In the van der Waals equation of state, the constant b represe...

Question

In the van der Waals equation of state, the constant b represents the amount of “unavailable volume” (per mole) occupied by the molecules themselves. Thus V is replaced by (V-nb), where n is the number of moles. For oxygen, b is about 3.2 x 10^-5 m³ /mol. Estimate the diameter of an oxygen molecule.

Accepted Answer

Hey, everyone in this problem, we're told that the term V minus NB is used in Van der Waal's equation where V stands for total volume and B is the volume occupied by one mole of molecules themselves, making it unavailable for gas particles to move around it. Here, N denotes the number of moles and B is a constant specific to each gas. And for carbon dioxide, B equals approximately 4.3 multiplied by 10 to the exponent negative 5 m C per mole or asked for an estimation of carbon dioxide's molecular diameter. We're given four answer choices all in meters. Option A 3.2 multiplied by 10 to the exponent negative 10. Option B 5.1 multiplied by 10 to the exponent negative 10. Option C 4.7 multiplied by 10 to the exponent negative nine. And option D 6.9 multiplied by 10 to the exponent negative nine. So what we want to think about here is how can we get carbon dioxide's molecular diameter from the information we're given? And the key piece of information we're given here is the value of B OK. So we wanna figure out how can we relate this value of B to carbon dioxide's molecular diameter? Now, as the problem tells us B is going to be the volume occupied by one mole of molecule. And so we think about B that's gonna be equal to the number of molecules and multiplied by the volume of a single molecule. And we take the number molecules we multiply by the volume that's gonna give us that total volume occupied. And if we consider N to be the number of molecules in one mole, all right, then that is gonna give us that the value right. Now, if we think about the number of molecules in one mole, how can we figure that out? Well, recall that the number of molecules and is equal to the number of moles little N multiplied by Avogadro's number N A. So if we have one mole, a FN is equal to one, then the number of molecules is just going to be equal to Avogadro's number. And so our equation is going to be speed is equal to N A multiplied by the volume of a molecule. All right. So we have this equation, but again, we wanna find the molecular diameter. So we need to go a little bit further than having this volume. What we want to think about is how can we calculate this volume? And while we have this sphere, OK, we can assume that the molecule is spherical. What's the volume of a sphere four thirds pi are cubed. And so we can write that B will be equal to N A multiplied by four thirds pi R. All right. So this is in terms of the radius, we're looking for the diameter, recall that we can relate the radius through the diameter and it's just going to be half of that diameter. So we can write our equation as B is equal to N A multiplied by four thirds. Pi multiplied by D divided by two. Cute. So we have this equation now that contains D, the diameter, that value we're looking for. And everything else is the value we know we were given B, we know Avogadro's number and A that's a constant. And so what we're gonna do is we're gonna simplify our equation. We're gonna rearrange for the diameter D that we're looking for. And then we're gonna substitute in those values just to save a little bit of rearranging with the numbers in there. It'll be a little bit cleaner to do it. So simplifying what we have, we get that B is going to be equal to, oh, we have Avao's number N A multiplied by pi OK. And we're gonna divide this, hey, we have four thirds divided by two cubed, two cubed is going to give us eight. Hm. So we're gonna have that three and then we have four divided by eight which leaves us with a half or two in the Dino. So we have N A multiplied by pi divided by three multiplied by two. And all of this is multiplied by that DQ term. All right. So isolating four DD is going to be equal to, you can multiply both sides by six. So six B divide both sides by NA I, OK. And then we're gonna take that cube root on both sides. We have our expression for the diameter D. Now, in terms of these values, we know now it's just a matter of substituting in and solving. So D will be equal to the Q group of six multiplied by 4.3 multiplied by 10 to the exponent negative 5 m cubed per mole divided by 6.02 multiplied by 10 to the exponent 23 inverse moles multiplied by pi. And when we work all of this out on our calculator, we get that this diameter D is going to be approximately 5.147 83 multiplied by 10 to the exponent negative 10 m. And that is the value we were looking. So we can go back up to our answer choices. We're gonna run round our answer choice to two significant digits. When we do that, we can see that the correct answer is going to correspond with answer choice B estimate that carbon dioxide's molecular diameter is about 5.1 multiplied by 10 to the exponent negative 10 m. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Van der Waals Equation

Unavailable Volume

Molecular Diameter Estimation

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