and those ones are kind of pure numbers, while these are from measurements. So this is the fewest number of significant figures we have, two.

So, 2 divided by that—let’s see—it’s going to be 0.050 moles of electrons. You can see right from this calculation that the one gram of calcium can transfer more electrons than one gram of sodium. So, we see very clearly that calcium—so one gram of calcium can transfer more electrons than 1.0 gram of sodium. You can see it right over here; we have done the calculation.